Precalculus — Chapter 1: Fundamentals

1.1 Real Numbers 95 problems

Concepts

1.Provide one example for each: (a) a natural number, (b) an integer that is not a natural number, (c) a rational number that is not an integer, (d) an irrational number.
2.Fill in each blank and identify the property of real numbers applied: (a) $ab = \underline{\;\;\;}$; (b) $a + (b + c) = \underline{\;\;\;}$; (c) $a(b + c) = \underline{\;\;\;}$.
3.Describe the set of all real numbers strictly between 2 and 7 (a) in set-builder notation and (b) in interval notation.
4.The notation $|x|$ represents the ______ of $x$. When $x \neq 0$, the sign of $|x|$ is always ______.
5.On the real line, the distance between $a$ and $b$ is $d(a,b) = \underline{\;\;\;}$. Using this, the distance between $-5$ and $2$ equals ______.
6.Yes or No (give a reason if No; assume $a,b$ are nonzero reals): (a) Must the sum of two rational numbers be rational? (b) Must the sum of two irrational numbers be irrational?
7.Yes or No (give a reason if No): (a) Does $a - b = b - a$? (b) Does $-2(a - 5) = -2a - 10$?
8.Yes or No (give a reason if No): (a) Is the distance between two distinct real numbers always positive? (b) Is the distance from $a$ to $b$ equal to the distance from $b$ to $a$?

Skills

9.For the set $\left\{-1.5,\ 0,\ \tfrac{5}{2},\ \sqrt{7},\ 2.71,\ -\pi,\ 3.1\overline{4},\ 100,\ -8\right\}$, list the elements that are (a) natural numbers, (b) integers, (c) rational numbers, (d) irrational numbers.
10.For the set $\left\{1.3,\ 1.3333\ldots,\ \sqrt{5},\ 5.34,\ -500,\ 1\tfrac{2}{3},\ \sqrt{16},\ \tfrac{246}{579},\ -\tfrac{20}{5}\right\}$, list the elements that are (a) natural numbers, (b) integers, (c) rational numbers, (d) irrational numbers.
11.Name the property of real numbers shown: $3 + 7 = 7 + 3$.
12.Name the property of real numbers shown: $4(2 + 3) = (2 + 3)4$.
13.Name the property of real numbers shown: $(x + 2y) + 3z = x + (2y + 3z)$.
14.Name the property of real numbers shown: $2(A + B) = 2A + 2B$.
15.Name the property of real numbers shown: $(5x + 1)3 = 15x + 3$.
16.Name the property of real numbers shown: $(x + a)(x + b) = (x + a)x + (x + a)b$.
17.Name the property of real numbers shown: $2x(3 + y) = (3 + y)2x$.
18.Name the property of real numbers shown: $7(a + b + c) = 7(a + b) + 7c$.
19.Rewrite using the Commutative Property of Addition: $x + 3$.
20.Rewrite using the Associative Property of Multiplication: $7(3x)$.
21.Rewrite using the Distributive Property: $4(A + B)$.
22.Rewrite using the Distributive Property: $5x + 5y$.
23.Use the properties of real numbers to remove the parentheses: $3(x + y)$.
24.Use the properties of real numbers to remove the parentheses: $(a - b)8$.
25.Use the properties of real numbers to remove the parentheses: $4(2m)$.
26.Use the properties of real numbers to remove the parentheses: $\tfrac{4}{3}(-6y)$.
27.Use the properties of real numbers to remove the parentheses: $-\tfrac{5}{2}(2x - 4y)$.
28.Use the properties of real numbers to remove the parentheses: $(3a)(b + c - 2d)$.
29.Carry out the operations: (a) $\tfrac{3}{10} + \tfrac{4}{15}$; (b) $\tfrac{1}{4} + \tfrac{1}{5}$.
30.Carry out the operations: (a) $\tfrac{2}{3} - \tfrac{3}{5}$; (b) $1 + \tfrac{5}{8} - \tfrac{1}{6}$.
31.Carry out the operations: (a) $\tfrac{2}{3}\left(6 - \tfrac{3}{2}\right)$; (b) $\left(3 + \tfrac{1}{4}\right)\left(1 - \tfrac{4}{5}\right)$.
32.Carry out the operations: (a) $\dfrac{\tfrac{2}{3} - \tfrac{2}{3}}{2}$; (b) $\dfrac{\tfrac{2}{5} + \tfrac{1}{2}}{\tfrac{1}{10} + \tfrac{3}{15}}$.
33.Insert the correct symbol ($<$, $>$, or $=$): (a) $3 \;\square\; \tfrac{7}{2}$; (b) $-3 \;\square\; -\tfrac{7}{2}$; (c) $3.5 \;\square\; \tfrac{7}{2}$.
34.Insert the correct symbol ($<$, $>$, or $=$): (a) $\tfrac{2}{3} \;\square\; 0.67$; (b) $\tfrac{2}{3} \;\square\; -0.67$; (c) $|0.67| \;\square\; |-0.67|$.
35.State whether the inequality is true or false: (a) $-3 < -4$; (b) $3 < 4$.
36.State whether the inequality is true or false: (a) $\sqrt{3} > 1.7325$; (b) $1.732 \ge \sqrt{3}$.
37.State whether the inequality is true or false: (a) $\tfrac{10}{2} \ge 5$; (b) $\tfrac{6}{10} \ge \tfrac{5}{6}$.
38.State whether the inequality is true or false: (a) $\tfrac{7}{11} \ge \tfrac{8}{13}$; (b) $-\tfrac{3}{5} > -\tfrac{3}{4}$.
39.Write each statement as an inequality: (a) $x$ is positive; (b) $t$ is less than 4; (c) $a$ is at least $\pi$; (d) $x$ is less than $\tfrac{1}{3}$ and greater than $-5$; (e) the distance from $p$ to 3 is no more than 5.
40.Write each statement as an inequality: (a) $y$ is negative; (b) $z$ is greater than 1; (c) $b$ is at most 8; (d) $w$ is positive and no greater than 17; (e) $y$ is at least 2 units away from $\pi$.
41.With $A = \{1,2,3,4,5,6,7\}$, $B = \{2,4,6,8\}$, $C = \{7,8,9,10\}$, find: (a) $A \cup B$; (b) $A \cap B$.
42.With $A = \{1,2,3,4,5,6,7\}$, $B = \{2,4,6,8\}$, $C = \{7,8,9,10\}$, find: (a) $B \cup C$; (b) $B \cap C$.
43.With $A = \{1,2,3,4,5,6,7\}$, $B = \{2,4,6,8\}$, $C = \{7,8,9,10\}$, find: (a) $A \cup C$; (b) $A \cap C$.
44.With $A = \{1,2,3,4,5,6,7\}$, $B = \{2,4,6,8\}$, $C = \{7,8,9,10\}$, find: (a) $A \cup B \cup C$; (b) $A \cap B \cap C$.
45.With $A = \{x \mid x \ge -2\}$, $B = \{x \mid x < 4\}$, $C = \{x \mid -1 < x \le 5\}$, find: (a) $B \cup C$; (b) $B \cap C$.
46.With $A = \{x \mid x \ge -2\}$, $B = \{x \mid x < 4\}$, $C = \{x \mid -1 < x \le 5\}$, find: (a) $A \cap C$; (b) $A \cap B$.
47.Write the interval as an inequality, then graph it: $(-3, 0)$.
48.Write the interval as an inequality, then graph it: $(2, 8]$.
49.Write the interval as an inequality, then graph it: $[2, 8)$.
50.Write the interval as an inequality, then graph it: $\left[-6, -\tfrac{1}{2}\right]$.
51.Write the interval as an inequality, then graph it: $[2, \infty)$.
52.Write the interval as an inequality, then graph it: $(-\infty, 1)$.
53.Write the inequality in interval notation, then graph the interval: $x \le 1$.
54.Write the inequality in interval notation, then graph the interval: $1 \le x \le 2$.
55.Write the inequality in interval notation, then graph the interval: $-2 < x \le 1$.
56.Write the inequality in interval notation, then graph the interval: $x \ge -5$.
57.Write the inequality in interval notation, then graph the interval: $x > -1$.
58.Write the inequality in interval notation, then graph the interval: $-5 < x < 2$.
59.Express each graphed set in interval notation: (a) the segment shown from $-3$ to $5$ with the relevant endpoints; (b) the second segment shown from $-3$ to $5$ with its endpoints. (Refer to the number-line graphs.)
60.Express each graphed set in interval notation: (a) the segment shown from $0$ to $2$ with the relevant endpoints; (b) the segment shown beginning near $-2$ through $0$. (Refer to the number-line graphs.)
61.Graph the set: $(-2, 0) \cup (-1, 1)$.
62.Graph the set: $(-2, 0) \cap (-1, 1)$.
63.Graph the set: $[-4, 6] \cap [0, 8)$.
64.Graph the set: $[-4, 6] \cup [0, 8)$.
65.Graph the set: $(-\infty, -4) \cup (4, \infty)$.
66.Graph the set: $(-\infty, 6] \cup (2, 10)$.
67.Evaluate each expression: (a) $|100|$; (b) $|-73|$.
68.Evaluate each expression: (a) $|\sqrt{5} - 5|$; (b) $|10 - \pi|$.
69.Evaluate each expression: (a) $\big|\,|-6| - |-4|\,\big|$; (b) $\dfrac{-1}{|-1|}$.
70.Evaluate each expression: (a) $\big|\,2 - |-12|\,\big|$; (b) $-1 - \big|\,1 - |-1|\,\big|$.
71.Evaluate each expression: (a) $|(-2)\cdot 6|$; (b) $\left|\left(-\tfrac{1}{3}\right)(-15)\right|$.
72.Evaluate each expression: (a) $\left|\dfrac{-6}{24}\right|$; (b) $\left|\dfrac{7 - 12}{12 - 7}\right|$.
73.Find the distance between the two marked points shown on the number line (points indicated at $-2$ and $3$ on the displayed scale).
74.Find the distance between the two marked points shown on the number line (points indicated at $-1$ and $2$ on the displayed scale).
75.Find the distance between the given numbers: (a) $2$ and $17$; (b) $-3$ and $21$; (c) $\tfrac{11}{8}$ and $-\tfrac{3}{10}$.
76.Find the distance between the given numbers: (a) $\tfrac{7}{15}$ and $-\tfrac{1}{21}$; (b) $-38$ and $-57$; (c) $-2.6$ and $-1.8$.

Skills Plus

77.Write each repeating decimal as a fraction: (a) $0.\overline{7}$; (b) $0.2\overline{8}$; (c) $0.\overline{57}$.
78.Write each repeating decimal as a fraction: (a) $5.\overline{23}$; (b) $1.3\overline{7}$; (c) $2.1\overline{35}$.
79.Rewrite the quantity without absolute-value bars: $|\pi - 3|$.
80.Rewrite the quantity without absolute-value bars: $|1 - \sqrt{2}|$.
81.Rewrite without absolute-value bars: $|a - b|$, given that $a < b$.
82.Rewrite without absolute-value bars: $a + b + |a - b|$, given that $a < b$.
83.Let $a > 0$, $b < 0$, and $c < 0$. Determine the sign of: (a) $-a$; (b) $bc$; (c) $a - b$; (d) $ab + ac$.
84.Let $a > 0$, $b < 0$, and $c < 0$. Determine the sign of: (a) $-b$; (b) $a + bc$; (c) $c - a$; (d) $ab^2$.

Applications

85.Garden area: A rectangular vegetable garden is 20 ft by 30 ft, giving area $20 \times 30 = 600\ \text{ft}^2$. Extending its length by $x$ makes the area $A = 20(30 + x)$. Which property of real numbers shows this equals $A = 600 + 20x$? figure — add your own
86.Temperature comparison: A bar graph gives daily high temperatures for Omak, WA $(T_O)$ and Geneseo, NY $(T_G)$ over one week in June. For each day, compute $T_O - T_G$ and $|T_O - T_G|$, and decide which of the two quantities conveys more information. figure — add your own
87.Shipping limits: A parcel is accepted only if its length plus girth is at most 108 in, i.e. $L + 2(x + y) \le 108$. (a) Will a box that is 6 in wide, 8 in deep, and 5 ft long be accepted? What about one measuring 2 ft by 2 ft by 4 ft? (b) For a square base of 9 in by 9 in, what is the largest allowable length? figure — add your own

Discuss · Discover · Prove · Write

88.Discuss: Explain why the sum, difference, and product of any two rational numbers are rational. Is the product of two irrationals necessarily irrational? What about their sum?
89.Discover / Prove: Is $\tfrac{1}{2} + \sqrt{2}$ rational or irrational? Is $\tfrac{1}{2}\cdot\sqrt{2}$ rational or irrational? After experimenting, prove: (a) the sum of a rational $r$ and an irrational $t$ is irrational; (b) the product of a nonzero rational $r$ and an irrational $t$ is irrational. (Hint for (a): assume $r + t = q$ is rational and derive a contradiction.)
90.Discover: Fill in tables of $1/x$ for $x = 1, 2, 10, 100, 1000$ and for $x = 1.0, 0.5, 0.1, 0.01, 0.001$. Describe what happens to $1/x$ as $x$ grows large and as $x$ becomes small.
91.Discover: Using the given figures, explain how to locate $\sqrt{2}$ on the number line. Can $\sqrt{5}$ be located similarly? How does the rolling-circle figure help locate $\pi$? List other irrational numbers that can be located on the line. figure — add your own
92.Prove: Define $\max(a,b)$ and $\min(a,b)$ as the larger and smaller of $a$ and $b$. Prove: (a) $\max(a,b) = \dfrac{a + b + |a - b|}{2}$; (b) $\min(a,b) = \dfrac{a + b - |a - b|}{2}$. (Hint: take cases.)
93.Write: Compose a paragraph describing real-world settings where natural numbers, integers, rational numbers, and irrational numbers each arise, with an example for each type.
94.Discuss: Addition and multiplication are commutative. Decide whether each of the following is commutative: (a) subtraction; (b) division of nonzero reals; (c) putting on socks then shoes; (d) putting on a hat then a coat; (e) washing then drying laundry.
95.Prove: The Triangle Inequality states $|x + y| \le |x| + |y|$. (a) Verify it for $x = 2, y = 3$; for $x = -2, y = -3$; and for $x = -2, y = 3$. (b) Prove it for all real $x, y$. (Hint: take cases.)

1.2 Exponents and Radicals 98 problems

Skills

9.Complete the table row: rewrite the radical $\dfrac{1}{\sqrt{3}}$ using a rational exponent (and vice versa where a column is blank).
10.Complete the table row: rewrite $\sqrt[3]{7^{2}}$ using a rational exponent.
11.Complete the table row: rewrite the exponential expression $4^{2/3}$ as a radical.
12.Complete the table row: rewrite the exponential expression $10^{-3/2}$ as a radical.
13.Complete the table row: rewrite the radical $\sqrt[5]{5^{3}}$ using a rational exponent.
14.Complete the table row: rewrite the exponential expression $2^{-1.5}$ as a radical.
15.Complete the table row: rewrite the exponential expression $a^{2/5}$ as a radical.
16.Complete the table row: rewrite the radical $\dfrac{1}{\sqrt{x^{5}}}$ using a rational exponent.
17.Evaluate: (a) $-2^{6}$; (b) $(-2)^{6}$; (c) $\left(\tfrac{1}{5}\right)^{2}\cdot(-3)^{3}$.
18.Evaluate: (a) $(-5)^{3}$; (b) $-5^{3}$; (c) $(-5)^{2}\cdot\left(\tfrac{2}{3}\right)^{2}$.
19.Evaluate: (a) $\left(\tfrac{5}{3}\right)^{0}\cdot 2^{-1}$; (b) $\dfrac{2^{-3}}{3^{0}}$; (c) $\left(\tfrac{2}{3}\right)^{-2}$.
20.Evaluate: (a) $-2^{3}\cdot(-2)^{0}$; (b) $-2^{-3}\cdot(-2)^{0}$; (c) $\left(\tfrac{-3}{5}\right)^{-3}$.
21.Evaluate: (a) $5^{3}\cdot 5$; (b) $5^{4}\cdot 5^{-2}$; (c) $(2^{2})^{3}$.
22.Evaluate: (a) $3^{8}\cdot 3^{5}$; (b) $\dfrac{10^{7}}{10^{4}}$; (c) $(3^{5})^{4}$.
23.Evaluate: (a) $3\sqrt[3]{16}$; (b) $\dfrac{\sqrt{18}}{\sqrt{81}}$; (c) $\sqrt{\tfrac{27}{4}}$.
24.Evaluate: (a) $2\sqrt[3]{81}$; (b) $\dfrac{\sqrt{18}}{\sqrt{25}}$; (c) $\sqrt{\tfrac{12}{49}}$.
25.Evaluate: (a) $\sqrt{3}\,\sqrt{15}$; (b) $\dfrac{\sqrt{48}}{\sqrt{3}}$; (c) $\sqrt[3]{24}\,\sqrt[3]{18}$.
26.Evaluate: (a) $\sqrt{10}\,\sqrt{32}$; (b) $\dfrac{\sqrt{54}}{\sqrt{6}}$; (c) $\sqrt[3]{15}\,\sqrt[3]{75}$.
27.Evaluate: (a) $\dfrac{\sqrt{132}}{\sqrt{3}}$; (b) $\sqrt[3]{2}\,\sqrt[3]{32}$; (c) $\sqrt[4]{\tfrac{1}{4}}\,\sqrt[4]{\tfrac{1}{64}}$.
28.Evaluate: (a) $\sqrt[5]{\tfrac{7}{8}}\,\sqrt[5]{\tfrac{1}{4}}$; (b) $\sqrt[5]{\tfrac{1}{2}}\,\sqrt[5]{128}$; (c) $\dfrac{\sqrt[3]{4}}{\sqrt[3]{108}}$.
29.Simplify, eliminating negative exponents: (a) $x^{3}\cdot x^{4}$; (b) $(2y^{2})^{3}$; (c) $y^{-2}y^{7}$.
30.Simplify, eliminating negative exponents: (a) $y^{5}\cdot y^{2}$; (b) $(8x)^{2}$; (c) $x^{4}x^{-3}$.
31.Simplify, eliminating negative exponents: (a) $x^{-5}\cdot x^{3}$; (b) $w^{-2}w^{-4}w^{5}$; (c) $\dfrac{x^{16}}{x^{10}}$.
32.Simplify, eliminating negative exponents: (a) $y^{2}\cdot y^{-5}$; (b) $z^{5}z^{-3}z^{-4}$; (c) $\dfrac{y^{7}y^{0}}{y^{10}}$.
33.Simplify, eliminating negative exponents: (a) $\dfrac{a^{9}a^{-2}}{a}$; (b) $(a^{2}a^{4})^{3}$; (c) $\left(\tfrac{x}{2}\right)^{3}(5x^{6})$.
34.Simplify, eliminating negative exponents: (a) $\dfrac{z^{2}z^{4}}{z^{3}z^{-1}}$; (b) $(2a^{3}a^{2})^{4}$; (c) $(-3z^{2})^{3}(2z^{3})$.
35.Simplify, eliminating negative exponents: (a) $(3x^{3}y^{2})(2y^{3})$; (b) $(5w^{-2}z^{-2})(z^{3})$.
36.Simplify, eliminating negative exponents: (a) $(8m^{-2}n^{4})\left(\tfrac{1}{2}n^{-2}\right)$; (b) $(3a^{4}b^{-2})^{3}(a^{2}b^{-1})$.
37.Simplify, eliminating negative exponents: (a) $\dfrac{x^{2}y^{-1}}{x^{-5}}$; (b) $\left(\dfrac{a^{3}}{2b^{2}}\right)^{3}$.
38.Simplify, eliminating negative exponents: (a) $\dfrac{y^{-2}z^{-3}}{y^{-1}}$; (b) $\left(\dfrac{x^{3}y^{-2}}{x^{-3}y^{2}}\right)^{-2}$.
39.Simplify, eliminating negative exponents: (a) $\left(\dfrac{a^{2}}{b}\right)^{5}\left(\dfrac{a^{3}b^{2}}{c^{3}}\right)^{3}$; (b) $\dfrac{(u^{-1}v^{2})^{2}}{(u^{3}v^{-2})^{3}}$.
40.Simplify, eliminating negative exponents: (a) $\left(\dfrac{x^{4}z^{2}}{4y^{5}}\right)\left(\dfrac{2x^{3}y^{2}}{z^{3}}\right)^{2}$; (b) $\dfrac{(rs^{2})^{3}}{(r^{-3}s^{2})^{2}}$.
41.Simplify, eliminating negative exponents: (a) $\dfrac{8a^{3}b^{-4}}{2a^{-5}b^{5}}$; (b) $\left(\dfrac{y}{5x^{-2}}\right)^{-3}$.
42.Simplify, eliminating negative exponents: (a) $\dfrac{5xy^{-2}}{x^{-1}y^{-3}}$; (b) $\left(\dfrac{2a^{-1}b}{a^{2}b^{-3}}\right)^{-3}$.
43.Simplify, eliminating negative exponents: (a) $\left(\dfrac{3a}{b^{3}}\right)^{-1}$; (b) $\left(\dfrac{q^{-1}r^{-1}s^{-2}}{r^{-5}sq^{-8}}\right)^{-1}$.
44.Simplify, eliminating negative exponents: (a) $\left(\dfrac{s^{2}t^{-4}}{5s^{-1}t}\right)^{-2}$; (b) $\left(\dfrac{xy^{-2}z^{-3}}{x^{2}y^{3}z^{-4}}\right)^{-3}$.
45.Simplify (letters denote positive reals): (a) $\sqrt[4]{x^{4}}$; (b) $\sqrt[5]{16x^{8}}$.
46.Simplify (letters denote positive reals): (a) $\sqrt[5]{x^{10}}$; (b) $\sqrt[3]{x^{3}y^{6}}$.
47.Simplify (letters denote positive reals): (a) $\sqrt[6]{64a^{6}b^{7}}$; (b) $\sqrt[3]{a^{2}b}\,\sqrt[3]{64a^{4}b}$.
48.Simplify (letters denote positive reals): (a) $\sqrt[4]{x^{4}y^{2}z^{2}}$; (b) $\sqrt[3]{\sqrt{64x^{6}}}$.
49.Simplify: (a) $\sqrt{32} + \sqrt{18}$; (b) $\sqrt{75} + \sqrt{48}$.
50.Simplify: (a) $\sqrt[3]{125} + \sqrt[3]{45}$; (b) $\sqrt[3]{54} - \sqrt[3]{16}$.
51.Simplify: (a) $\sqrt{9a^{3}} + \sqrt{a}$; (b) $\sqrt{16x} + \sqrt{x^{5}}$.
52.Simplify: (a) $\sqrt[4]{x^{4}} + \sqrt[4]{8x}$; (b) $4\sqrt{18rt^{3}} + 5\sqrt{32r^{3}t^{5}}$.
53.Simplify: (a) $\sqrt{81x^{2} + 81}$; (b) $\sqrt{36x^{2} + 36y^{2}}$.
54.Simplify: (a) $\sqrt{27a^{2} + 63a}$; (b) $\sqrt{75t + 100t^{2}}$.
55.Evaluate: (a) $16^{1/4}$; (b) $-8^{1/3}$; (c) $9^{-1/2}$.
56.Evaluate: (a) $27^{1/3}$; (b) $(-8)^{1/3}$; (c) $-\left(\tfrac{1}{8}\right)^{1/3}$.
57.Evaluate: (a) $32^{2/5}$; (b) $\left(\tfrac{4}{9}\right)^{-1/2}$; (c) $\left(\tfrac{16}{81}\right)^{3/4}$.
58.Evaluate: (a) $125^{2/3}$; (b) $\left(\tfrac{25}{64}\right)^{3/2}$; (c) $27^{-4/3}$.
59.Evaluate: (a) $5^{2/3}\cdot 5^{1/3}$; (b) $\dfrac{3^{3/5}}{3^{2/5}}$; (c) $(\sqrt[3]{4})^{3}$.
60.Evaluate: (a) $3^{2/7}\cdot 3^{12/7}$; (b) $\dfrac{7^{2/3}}{7^{5/3}}$; (c) $(\sqrt[5]{6})^{-10}$.
61.Simplify (letters denote positive reals; eliminate negative exponents): (a) $x^{3/4}x^{5/4}$; (b) $y^{2/3}y^{4/3}$.
62.Simplify: (a) $(4b)^{1/2}(8b^{1/4})$; (b) $(3a^{3/4})^{2}(5a^{1/2})$.
63.Simplify: (a) $\dfrac{w^{4/3}w^{2/3}}{w^{1/3}}$; (b) $\dfrac{a^{5/4}(2a^{3/4})^{3}}{a^{1/4}}$.
64.Simplify: (a) $(8y^{3})^{-2/3}$; (b) $(u^{4}b^{6})^{-1/3}$.
65.Simplify: (a) $(8a^{6}b^{3/2})^{2/3}$; (b) $(4a^{6}b^{8})^{3/2}$.
66.Simplify: (a) $(x^{-5}y^{1/3})^{-3/5}$; (b) $(4r^{8}s^{-1/2})^{1/2}(32s^{-5/4})^{-1/5}$.
67.Simplify: (a) $\dfrac{(8s^{3}t^{3})^{2/3}}{(s^{4}t^{-8})^{1/4}}$; (b) $\dfrac{(32x^{5}y^{-3/2})^{2/5}}{(x^{5/3}y^{2/3})^{3/5}}$.
68.Simplify: (a) $\left(\dfrac{x^{8}y^{-4}}{16y^{4/3}}\right)^{-1/4}$; (b) $\left(\dfrac{4s^{3}t^{4}}{s^{2}t^{9/2}}\right)^{-1/2}$.
69.Simplify: (a) $\left(\dfrac{x^{3/2}}{y^{-1/2}}\right)^{4}\left(\dfrac{x^{-2}}{y^{3}}\right)$; (b) $\left(\dfrac{4y^{3}z^{2/3}}{x^{1/2}}\right)^{2}\left(\dfrac{x^{-3}y^{6}}{8z^{4}}\right)^{1/3}$.
70.Simplify: (a) $\left(\dfrac{a^{1/6}b^{-3}}{x^{-1}y}\right)^{3}\left(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\right)$; (b) $\dfrac{(9st)^{3/2}}{(27s^{3}t^{-4})^{2/3}}\left(\dfrac{3s^{-2}}{4t^{-1}}\right)^{-1}$.
71.Simplify and remove negative exponents (letters positive): (a) $\sqrt{x^{3}}$; (b) $\sqrt[5]{x^{6}}$.
72.Simplify: (a) $\sqrt{x^{5}}$; (b) $\sqrt[4]{x^{6}}$.
73.Simplify: (a) $\sqrt[6]{y^{5}}\,\sqrt[3]{y^{2}}$; (b) $(5\sqrt[3]{x})(2\sqrt[4]{x})$.
74.Simplify: (a) $\sqrt[4]{b^{3}}\,\sqrt{b}$; (b) $(2\sqrt{a})(\sqrt[3]{a^{2}})$.
75.Simplify: (a) $\sqrt{4st}\,\sqrt[6]{s^{3}t^{2}}$; (b) $\dfrac{\sqrt[4]{x^{7}}}{\sqrt[4]{x^{3}}}$.
76.Simplify: (a) $\sqrt[5]{x^{3}y^{2}}\,\sqrt[10]{x^{4}y^{16}}$; (b) $\dfrac{\sqrt[3]{8x^{2}}}{\sqrt{x}}$.
77.Simplify: (a) $\sqrt[3]{y\sqrt{y}}$; (b) $\sqrt{\dfrac{16u^{3}v}{uw^{5}}}$.
78.Simplify: (a) $\sqrt{s\sqrt{s^{3}}}$; (b) $\sqrt[3]{\dfrac{54x^{2}y^{4}}{2x^{5}y}}$.
79.Rationalize the denominator: (a) $\dfrac{1}{\sqrt{6}}$; (b) $\sqrt{\tfrac{3}{2}}$; (c) $\dfrac{9}{\sqrt[4]{2}}$.
80.Rationalize the denominator: (a) $\dfrac{12}{\sqrt{3}}$; (b) $\sqrt{\tfrac{12}{5}}$; (c) $\dfrac{8}{\sqrt[5]{5^{2}}}$.
81.Rationalize the denominator: (a) $\dfrac{1}{\sqrt{5x}}$; (b) $\sqrt{\tfrac{x}{5}}$; (c) $\sqrt[5]{\dfrac{1}{x^{3}}}$.
82.Rationalize the denominator: (a) $\sqrt{\dfrac{s}{3t}}$; (b) $\dfrac{a}{\sqrt[5]{b^{2}}}$; (c) $\dfrac{1}{c^{3/5}}$.
83.Write in scientific notation: (a) 69,300,000; (b) 7,200,000,000,000; (c) 0.000028536; (d) 0.0001213.
84.Write in scientific notation: (a) 129,540,000; (b) 7,259,000,000; (c) 0.0000000014; (d) 0.0007029.
85.Write in decimal (ordinary) notation: (a) $3.19\times 10^{5}$; (b) $2.721\times 10^{8}$; (c) $2.670\times 10^{-8}$; (d) $9.999\times 10^{-9}$.
86.Write in decimal (ordinary) notation: (a) $7.1\times 10^{14}$; (b) $6\times 10^{12}$; (c) $8.55\times 10^{-3}$; (d) $6.257\times 10^{-10}$.
87.Express each quantity in scientific notation: (a) one light-year $\approx$ 5,900,000,000,000 mi; (b) an electron's diameter $\approx$ 0.0000000000004 cm; (c) a water drop holds more than 33 billion billion molecules.
88.Express each quantity in scientific notation: (a) the Earth–Sun distance $\approx$ 93 million miles; (b) an oxygen molecule's mass $\approx$ 0.000000000000000000000053 g; (c) Earth's mass $\approx$ 5,970,000,000,000,000,000,000,000,000 kg.
89.Using scientific notation and the Laws of Exponents, evaluate and round to the appropriate significant digits: $(7.2\times 10^{-9})(1.806\times 10^{-12})$.
90.Evaluate (scientific notation): $(1.062\times 10^{24})(8.61\times 10^{19})$.
91.Evaluate (scientific notation): $\dfrac{1.295643\times 10^{9}}{(3.610\times 10^{-17})(2.511\times 10^{6})}$.
92.Evaluate (scientific notation): $\dfrac{(73.1)(1.6341\times 10^{28})}{0.0000000019}$.
93.Evaluate (scientific notation): $\dfrac{(0.0000162)(0.01582)}{(594{,}621{,}000)(0.0058)}$.
94.Evaluate (scientific notation): $\dfrac{(3.542\times 10^{-6})^{9}}{(5.05\times 10^{4})^{12}}$.

Skills Plus

95.Let $a > 0$, $b < 0$, $c < 0$. Find the sign of each: (a) $b^{5}$; (b) $b^{10}$; (c) $ab^{2}c^{3}$; (d) $(b - a)^{3}$; (e) $(b - a)^{4}$; (f) $\dfrac{a^{3}c^{3}}{b^{6}c^{6}}$.
96.Without a calculator, decide which value is larger in each pair: (a) $2^{1/2}$ or $2^{1/3}$; (b) $\left(\tfrac{1}{2}\right)^{1/2}$ or $\left(\tfrac{1}{2}\right)^{1/3}$; (c) $7^{1/4}$ or $4^{1/3}$; (d) $\sqrt[3]{5}$ or $\sqrt{3}$.

Applications

97.Distance to the nearest star: Proxima Centauri lies about 4.3 light-years away. Using the light-year value from Exercise 87(a), express this distance in miles.
98.Speed of light: Light travels about 186,000 mi/s. Using the Earth–Sun distance from Exercise 88(a), find how long a light ray takes to reach Earth from the Sun.
99.Ocean volume: The oceans average $3.7\times 10^{3}$ m deep and cover $3.6\times 10^{14}\ \text{m}^2$. Find the total ocean volume in liters (1 m$^3$ = 1000 L). figure — add your own
100.National debt: In July 2013 the U.S. population was $3.164\times 10^{8}$ and the national debt was $1.674\times 10^{13}$ dollars. What was each person's share of the debt?
101.Counting molecules: A sealed hospital room 5 m wide, 10 m long, and 3 m high is filled with pure oxygen. With 1000 L per cubic meter and 22.4 L of gas containing $6.02\times 10^{23}$ molecules (Avogadro's number), how many oxygen molecules are in the room?
102.Seeing distance: Earth's curvature limits the distance $D$ visible from a height $h$ by $D = \sqrt{2rh + h^{2}}$, where $r = 3960$ mi is Earth's radius and $D, h$ are in miles. How far can you see from the CN Tower observation deck, 1135 ft above the ground? figure — add your own
103.Skid speed: Police estimate a car's speed $s$ (mi/h) from a skid of $d$ feet using $s = \sqrt{30 f d}$, where $f$ is the road's friction coefficient (typical values are tabulated for dry/wet tar, concrete, and gravel). (a) How fast was a car going if it skidded 65 ft on wet concrete? (b) How far will a car traveling 50 mi/h skid on wet tar? figure — add your own
104.Earth–Sun distance via Kepler's Third Law: A planet's average distance to the Sun is $d = \left(\dfrac{GM}{4\pi^{2}}\right)^{1/3} T^{2/3}$, where $M = 1.99\times 10^{30}$ kg, $G = 6.67\times 10^{-11}\ \text{N·m}^2/\text{kg}^2$, and $T$ is the orbital period in seconds. Using Earth's orbital period of about 365.25 days, find Earth's distance from the Sun.

Discuss · Discover · Prove · Write

105.Discuss: If a suitcase holds one million $(10^{6})$ dollars and you spend a thousand $(10^{3})$ dollars per day, how many years to use it all? At the same rate, how many years to empty a suitcase holding a billion $(10^{9})$ dollars?
106.Discuss: Use the Laws of Exponents to evaluate mentally: (a) $\dfrac{18^{5}}{9^{5}}$; (b) $20^{6}\cdot(0.5)^{6}$.

1.3 Algebraic Expressions 142 problems

Concepts

1.For the polynomial $2x^{5} + 6x^{4} + 4x^{3}$: (a) How many terms does it have, and what are they? (b) What factor is common to every term, and what is the factored form of $2x^{5} + 6x^{4} + 4x^{3}$?
2.To factor $x^{2} + 7x + 10$, find two integers whose product is ______ and whose sum is ______. Those integers are ______ and ______, so the trinomial factors as ______.
3.The greatest common factor of $3x^{3} + x^{2}$ is ______, and the expression factors as ______ ( ______ + ______ ).
4.The square-of-a-sum formula is $(A + B)^{2} = \underline{\;\;\;}$. Apply it: $(2x + 3)^{2} = \underline{\;\;\;}$.
5.The sum-times-difference formula is $(A + B)(A - B) = \underline{\;\;\;}$. Apply it: $(5 + x)(5 - x) = \underline{\;\;\;}$.
6.The difference-of-squares factoring formula is $A^{2} - B^{2} = \underline{\;\;\;}$. Apply it: $4x^{2} - 25 = \underline{\;\;\;}$.
7.The perfect-square factoring formula is $A^{2} + 2AB + B^{2} = \underline{\;\;\;}$. Apply it: $x^{2} + 10x + 25 = \underline{\;\;\;}$.
8.Yes or No (give a reason if No): (a) Does $(x + 5)^{2} = x^{2} + 25$? (b) When you expand $(x + a)^{2}$ with $a \neq 0$, are there three terms? (c) Does $(x + 5)(x - 5) = x^{2} - 25$? (d) When you expand $(x + a)(x - a)$ with $a \neq 0$, are there two terms?

Skills

9.Classify $5x^{3} + 6$ as a monomial, binomial, or trinomial; list its terms and give its degree.
10.Classify $-2x^{2} + 5x - 3$ as a monomial, binomial, or trinomial; list its terms and give its degree.
11.Classify $-8$ as a monomial, binomial, or trinomial; list its terms and give its degree.
12.Classify $\tfrac{1}{2}x^{7}$ as a monomial, binomial, or trinomial; list its terms and give its degree.
13.Classify $x - x^{2} + x^{3} - x^{4}$ as a monomial, binomial, or trinomial; list its terms and give its degree.
14.Classify $\sqrt{2}x - \sqrt{3}$ as a monomial, binomial, or trinomial; list its terms and give its degree.
15.Find the sum, difference, or product: $(12x - 7) - (5x - 12)$.
16.Find the sum, difference, or product: $(5 - 3x) + (2x - 8)$.
17.Find the sum, difference, or product: $(-2x^{2} - 3x + 1) + (3x^{2} + 5x - 4)$.
18.Find the sum, difference, or product: $(3x^{2} + x + 1) - (2x^{2} - 3x - 5)$.
19.Find the sum, difference, or product: $(5x^{3} + 4x^{2} - 3x) - (x^{2} + 7x + 2)$.
20.Find the sum, difference, or product: $3(x - 1) + 4(x + 2)$.
21.Find the sum, difference, or product: $8(2x + 5) - 7(x - 9)$.
22.Find the sum, difference, or product: $4(x^{2} - 3x + 5) - 3(x^{2} - 2x + 1)$.
23.Find the sum, difference, or product: $2(2 - 5t) + t^{2}(t - 1) - (t^{4} - 1)$.
24.Find the sum, difference, or product: $5(3t - 4) - (t^{2} + 2) - 2t(t - 3)$.
25.Multiply using FOIL and simplify: $(3t - 2)(7t - 4)$.
26.Multiply using FOIL and simplify: $(4s - 1)(2s + 5)$.
27.Multiply using FOIL and simplify: $(3x + 5)(2x - 1)$.
28.Multiply using FOIL and simplify: $(7y - 3)(2y - 1)$.
29.Multiply using FOIL and simplify: $(x + 3y)(2x - y)$.
30.Multiply using FOIL and simplify: $(4x - 5y)(3x - y)$.
31.Multiply using a Special Product Formula and simplify: $(5x + 1)^{2}$.
32.Multiply using a Special Product Formula and simplify: $(2 - 7y)^{2}$.
33.Multiply using a Special Product Formula and simplify: $(2u + v)^{2}$.
34.Multiply using a Special Product Formula and simplify: $(x - 3y)^{2}$.
35.Multiply using a Special Product Formula and simplify: $(2x + 3y)^{2}$.
36.Multiply using a Special Product Formula and simplify: $(r - 2s)^{2}$.
37.Multiply using a Special Product Formula and simplify: $(x + 6)(x - 6)$.
38.Multiply using a Special Product Formula and simplify: $(5 - y)(5 + y)$.
39.Multiply using a Special Product Formula and simplify: $(3x - 4)(3x + 4)$.
40.Multiply using a Special Product Formula and simplify: $(2y + 5)(2y - 5)$.
41.Multiply using a Special Product Formula and simplify: $(\sqrt{x} + 2)(\sqrt{x} - 2)$.
42.Multiply using a Special Product Formula and simplify: $(\sqrt{y} + \sqrt{2})(\sqrt{y} - \sqrt{2})$.
43.Multiply using a Special Product Formula and simplify: $(y + 2)^{3}$.
44.Multiply using a Special Product Formula and simplify: $(x - 3)^{3}$.
45.Multiply using a Special Product Formula and simplify: $(1 - 2r)^{3}$.
46.Multiply using a Special Product Formula and simplify: $(3 + 2y)^{3}$.
47.Perform the operations and simplify: $(x + 2)(x^{2} + 2x + 3)$.
48.Perform the operations and simplify: $(x + 1)(2x^{2} - x + 1)$.
49.Perform the operations and simplify: $(2x - 5)(x^{2} - x + 1)$.
50.Perform the operations and simplify: $(1 + 2x)(x^{2} - 3x + 1)$.
51.Perform the operations and simplify: $\sqrt{x}(x - \sqrt{x})$.
52.Perform the operations and simplify: $x^{3/2}(\sqrt{x} - 1/\sqrt{x})$.
53.Perform the operations and simplify: $y^{1/3}(y^{2/3} + y^{5/3})$.
54.Perform the operations and simplify: $x^{1/4}(2x^{3/4} - x^{1/4})$.
55.Perform the operations and simplify: $(x^{2} - a^{2})(x^{2} + a^{2})$.
56.Perform the operations and simplify: $(x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2})$.
57.Perform the operations and simplify: $(\sqrt{a} - b)(\sqrt{a} + b)$.
58.Perform the operations and simplify: $(\sqrt{h^{2} + 1} + 1)(\sqrt{h^{2} + 1} - 1)$.
59.Perform the operations and simplify: $\big((x - 1) + x^{2}\big)\big((x - 1) - x^{2}\big)$.
60.Perform the operations and simplify: $(x + (2 + x^{2}))(x - (2 + x^{2}))$.
61.Perform the operations and simplify: $(2x + y - 3)(2x + y + 3)$.
62.Perform the operations and simplify: $(x + y + z)(x - y - z)$.
63.Factor out the common factor: $-2x^{3} + x$.
64.Factor out the common factor: $3x^{4} - 6x^{3} - x^{2}$.
65.Factor out the common factor: $y(y - 6) + 9(y - 6)$.
66.Factor out the common factor: $(z + 2)^{2} - 5(z + 2)$.
67.Factor out the common factor: $2x^{2}y - 6xy^{2} + 3xy$.
68.Factor out the common factor: $-7x^{4}y^{2} + 14xy^{3} + 21xy^{4}$.
69.Factor the trinomial: $x^{2} + 8x + 7$.
70.Factor the trinomial: $x^{2} + 4x - 5$.
71.Factor the trinomial: $8x^{2} - 14x - 15$.
72.Factor the trinomial: $6y^{2} + 11y - 21$.
73.Factor the trinomial: $3x^{2} - 16x + 5$.
74.Factor the trinomial: $5x^{2} - 7x - 6$.
75.Factor the trinomial: $(3x + 2)^{2} + 8(3x + 2) + 12$.
76.Factor the trinomial: $2(a + b)^{2} + 5(a + b) - 3$.
77.Use a Special Factoring Formula to factor: $9a^{2} - 16$.
78.Use a Special Factoring Formula to factor: $(x + 3)^{2} - 4$.
79.Use a Special Factoring Formula to factor: $27x^{3} + y^{3}$.
80.Use a Special Factoring Formula to factor: $a^{3} - b^{6}$.
81.Use a Special Factoring Formula to factor: $8s^{3} - 125t^{3}$.
82.Use a Special Factoring Formula to factor: $1 + 1000y^{3}$.
83.Use a Special Factoring Formula to factor: $x^{2} + 12x + 36$.
84.Use a Special Factoring Formula to factor: $16z^{2} - 24z + 9$.
85.Factor by grouping: $x^{3} + 4x^{2} + x + 4$.
86.Factor by grouping: $3x^{3} - x^{2} + 6x - 2$.
87.Factor by grouping: $5x^{3} - x^{2} + 5x - 1$.
88.Factor by grouping: $18x^{3} + 9x^{2} + 2x + 1$.
89.Factor by grouping: $x^{3} + x^{2} + x + 1$.
90.Factor by grouping: $x^{5} + x^{4} + x + 1$.
91.Factor completely, first removing the lowest power: $x^{5/2} - x^{1/2}$.
92.Factor completely, first removing the lowest power: $3x^{-1/2} + 4x^{1/2} + x^{3/2}$.
93.Factor completely, first removing the lowest power: $x^{-3/2} + 2x^{-1/2} + x^{1/2}$.
94.Factor completely, first removing the lowest power: $(x - 1)^{7/2} - (x - 1)^{3/2}$.
95.Factor completely, first removing the lowest power: $(x^{2} + 1)^{1/2} + 2(x^{2} + 1)^{-1/2}$.
96.Factor completely, first removing the lowest power: $x^{-1/2}(x + 1)^{1/2} + x^{1/2}(x + 1)^{-1/2}$.
97.Factor completely: $12x^{3} + 18x$.
98.Factor completely: $30x^{3} + 15x^{4}$.
99.Factor completely: $x^{2} - 2x - 8$.
100.Factor completely: $x^{2} - 14x + 48$.
101.Factor completely: $2x^{2} + 5x + 3$.
102.Factor completely: $2x^{2} + 7x - 4$.
103.Factor completely: $9x^{2} - 36x - 45$.
104.Factor completely: $8x^{2} + 10x + 3$.
105.Factor completely: $49 - 4y^{2}$.
106.Factor completely: $4t^{2} - 9s^{2}$.
107.Factor completely: $t^{2} - 6t + 9$.
108.Factor completely: $x^{2} + 10x + 25$.
109.Factor completely: $4x^{2} + 4xy + y^{2}$.
110.Factor completely: $r^{2} - 6rs + 9s^{2}$.
111.Factor completely: $(a + b)^{2} - (a - b)^{2}$.
112.Factor completely: $\left(1 + \tfrac{1}{x}\right)^{2} - \left(1 - \tfrac{1}{x}\right)^{2}$.
113.Factor completely: $x^{2}(x^{2} - 1) - 9(x^{2} - 1)$.
114.Factor completely: $(a^{2} - 1)b^{2} - 4(a^{2} - 1)$.
115.Factor completely: $8x^{3} - 125$.
116.Factor completely: $x^{6} + 64$.
117.Factor completely: $x^{3} + 2x^{2} + x$.
118.Factor completely: $3x^{3} - 27x$.
119.Factor completely: $x^{4}y^{3} - x^{2}y^{5}$.
120.Factor completely: $18y^{3}x^{2} - 2xy^{4}$.
121.Factor completely: $3x^{3} - x^{2} - 12x + 4$.
122.Factor completely: $9x^{3} + 18x^{2} - x - 2$.
123.Factor completely: $(x - 1)(x + 2)^{2} - (x - 1)^{2}(x + 2)$.
124.Factor completely: $y^{4}(y + 2)^{3} + y^{5}(y + 2)^{4}$.
125.Factor completely: $(a^{2} + 1)^{2} - 7(a^{2} + 1) + 10$.
126.Factor completely: $(a^{2} + 2a)^{2} - 2(a^{2} + 2a) - 3$.
127.Factor completely (this type arises in calculus via the Product Rule): $5(x^{2} + 4)^{4}(2x)(x - 2)^{4} + (x^{2} + 4)^{5}(4)(x - 2)^{3}$.
128.Factor completely: $3(2x - 1)^{2}(2)(x + 3)^{1/2} + (2x - 1)^{3}\left(\tfrac{1}{2}\right)(x + 3)^{-1/2}$.
129.Factor completely: $(x^{2} + 3)^{-1/3} - \tfrac{2}{3}x^{2}(x^{2} + 3)^{-4/3}$.
130.Factor completely: $\tfrac{1}{2}x^{-1/2}(3x + 4)^{1/2} - \tfrac{3}{2}x^{1/2}(3x + 4)^{-1/2}$.

Skills Plus

131.Verify the identities: (a) $ab = \tfrac{1}{2}\big[(a + b)^{2} - (a^{2} + b^{2})\big]$; (b) $(a^{2} + b^{2})^{2} - (a^{2} - b^{2})^{2} = 4a^{2}b^{2}$.
132.Verify the identity: $(a^{2} + b^{2})(c^{2} + d^{2}) = (ac + bd)^{2} + (ad - bc)^{2}$.
133.Factor completely: $4a^{2}c^{2} - (a^{2} - b^{2} + c^{2})^{2}$.
134.Some quartics $x^{4} + ax^{2} + b$ factor easily (e.g. $x^{4} + 3x^{2} - 4 = (x^{2} + 4)(x^{2} - 1)$), but $x^{4} + 3x^{2} + 4$ does not. Using the add-and-subtract method shown, factor each: (a) $x^{4} + x^{2} - 2$; (b) $x^{4} + 2x^{2} + 9$; (c) $x^{4} + 4x^{2} + 16$; (d) $x^{4} + 2x^{2} + 1$.

Applications

135.Concrete volume: A culvert is a cylindrical concrete shell (a cylinder with a cylindrical hole). With outer radius $R$, inner radius $r$, and height $h$, its volume is $V = \pi R^{2}h - \pi r^{2}h$. Factor this to show $V = 2\pi \cdot (\text{average radius}) \cdot (\text{height}) \cdot (\text{thickness})$, and use the unrolled-shell diagram to explain why this is reasonable geometrically. figure — add your own
136.Mowing a field: A square field measures $b$ ft on each side; a border strip of width $x$ is mowed around the edge while the center is left as habitat. (a) Explain why the mowed area is $b^{2} - (b - 2x)^{2}$. (b) Factor that expression to show the mowed area also equals $4x(b - x)$. figure — add your own

Discuss · Discover · Prove · Write

137.Discover: Form several pairs of polynomials and compute each pair's sum and product. From your observations, describe (a) how the degree of a product relates to the degrees of the factors, and (b) how the degree of a sum relates to the degrees of the addends.
138.Discuss: Use the difference-of-squares formula to evaluate mentally: (a) $528^{2} - 527^{2}$; (b) $122^{2} - 120^{2}$; (c) $1020^{2} - 1010^{2}$. Then invent similar problems you can do in your head.
139.Discuss: Use $(A + B)(A - B) = A^{2} - B^{2}$ to evaluate mentally: (a) $501\cdot 499$; (b) $79\cdot 61$; (c) $2007\cdot 1993$. Then invent similar products.
140.Discover: (a) Factor $A^{4} - B^{4}$ and $A^{6} - B^{6}$ completely. (b) Verify $18{,}335 = 12^{4} - 7^{4}$ and $2{,}868{,}335 = 12^{6} - 7^{6}$. (c) Use parts (a) and (b) to factor these integers, and show that all the factors in each are prime.
141.Discover: (a) Verify by expanding: $A^{2} - 1 = (A - 1)(A + 1)$, $A^{3} - 1 = (A - 1)(A^{2} + A + 1)$, $A^{4} - 1 = (A - 1)(A^{3} + A^{2} + A + 1)$. (b) Predict how $A^{5} - 1$ factors, verify it, then generalize to a factoring formula for $A^{n} - 1$ with $n$ a positive integer.
142.Prove (by expanding the right side): (a) Difference of cubes, $A^{3} - B^{3} = (A - B)(A^{2} + AB + B^{2})$; (b) Sum of cubes, $A^{3} + B^{3} = (A + B)(A^{2} - AB + B^{2})$.

1.4 Rational Expressions 100 problems

Concepts

1.Which of these are rational expressions? (a) $\dfrac{3x}{x^{2} - 1}$; (b) $\dfrac{\sqrt{x} + 1}{2x + 3}$; (c) $\dfrac{x(x^{2} - 1)}{x + 3}$.
2.Simplifying a rational expression means cancelling ______ common to the ______ and ______. So $\dfrac{(x + 1)(x + 2)}{(x + 3)(x + 2)}$ simplifies to ______.
3.To multiply two rational expressions, multiply their ______ together and their ______ together. So $\dfrac{2}{x + 1}\cdot\dfrac{x}{x + 3}$ equals ______.
4.For $\dfrac{1}{x} - \dfrac{2}{x + 1} - \dfrac{x}{(x + 1)^{2}}$: (a) how many terms are there? (b) find the least common denominator; (c) carry out the subtraction and simplify.
5.Yes or No (give a reason if No; ignore values making a denominator zero): (a) Is $\dfrac{x(x + 1)}{(x + 1)^{2}}$ equal to $\dfrac{x}{x + 1}$? (b) Is $\sqrt{x^{2} + 25}$ equal to $x + 5$?
6.Yes or No (give a reason if No): (a) Is $\dfrac{3 + a}{3}$ equal to $1 + \dfrac{a}{3}$? (b) Is $\dfrac{2}{4 + x}$ equal to $\dfrac{1}{2} + \dfrac{2}{x}$?

Skills

7.Find the domain of the expression $4x^{2} - 10x + 3$.
8.Find the domain of the expression $-x^{4} + x^{3} + 9x$.
9.Find the domain of $\dfrac{x^{2} - 1}{x - 3}$.
10.Find the domain of $\dfrac{2t^{2} - 5}{3t + 6}$.
11.Find the domain of $\sqrt{x + 3}$.
12.Find the domain of $\dfrac{1}{\sqrt{x} - 1}$.
13.Find the domain of $\dfrac{x^{2} + 1}{x^{2} - x - 2}$.
14.Find the domain of $\dfrac{\sqrt{2x}}{x + 1}$.
15.Simplify the rational expression: $\dfrac{5(x - 3)(2x + 1)}{10(x - 3)^{2}}$.
16.Simplify the rational expression: $\dfrac{4(x^{2} - 1)}{12(x + 2)(x - 1)}$.
17.Simplify the rational expression: $\dfrac{x - 2}{x^{2} - 4}$.
18.Simplify the rational expression: $\dfrac{x^{2} - x - 2}{x^{2} - 1}$.
19.Simplify the rational expression: $\dfrac{x^{2} + 5x + 6}{x^{2} + 8x + 15}$.
20.Simplify the rational expression: $\dfrac{x^{2} - x - 12}{x^{2} + 5x + 6}$.
21.Simplify the rational expression: $\dfrac{y^{2} + y}{y^{2} - 1}$.
22.Simplify the rational expression: $\dfrac{y^{2} - 3y - 18}{2y^{2} + 7y + 3}$.
23.Simplify the rational expression: $\dfrac{2x^{3} - x^{2} - 6x}{2x^{2} - 7x + 6}$.
24.Simplify the rational expression: $\dfrac{1 - x^{2}}{x^{3} - 1}$.
25.Multiply or divide and simplify: $\dfrac{4x}{x^{2} - 4}\cdot\dfrac{x + 2}{16x}$.
26.Multiply or divide and simplify: $\dfrac{x^{2} - 25}{x^{2} - 16}\cdot\dfrac{x + 4}{x + 5}$.
27.Multiply or divide and simplify: $\dfrac{x^{2} + 2x - 15}{x^{2} - 25}\cdot\dfrac{x - 5}{x + 2}$.
28.Multiply or divide and simplify: $\dfrac{x^{2} + 2x - 3}{x^{2} - 2x - 3}\cdot\dfrac{3 - x}{3 + x}$.
29.Multiply or divide and simplify: $\dfrac{t - 3}{t^{2} + 9}\cdot\dfrac{t + 3}{t^{2} - 9}$.
30.Multiply or divide and simplify: $\dfrac{x^{2} - x - 6}{x^{2} + 2x}\cdot\dfrac{x^{3} + x^{2}}{x^{2} - 2x - 3}$.
31.Multiply or divide and simplify: $\dfrac{x^{2} + 7x + 12}{x^{2} + 3x + 2}\cdot\dfrac{x^{2} + 5x + 6}{x^{2} + 6x + 9}$.
32.Multiply or divide and simplify: $\dfrac{x^{2} + 2xy + y^{2}}{x^{2} - y^{2}}\cdot\dfrac{2x^{2} - xy - y^{2}}{x^{2} - xy - 2y^{2}}$.
33.Multiply or divide and simplify: $\dfrac{x + 3}{4x^{2} - 9}\div\dfrac{x^{2} + 7x + 12}{2x^{2} + 7x - 15}$.
34.Multiply or divide and simplify: $\dfrac{2x + 1}{2x^{2} + x - 15}\div\dfrac{6x^{2} - x - 2}{x + 3}$.
35.Multiply or divide and simplify: $\dfrac{\tfrac{x^{3}}{x + 1}}{\tfrac{x}{x^{2} + 2x + 1}}$.
36.Multiply or divide and simplify: $\dfrac{\tfrac{2x^{2} - 3x - 2}{x^{2} - 1}}{\tfrac{2x^{2} + 5x + 2}{x^{2} + x - 2}}$.
37.Multiply or divide and simplify: $\dfrac{x/y}{z}$.
38.Multiply or divide and simplify: $\dfrac{x}{y/z}$.
39.Add or subtract and simplify: $1 + \dfrac{1}{x + 3}$.
40.Add or subtract and simplify: $\dfrac{3x - 2}{x + 1} - 2$.
41.Add or subtract and simplify: $\dfrac{1}{x + 5} + \dfrac{2}{x - 3}$.
42.Add or subtract and simplify: $\dfrac{1}{x + 1} + \dfrac{1}{x - 1}$.
43.Add or subtract and simplify: $\dfrac{3}{x + 1} - \dfrac{1}{x + 2}$.
44.Add or subtract and simplify: $\dfrac{x}{x - 4} - \dfrac{3}{x + 6}$.
45.Add or subtract and simplify: $\dfrac{5}{2x - 3} - \dfrac{3}{(2x - 3)^{2}}$.
46.Add or subtract and simplify: $\dfrac{x}{(x + 1)^{2}} + \dfrac{2}{x + 1}$.
47.Add or subtract and simplify: $u + 1 + \dfrac{u}{u + 1}$.
48.Add or subtract and simplify: $\dfrac{2}{a^{2}} - \dfrac{3}{ab} + \dfrac{4}{b^{2}}$.
49.Add or subtract and simplify: $\dfrac{1}{x^{2}} + \dfrac{1}{x^{2} + x}$.
50.Add or subtract and simplify: $\dfrac{1}{x} + \dfrac{1}{x^{2}} + \dfrac{1}{x^{3}}$.
51.Add or subtract and simplify: $\dfrac{2}{x + 3} - \dfrac{1}{x^{2} + 7x + 12}$.
52.Add or subtract and simplify: $\dfrac{x}{x^{2} - 4} + \dfrac{1}{x - 2}$.
53.Add or subtract and simplify: $\dfrac{1}{x + 3} + \dfrac{1}{x^{2} - 9}$.
54.Add or subtract and simplify: $\dfrac{x}{x^{2} + x - 2} - \dfrac{2}{x^{2} - 5x + 4}$.
55.Add or subtract and simplify: $\dfrac{2}{x} + \dfrac{3}{x - 1} - \dfrac{4}{x^{2} - x}$.
56.Add or subtract and simplify: $\dfrac{x}{x^{2} - x - 6} - \dfrac{1}{x + 2} - \dfrac{2}{x - 3}$.
57.Add or subtract and simplify: $\dfrac{1}{x^{2} + 3x + 2} - \dfrac{1}{x^{2} - 2x - 3}$.
58.Add or subtract and simplify: $\dfrac{1}{x + 1} - \dfrac{2}{(x + 1)^{2}} + \dfrac{3}{x^{2} - 1}$.
59.Simplify the compound fraction: $\dfrac{1 + \tfrac{1}{x}}{\tfrac{1}{x} - 2}$.
60.Simplify the compound fraction: $\dfrac{1 - \tfrac{2}{y}}{\tfrac{3}{y} - 1}$.
61.Simplify the compound fraction: $\dfrac{1 + \tfrac{1}{x + 2}}{1 - \tfrac{1}{x + 2}}$.
62.Simplify the compound fraction: $\dfrac{1 + \tfrac{1}{c - 1}}{1 - \tfrac{1}{c - 1}}$.
63.Simplify the compound fraction: $\dfrac{\tfrac{1}{x - 1} + \tfrac{1}{x + 3}}{x + 1}$.
64.Simplify the compound fraction: $\dfrac{\tfrac{x - 3}{x - 4} - \tfrac{x + 2}{x + 1}}{x + 3}$.
65.Simplify the compound fraction: $\dfrac{x - \tfrac{x}{y}}{y - \tfrac{y}{x}}$.
66.Simplify the compound fraction: $\dfrac{x + \tfrac{y}{x}}{y + \tfrac{x}{y}}$.
67.Simplify the compound fraction: $\dfrac{\tfrac{x}{y} - \tfrac{y}{x}}{\tfrac{1}{x^{2}} - \tfrac{1}{y^{2}}}$.
68.Simplify the compound fraction: $x - \dfrac{y}{\tfrac{x}{y} + \tfrac{y}{x}}$.
69.Simplify the compound fraction: $\dfrac{x^{-2} - y^{-2}}{x^{-1} + y^{-1}}$.
70.Simplify the compound fraction: $\dfrac{x^{-1} + y^{-1}}{(x + y)^{-1}}$.
71.Simplify the compound fraction: $1 - \dfrac{1}{1 - \tfrac{1}{x}}$.
72.Simplify the compound fraction: $1 + \dfrac{1}{1 + \tfrac{1}{1 + x}}$.
73.Simplify (this type arises in calculus): $\dfrac{\tfrac{1}{1 + x + h} - \tfrac{1}{1 + x}}{h}$.
74.Simplify (this type arises in calculus): $\dfrac{\tfrac{1}{\sqrt{x + h}} - \tfrac{1}{\sqrt{x}}}{h}$.
75.Simplify (this type arises in calculus): $\dfrac{\tfrac{1}{(x + h)^{2}} - \tfrac{1}{x^{2}}}{h}$.
76.Simplify (this type arises in calculus): $\dfrac{(x + h)^{3} - 7(x + h) - (x^{3} - 7x)}{h}$.
77.Simplify (this type arises in calculus): $\sqrt{1 + \left(\dfrac{x}{\sqrt{1 - x^{2}}}\right)^{2}}$.
78.Simplify (this type arises in calculus): $\sqrt{1 + \left(x^{3} - \dfrac{1}{4x^{3}}\right)^{2}}$.
79.Simplify (this type arises in calculus via the Quotient Rule): $\dfrac{3(x + 2)^{2}(x - 3)^{2} - (x + 2)^{3}(2)(x - 3)}{(x - 3)^{4}}$.
80.Simplify (Quotient Rule): $\dfrac{2x(x + 6)^{4} - x^{2}(4)(x + 6)^{3}}{(x + 6)^{8}}$.
81.Simplify (Quotient Rule): $\dfrac{2(1 + x)^{1/2} - x(1 + x)^{-1/2}}{x + 1}$.
82.Simplify (Quotient Rule): $\dfrac{(1 - x^{2})^{1/2} + x^{2}(1 - x^{2})^{-1/2}}{1 - x^{2}}$.
83.Simplify (Quotient Rule): $\dfrac{3(1 + x)^{1/3} - x(1 + x)^{-2/3}}{(1 + x)^{2/3}}$.
84.Simplify (Quotient Rule): $\dfrac{(7 - 3x)^{1/2} + \tfrac{3}{2}x(7 - 3x)^{-1/2}}{7 - 3x}$.
85.Rationalize the denominator: $\dfrac{1}{5 - \sqrt{3}}$.
86.Rationalize the denominator: $\dfrac{3}{2 - \sqrt{5}}$.
87.Rationalize the denominator: $\dfrac{2}{\sqrt{2} + \sqrt{7}}$.
88.Rationalize the denominator: $\dfrac{1}{\sqrt{x} + 1}$.
89.Rationalize the denominator: $\dfrac{y}{\sqrt{3} + \sqrt{y}}$.
90.Rationalize the denominator: $\dfrac{2(x - y)}{\sqrt{x} - \sqrt{y}}$.
91.Rationalize the numerator: $\dfrac{1 - \sqrt{5}}{3}$.
92.Rationalize the numerator: $\dfrac{\sqrt{3} + \sqrt{5}}{2}$.
93.Rationalize the numerator: $\dfrac{\sqrt{r} + \sqrt{2}}{5}$.
94.Rationalize the numerator: $\dfrac{\sqrt{x} - \sqrt{x + h}}{h\sqrt{x}\sqrt{x + h}}$.
95.Rationalize the numerator: $\sqrt{x^{2} + 1} - x$.
96.Rationalize the numerator: $\sqrt{x + 1} - \sqrt{x}$.

Applications

97.Resistors in parallel: For two resistors $R_1$ and $R_2$ in parallel, the total resistance is $R = \dfrac{1}{\tfrac{1}{R_1} + \tfrac{1}{R_2}}$. (a) Simplify the expression for $R$. (b) Find $R$ when $R_1 = 10$ ohms and $R_2 = 20$ ohms. figure — add your own
98.Average cost: Producing $x$ shirts costs $500 + 6x + 0.01x^{2}$ dollars. (a) Explain why the average cost per shirt is $A = \dfrac{500 + 6x + 0.01x^{2}}{x}$. (b) Complete the table of average cost for $x = 10, 20, 50, 100, 200, 500, 1000$.

Discuss · Discover · Prove · Write

99.Discover: The expression $\dfrac{x^{2} - 9}{x - 3}$ is undefined at $x = 3$. Complete tables of its values as $x$ approaches 3 from below and above, decide what value it approaches, and explain why by factoring the numerator and simplifying.
100.Discuss / Write: In $\dfrac{2}{\sqrt{x}}$ we could remove the radical by squaring numerator and denominator. Is that the same as rationalizing the denominator? Explain.

1.5 Equations 141 problems

Concepts

1.Yes or No (give a reason if No): (a) Does adding the same number to both sides of an equation always yield an equivalent equation? (b) Does multiplying both sides by the same nonzero number always yield an equivalent equation? (c) Does squaring both sides always yield an equivalent equation?
2.State a sensible first step for solving each equation: (a) $(x + 5)^{2} = 64$; (b) $(x + 5)^{2} + 5 = 64$; (c) $x^{2} + x = 2$.
3.For $x^{2} - 4x - 5 = 0$, describe how you would solve it (a) by factoring, (b) by completing the square, (c) using the Quadratic Formula.
4.(a) The solutions of $x^{2}(x - 4) = 0$ are ______. (b) To solve $x^{3} - 4x^{2} = 0$, we ______ the left-hand side.
5.Solve $\sqrt{2x} + x = 0$ step by step: (a) isolate the radical; (b) square both sides; (c) state the solutions of the resulting quadratic; (d) state which of those satisfy the original equation.
6.The equation $(x + 1)^{2} - 5(x + 1) + 6 = 0$ is of ______ type. Setting $W = \underline{\;\;\;}$ gives the quadratic ______.
7.To clear denominators in $\dfrac{3}{x} + \dfrac{5}{x + 2} = 2$, multiply both sides by the lowest common denominator ______ to obtain the equivalent equation ______.
8.To remove the radical in $2x + 1 = \sqrt{x + 1}$, we ______ both sides to get the equation ______.

Skills

9.Determine whether each value solves $4x + 7 = 9x - 3$: (a) $x = -2$; (b) $x = 2$.
10.Determine whether each value solves $1 - [2 - (3 - x)] = 4x - (6 + x)$: (a) $x = 2$; (b) $x = 4$.
11.Determine whether each value solves $\dfrac{1}{x} - \dfrac{1}{x - 4} = 1$: (a) $x = 2$; (b) $x = 4$.
12.Determine whether each value solves $\dfrac{x^{3/2}}{x - 6} = x - 8$: (a) $x = 4$; (b) $x = 8$.
13.Solve the linear equation: $5x - 6 = 14$.
14.Solve the linear equation: $3x + 4 = 7$.
15.Solve the linear equation: $\tfrac{1}{2}x - 8 = 1$.
16.Solve the linear equation: $3 + \tfrac{1}{3}x = 5$.
17.Solve the linear equation: $-x + 3 = 4x$.
18.Solve the linear equation: $2x + 3 = 7 - 3x$.
19.Solve the linear equation: $\dfrac{x}{3} - 2 = \dfrac{5}{3}x + 7$.
20.Solve the linear equation: $\tfrac{2}{5}x - 1 = \tfrac{3}{10}x + 3$.
21.Solve the linear equation: $2(1 - x) = 3(1 + 2x) + 5$.
22.Solve the linear equation: $\tfrac{2}{3}y + \tfrac{1}{2}(y - 3) = \dfrac{y + 1}{4}$.
23.Solve the linear equation: $x - \tfrac{1}{3}x - \tfrac{1}{2}x - 5 = 0$.
24.Solve the linear equation: $2x - \dfrac{x}{2} + \dfrac{x + 1}{4} = 6x$.
25.Solve the linear equation: $\dfrac{1}{x} = \dfrac{4}{3x} + 1$.
26.Solve the linear equation: $\dfrac{2x - 1}{x + 2} = \dfrac{4}{5}$.
27.Solve the linear equation: $\dfrac{3}{x + 1} - \dfrac{1}{2} = \dfrac{1}{3x + 3}$.
28.Solve the linear equation: $\dfrac{4}{x - 1} + \dfrac{2}{x + 1} = \dfrac{35}{x^{2} - 1}$.
29.Solve the linear equation: $(t - 4)^{2} = (t + 4)^{2} + 32$.
30.Solve the linear equation: $\sqrt{3}x + \sqrt{12} = \dfrac{x + 5}{\sqrt{3}}$.
31.Solve for the indicated variable: $PV = nRT$; for $R$.
32.Solve for the indicated variable: $F = G\dfrac{mM}{r^{2}}$; for $m$.
33.Solve for the indicated variable: $P = 2l + 2w$; for $w$.
34.Solve for the indicated variable: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$; for $R_1$.
35.Solve for the indicated variable: $\dfrac{ax + b}{cx + d} = 2$; for $x$.
36.Solve for the indicated variable: $a - 2[b - 3(c - x)] = 6$; for $x$.
37.Solve for the indicated variable: $a^{2}x + (a - 1) = (a + 1)x$; for $x$.
38.Solve for the indicated variable: $\dfrac{a + 1}{b} = \dfrac{a - 1}{b} + \dfrac{b + 1}{a}$; for $a$.
39.Solve for the indicated variable: $V = \tfrac{1}{3}\pi r^{2}h$; for $r$.
40.Solve for the indicated variable: $F = G\dfrac{mM}{r^{2}}$; for $r$.
41.Solve for the indicated variable: $a^{2} + b^{2} = c^{2}$; for $b$.
42.Solve for the indicated variable: $A = P\left(1 + \dfrac{i}{100}\right)^{2}$; for $i$.
43.Solve for the indicated variable: $h = \tfrac{1}{2}gt^{2} + v_0 t$; for $t$.
44.Solve for the indicated variable: $S = \dfrac{n(n + 1)}{2}$; for $n$.
45.Solve by factoring: $x^{2} + x - 12 = 0$.
46.Solve by factoring: $x^{2} + 3x - 4 = 0$.
47.Solve by factoring: $x^{2} - 7x + 12 = 0$.
48.Solve by factoring: $x^{2} + 8x + 12 = 0$.
49.Solve by factoring: $4x^{2} - 4x - 15 = 0$.
50.Solve by factoring: $2y^{2} + 7y + 3 = 0$.
51.Solve by factoring: $3x^{2} + 5x = 2$.
52.Solve by factoring: $6x(x - 1) = 21 - x$.
53.Solve by factoring: $2x^{2} = 8$.
54.Solve by factoring: $3x^{2} - 27 = 0$.
55.Solve by factoring: $(2x - 5)^{2} = 81$.
56.Solve by factoring: $(5x + 1)^{2} + 3 = 10$.
57.Solve by completing the square: $x^{2} + 2x - 5 = 0$.
58.Solve by completing the square: $x^{2} - 4x + 2 = 0$.
59.Solve by completing the square: $x^{2} - 6x - 11 = 0$.
60.Solve by completing the square: $x^{2} + 3x - \tfrac{7}{4} = 0$.
61.Solve by completing the square: $2x^{2} + 8x + 1 = 0$.
62.Solve by completing the square: $3x^{2} - 6x - 1 = 0$.
63.Solve by completing the square: $4x^{2} - x = 0$.
64.Solve by completing the square: $x^{2} = \tfrac{3}{4}x - \tfrac{1}{8}$.
65.Solve the quadratic equation: $x^{2} - 2x - 15 = 0$.
66.Solve the quadratic equation: $x^{2} + 5x - 6 = 0$.
67.Solve the quadratic equation: $x^{2} - 13x + 42 = 0$.
68.Solve the quadratic equation: $x^{2} + 10x - 600 = 0$.
69.Solve the quadratic equation: $2x^{2} + x - 3 = 0$.
70.Solve the quadratic equation: $3x^{2} + 7x + 4 = 0$.
71.Solve the quadratic equation: $3x^{2} + 6x - 5 = 0$.
72.Solve the quadratic equation: $x^{2} - 6x + 1 = 0$.
73.Solve the quadratic equation: $9x^{2} + 12x + 4 = 0$.
74.Solve the quadratic equation: $4x^{2} - 4x + 1 = 0$.
75.Solve the quadratic equation: $4x^{2} + 16x - 9 = 0$.
76.Solve the quadratic equation: $0 = x^{2} - 4x + 1$.
77.Solve the quadratic equation: $7x^{2} - 2x + 4 = 0$.
78.Solve the quadratic equation: $w^{2} = 3(w - 1)$.
79.Solve the quadratic equation: $10y^{2} - 16y + 5 = 0$.
80.Solve the quadratic equation: $25x^{2} + 70x + 49 = 0$.
81.Use the discriminant to find the number of real solutions (do not solve): $x^{2} - 6x + 1 = 0$.
82.Use the discriminant to find the number of real solutions (do not solve): $3x^{2} = 6x - 9$.
83.Use the discriminant to find the number of real solutions (do not solve): $x^{2} + 2.20x + 1.21 = 0$.
84.Use the discriminant to find the number of real solutions (do not solve): $x^{2} + 2.21x + 1.21 = 0$.
85.Use the discriminant to find the number of real solutions (do not solve): $4x^{2} + 5x + \tfrac{13}{8} = 0$.
86.Use the discriminant to find the number of real solutions (do not solve): $x^{2} + rx - s = 0$ with $s > 0$.
87.Find all real solutions: $\dfrac{x^{2}}{x + 100} = 50$.
88.Find all real solutions: $\dfrac{1}{x} - \dfrac{2}{x^{2}} = 0$.
89.Find all real solutions: $\dfrac{1}{x - 1} + \dfrac{1}{x + 2} = \dfrac{5}{4}$.
90.Find all real solutions: $\dfrac{x + 5}{x - 2} = \dfrac{5}{x + 2} + \dfrac{28}{x^{2} - 4}$.
91.Find all real solutions: $\dfrac{10}{x} - \dfrac{12}{x - 3} + 4 = 0$.
92.Find all real solutions: $\dfrac{x}{2x + 7} - \dfrac{x + 1}{x + 3} = 1$.
93.Find all real solutions: $5 = \sqrt{4x - 3}$.
94.Find all real solutions: $\sqrt{8x - 1} = 3$.
95.Find all real solutions: $\sqrt{2x - 1} = \sqrt{3x - 5}$.
96.Find all real solutions: $\sqrt{3 + x} = \sqrt{x^{2} + 1}$.
97.Find all real solutions: $\sqrt{2x + 1} + 1 = x$.
98.Find all real solutions: $\sqrt{5 - x} + 1 = x - 2$.
99.Find all real solutions: $2x + \sqrt{x + 1} = 8$.
100.Find all real solutions: $x - \sqrt{9 - 3x} = 0$.
101.Find all real solutions: $\sqrt{3x + 1} = 2 + \sqrt{x + 1}$.
102.Find all real solutions: $\sqrt{1 + x} + \sqrt{1 - x} = 2$.
103.Find all real solutions: $x^{4} - 13x^{2} + 40 = 0$.
104.Find all real solutions: $x^{4} - 5x^{2} + 4 = 0$.
105.Find all real solutions: $2x^{4} + 4x^{2} + 1 = 0$.
106.Find all real solutions: $x^{6} - 2x^{3} - 3 = 0$.
107.Find all real solutions: $x^{4/3} - 5x^{2/3} + 6 = 0$.
108.Find all real solutions: $\sqrt{x} - 3\sqrt[4]{x} - 4 = 0$.
109.Find all real solutions: $4(x + 1)^{1/2} - 5(x + 1)^{3/2} + (x + 1)^{5/2} = 0$.
110.Find all real solutions: $x^{1/2} + 3x^{-1/2} = 10x^{-3/2}$.
111.Find all real solutions: $x^{1/2} - 3x^{1/3} = 3x^{1/6} - 9$.
112.Find all real solutions: $x - 5\sqrt{x} + 6 = 0$.
113.Find all real solutions: $|3x + 5| = 1$.
114.Find all real solutions: $|2x| = 3$.
115.Find all real solutions: $|x - 4| = 0.01$.
116.Find all real solutions: $|x - 6| = -1$.

Skills Plus

117.Find all real solutions: $\dfrac{1}{x^{3}} + \dfrac{4}{x^{2}} + \dfrac{4}{x} = 0$.
118.Find all real solutions: $4x^{-4} - 16x^{-2} + 4 = 0$.
119.Find all real solutions: $\sqrt{\sqrt{x + 5} + x} = 5$.
120.Find all real solutions: $\sqrt[3]{4x^{2} - 4x} = x$.
121.Find all real solutions: $x^{2}\sqrt{x + 3} = (x + 3)^{3/2}$.
122.Find all real solutions: $\sqrt{11 - x^{2}} - \dfrac{2}{\sqrt{11 - x^{2}}} = 1$.
123.Solve for $x$ (constants $a, b$ are positive reals): $x^{4} - 5ax^{2} + 4a^{2} = 0$.
124.Solve for $x$: $a^{3}x^{3} + b^{3} = 0$.
125.Solve for $x$: $\sqrt{x + a} + \sqrt{x - a} = \sqrt{2}\sqrt{x + 6}$.
126.Solve for $x$: $\sqrt{x} - a\sqrt[3]{x} + b\sqrt[6]{x} - ab = 0$.

Applications

127.Falling body: An object dropped from height $h_0$ has height $h = -16t^{2} + h_0$ after $t$ seconds (in feet). If a ball is dropped from 288 ft, how long until it reaches the ground?
128.Falling body: A ball is dropped from a 96-ft building. (a) How long to fall halfway to the ground? (b) How long to reach the ground?
129.Projectile (use $h = -16t^{2} + v_0 t$): A ball is thrown straight up at $v_0 = 40$ ft/s. (a) When is it 24 ft high? (b) When is it 48 ft high? (c) What is its greatest height? (d) When does it reach the top? (e) When does it land?
130.How fast must a ball be thrown straight up to reach a maximum height of 100 ft? (Hint: use the discriminant of $16t^{2} - v_0 t + h = 0$.)
131.Concrete shrinkage: A drying concrete beam shrinks by a factor $S = \dfrac{0.032w - 2.5}{10{,}000}$, where $w$ is the water content (kg/m$^3$). (a) A 12.025-m beam is cast with 250 kg/m$^3$ water; find $S$ and the dried length. (b) A 10.014-m wet beam should dry to 10.009 m $(S = 0.00050)$; what water content gives this? figure — add your own
132.Lens equation: A convex lens of focal length $F$ relates object distance $x$ and image distance $y$ by $\dfrac{1}{F} = \dfrac{1}{x} + \dfrac{1}{y}$. If $F = 4.8$ cm and the image is 4 cm closer to the lens than the object, how far is the object from the lens?
133.Fish population: A lake's fish count follows $F = 1000(30 + 17t - t^{2})$, with $t$ in years since Jan 1, 2002 (when $F$ was first measured). (a) On what date will the population again equal its Jan 1, 2002 value? (b) By what date will all the fish have died?
134.Fish population: A stocked pond's fish count is $P = 3t + 10\sqrt{t} + 140$, where $t$ is days since introduction. After how many days does the population reach 500?
135.Profit: A manufacturer's weekly profit from producing $x$ microwaves is $P = \tfrac{1}{10}x(300 - x)$, valid for $0 \le x \le 200$. How many ovens per week yield a profit of $1250?
136.Gravity dead spot: Along the Earth–Moon line, the net gravitational force on an object is $F = \dfrac{-K}{x^{2}} + \dfrac{0.012K}{(239 - x)^{2}}$, where $K > 0$ is constant and $x$ (thousands of miles) is the distance from Earth's center. How far from Earth's center is the no-net-force point? (Answer to the nearest thousand miles.) figure — add your own
137.Well depth: Dropping a stone of depth $d$, the fall time is $t_1 = \dfrac{\sqrt{d}}{4}$ (from $d = 16t_1^{2}$) and the sound's return time is $t_2 = \dfrac{d}{1090}$ (sound speed 1090 ft/s), so the total is $\dfrac{\sqrt{d}}{4} + \dfrac{d}{1090}$. If the total elapsed time is 3 s, how deep is the well? figure — add your own

Discuss · Discover · Prove · Write

138.Discuss: The equation $3x + k - 5 = kx - k + 1$ is a family of equations indexed by the parameter $k$. What value of $k$ makes each given $x$ a solution: (a) $x = 0$; (b) $x = 1$; (c) $x = 2$?
139.Discuss: The following chain of steps appears to prove that $1 = 0$, starting from $x = 1$, multiplying by $x$, subtracting $x$, factoring, dividing by $x - 1$, and simplifying. Identify the error.
140.Discover / Prove (solutions vs. coefficients): (a) Solve $x^{2} - 9x + 20 = 0$ and verify that the product of its solutions is the constant term 20 and the sum of its solutions is 9 (the negative of the coefficient of $x$). (b) Show the same relationship holds for $x^{2} - 2x - 8 = 0$ and $x^{2} + 4x + 2 = 0$. (c) Using the Quadratic Formula, prove that if $x^{2} + bx + c = 0$ has solutions $r_1$ and $r_2$, then $c = r_1 r_2$ and $b = -(r_1 + r_2)$.
141.Discuss (solving an equation in different ways): Some equations can be solved by more than one method. Solve each using both indicated approaches and confirm the answers agree: (a) $x - \sqrt{x} - 2 = 0$ — as a quadratic type, and by solving for the radical then squaring; (b) $\dfrac{12}{(x - 3)^{2}} + \dfrac{10}{x - 3} + 1 = 0$ — as a quadratic type, and by multiplying through by the LCD.

1.6 Complex Numbers 86 problems

Concepts

1.The imaginary unit $i$ satisfies $i^{2} = \underline{\;\;\;}$.
2.For the complex number $3 + 4i$, the real part is ______ and the imaginary part is ______.
3.(a) The complex conjugate of $3 + 4i$ is $\overline{3 + 4i} = \underline{\;\;\;}$. (b) $(3 + 4i)\,\overline{(3 + 4i)} = \underline{\;\;\;}$.
4.If $3 + 4i$ is a solution of a quadratic equation with real coefficients, then ______ is also a solution.
5.Yes or No (give a reason if No): Is every real number also a complex number?
6.Yes or No (give a reason if No): Is the sum of a complex number and its conjugate a real number?

Skills

7.Find the real and imaginary parts of $5 - 7i$.
8.Find the real and imaginary parts of $-6 + 4i$.
9.Find the real and imaginary parts of $\dfrac{-2 - 5i}{3}$.
10.Find the real and imaginary parts of $\dfrac{4 + 7i}{2}$.
11.Find the real and imaginary parts of $3$.
12.Find the real and imaginary parts of $-\tfrac{1}{2}$.
13.Find the real and imaginary parts of $-\tfrac{2}{3}i$.
14.Find the real and imaginary parts of $i\sqrt{3}$.
15.Find the real and imaginary parts of $\sqrt{3} + \sqrt{-4}$.
16.Find the real and imaginary parts of $2 - \sqrt{-5}$.
17.Evaluate the sum or difference, writing the result as $a + bi$: $(3 + 2i) + 5i$.
18.Evaluate and write as $a + bi$: $3i - (2 - 3i)$.
19.Evaluate and write as $a + bi$: $(5 - 3i) + (-4 - 7i)$.
20.Evaluate and write as $a + bi$: $(-3 + 4i) - (2 - 5i)$.
21.Evaluate and write as $a + bi$: $(-6 + 6i) + (9 - i)$.
22.Evaluate and write as $a + bi$: $(3 - 2i) + \left(-5 - \tfrac{1}{3}i\right)$.
23.Evaluate and write as $a + bi$: $\left(7 - \tfrac{1}{2}i\right) - \left(5 + \tfrac{3}{2}i\right)$.
24.Evaluate and write as $a + bi$: $(-4 + i) - (2 - 5i)$.
25.Evaluate and write as $a + bi$: $(-12 + 8i) - (7 + 4i)$.
26.Evaluate and write as $a + bi$: $6i - (4 - i)$.
27.Evaluate the product and write as $a + bi$: $4(-1 + 2i)$.
28.Evaluate the product and write as $a + bi$: $-2(3 - 4i)$.
29.Evaluate the product and write as $a + bi$: $(7 - i)(4 + 2i)$.
30.Evaluate the product and write as $a + bi$: $(5 - 3i)(1 + i)$.
31.Evaluate the product and write as $a + bi$: $(6 + 5i)(2 - 3i)$.
32.Evaluate the product and write as $a + bi$: $(-2 + i)(3 - 7i)$.
33.Evaluate the product and write as $a + bi$: $(2 + 5i)(2 - 5i)$.
34.Evaluate the product and write as $a + bi$: $(3 - 7i)(3 + 7i)$.
35.Evaluate the product and write as $a + bi$: $(2 + 5i)^{2}$.
36.Evaluate the product and write as $a + bi$: $(3 - 7i)^{2}$.
37.Evaluate the quotient and write as $a + bi$: $\dfrac{1}{i}$.
38.Evaluate the quotient and write as $a + bi$: $\dfrac{1}{1 + i}$.
39.Evaluate the quotient and write as $a + bi$: $\dfrac{2 - 3i}{1 - 2i}$.
40.Evaluate the quotient and write as $a + bi$: $\dfrac{5 - i}{3 + 4i}$.
41.Evaluate the quotient and write as $a + bi$: $\dfrac{10i}{1 - 2i}$.
42.Evaluate the quotient and write as $a + bi$: $(2 - 3i)^{-1}$.
43.Evaluate the quotient and write as $a + bi$: $\dfrac{4 + 6i}{3i}$.
44.Evaluate the quotient and write as $a + bi$: $\dfrac{-3 + 5i}{15i}$.
45.Evaluate and write as $a + bi$: $\dfrac{1}{1 + i} - \dfrac{1}{1 - i}$.
46.Evaluate and write as $a + bi$: $\dfrac{(1 + 2i)(3 - i)}{2 + i}$.
47.Evaluate the power and write as $a + bi$: $i^{3}$.
48.Evaluate the power and write as $a + bi$: $i^{10}$.
49.Evaluate the power and write as $a + bi$: $(3i)^{5}$.
50.Evaluate the power and write as $a + bi$: $(2i)^{4}$.
51.Evaluate the power and write as $a + bi$: $i^{1000}$.
52.Evaluate the power and write as $a + bi$: $i^{1002}$.
53.Evaluate the radical expression and write as $a + bi$: $\sqrt{-49}$.
54.Evaluate and write as $a + bi$: $\sqrt{\dfrac{-81}{16}}$.
55.Evaluate and write as $a + bi$: $\sqrt{-3}\,\sqrt{-12}$.
56.Evaluate and write as $a + bi$: $\sqrt{\tfrac{1}{3}}\,\sqrt{-27}$.
57.Evaluate and write as $a + bi$: $(3 - \sqrt{-5})(1 + \sqrt{-1})$.
58.Evaluate and write as $a + bi$: $(\sqrt{3} - \sqrt{-4})(\sqrt{6} - \sqrt{-8})$.
59.Evaluate and write as $a + bi$: $\dfrac{2 + \sqrt{-8}}{1 + \sqrt{-2}}$.
60.Evaluate and write as $a + bi$: $\dfrac{\sqrt{-36}}{\sqrt{-2}\,\sqrt{-9}}$.
61.Find all solutions and write them as $a + bi$: $x^{2} + 49 = 0$.
62.Find all solutions and write them as $a + bi$: $3x^{2} + 1 = 0$.
63.Find all solutions and write them as $a + bi$: $x^{2} - x + 2 = 0$.
64.Find all solutions and write them as $a + bi$: $x^{2} + 2x + 2 = 0$.
65.Find all solutions and write them as $a + bi$: $x^{2} + 3x + 7 = 0$.
66.Find all solutions and write them as $a + bi$: $x^{2} - 6x + 10 = 0$.
67.Find all solutions and write them as $a + bi$: $x^{2} + x + 1 = 0$.
68.Find all solutions and write them as $a + bi$: $x^{2} - 3x + 3 = 0$.
69.Find all solutions and write them as $a + bi$: $2x^{2} - 2x + 1 = 0$.
70.Find all solutions and write them as $a + bi$: $t + 3 + \dfrac{3}{t} = 0$.
71.Find all solutions and write them as $a + bi$: $6x^{2} + 12x + 7 = 0$.
72.Find all solutions and write them as $a + bi$: $x^{2} + \tfrac{1}{2}x + 1 = 0$.

Skills Plus

73.With $z = 3 - 4i$ and $w = 5 + 2i$, evaluate $\bar{z} + \bar{w}$.
74.With $z = 3 - 4i$ and $w = 5 + 2i$, evaluate $\overline{z + w}$.
75.With $z = 3 - 4i$ and $w = 5 + 2i$, evaluate $z\cdot\bar{z}$.
76.With $z = 3 - 4i$ and $w = 5 + 2i$, evaluate $\bar{z}\cdot\bar{w}$.
77.For $z = a + bi$ and $w = c + di$, show that $\bar{z} + \bar{w} = \overline{z + w}$.
78.For $z = a + bi$ and $w = c + di$, show that $\overline{zw} = \bar{z}\cdot\bar{w}$.
79.For $z = a + bi$, show that $(\bar{z})^{2} = \overline{z^{2}}$.
80.For $z = a + bi$, show that $\bar{\bar{z}} = z$.
81.Show that $z + \bar{z}$ is a real number.
82.Show that $z - \bar{z}$ is a pure imaginary number.
83.Show that $z\cdot\bar{z}$ is a real number.
84.Show that $z = \bar{z}$ if and only if $z$ is real.

Discuss · Discover · Prove · Write

85.Prove: Suppose $ax^{2} + bx + c = 0$ has real coefficients and complex (non-real) roots. Explain why the roots must be complex conjugates of each other. (Hint: consider how the Quadratic Formula produces them.)
86.Discuss: Compute the first twelve powers $i, i^{2}, i^{3}, \ldots, i^{12}$. Describe the repeating pattern and explain how to find any whole-number power of $i$. Use your method to compute $i^{4446}$.

1.7 Modeling with Equations 94 problems

Concepts

1.In your own words, explain what it means for an equation to model a real-world situation, and give an example.
2.In the simple-interest formula $I = Prt$, state what $P$, $r$, and $t$ each represent.
3.Give an area formula for: (a) a square of side $x$, $A = \underline{\;\;}$; (b) a rectangle of length $l$ and width $w$, $A = \underline{\;\;}$; (c) a circle of radius $r$, $A = \underline{\;\;}$.
4.Balsamic vinegar is 5% acetic acid, so a 32-oz bottle contains ______ ounces of acetic acid.
5.A painter paints a wall in $x$ hours, so the fraction of the wall painted in 1 hour is ______.
6.The formula $d = rt$ models distance $d$ traveled at constant rate $r$ over time $t$. Solve for $r$ and for $t$: $r = \underline{\;\;}$, $t = \underline{\;\;}$.

Skills

7.Express in terms of the variable: the sum of three consecutive integers, with $n$ the first of them.
8.Express in terms of the variable: the sum of three consecutive integers, with $n$ the middle one.
9.Express in terms of the variable: the sum of three consecutive even integers, with $n$ the first of them.
10.Express in terms of the variable: the sum of the squares of two consecutive integers, with $n$ the first of them.
11.Express in terms of the variable: the average of three test scores when the first two are 78 and 82, with $s$ the third score.
12.Express in terms of the variable: the average of four quiz scores when the first three are each 8, with $q$ the fourth score.
13.Express in terms of the variable: the interest earned after 1 year on an investment at $2\tfrac{1}{2}\%$ simple interest, with $x$ the number of dollars invested.
14.Express in terms of the variable: the total rent paid for an apartment costing $795 a month, with $n$ the number of months.
15.Express in terms of the variable: the area (in ft$^2$) of a rectangle four times as long as it is wide, with $w$ the width (in ft).
16.Express in terms of the variable: the perimeter (in cm) of a rectangle 6 cm longer than it is wide, with $w$ the width (in cm).
17.Express in terms of the variable: the time (in hours) to travel a given distance at 55 mi/h, with $d$ the distance (in mi).
18.Express in terms of the variable: the distance (in mi) a car travels in 45 min, with $s$ the speed (in mi/h).
19.Express in terms of the variable: the salt concentration (in oz/gal) when pure water is added to 3 gal of brine containing 25 oz of salt, with $x$ the volume of pure water added (in gal).
20.Express in terms of the variable: the value (in cents) of the change in a purse with twice as many nickels as pennies, four more dimes than nickels, and as many quarters as dimes and nickels combined, with $p$ the number of pennies.

Applications

21.Truck rental: A company charges $65 per day plus $0.20 per mile. Michael's bill for a 3-day rental was $275. How many miles did he drive?
22.Cell phone cost: A plan charges $10 monthly for the first 1000 texts and $0.10 per additional text. Miriam's June bill was $38.50. How many texts did she send?
23.Exam average: Linh scored 82, 75, and 71 on three midterms. If the final counts double a midterm, what final score gives an average of 80? (Max possible per test is 100.)
24.Class average: In a class of 25 the average is 84. Six students scored 100 each and three scored 60 each. What is the average of the remaining students?
25.Investments: Phyllis invested $12,000, part at $4\tfrac{1}{2}\%$ and the rest at 4% simple annual interest. After 1 year the total interest was $525. How much was invested at each rate?
26.Investments: Ben invests $4000 at 4% per year. How much more must he invest at $5\tfrac{1}{2}\%$ so that his yearly interest equals $4\tfrac{1}{2}\%$ of the total invested?
27.Investments: What annual interest rate on a $3500 investment yields $262.50 of interest after 1 year?
28.Investments: Jack invests $1000 at one rate and $2000 at a rate one-half percent higher. If his total 1-year interest is $190, what rate applies to the $1000?
29.Salary: An executive earns a monthly salary plus an $8500 Christmas bonus. If her yearly total is $97,300, what is her monthly salary?
30.Salary: A woman earns 15% more than her husband; together they make $69,875 per year. What is the husband's annual salary?
31.Overtime: Helen earns $7.50/hour, with overtime (beyond 35 hours/week) paid at $1\tfrac{1}{2}$ times her rate. One week her gross pay was $352.50. How many overtime hours did she work?
32.Labor cost: A plumber bills $45/hour and his assistant $25/hour. The plumber worked twice as long as the assistant, and the total labor charge was $4025. How long did each work?
33.Riddle: A movie star says, 'Seven years ago I was eleven times as old as my daughter; now I am four times as old as she is.' How old is the star?
34.Home runs: Hank Aaron hit 41 more career home runs than Babe Ruth, and together they hit 1469. How many did Babe Ruth hit?
35.Coins: A purse has equal numbers of pennies, nickels, and dimes, totaling $1.44. How many of each coin are there?
36.Coins: Mary has $3.00 in nickels, dimes, and quarters, with twice as many dimes as quarters and five more nickels than dimes. How many of each does she have?
37.Garden length: A rectangular garden is 25 ft wide with area 1125 ft$^2$. What is its length? figure — add your own
38.Pasture width: A pasture twice as long as it is wide has area 115,200 ft$^2$. How wide is it?
39.Lot dimensions: On a square plot, a building 60 ft by 40 ft sits in one corner; the remaining land is a 12,000-ft$^2$ parking lot. What are the plot's dimensions?
40.Lot dimensions: A half-acre building lot is five times as long as it is wide. Find its dimensions. (1 acre = 43,560 ft$^2$.)
41.Garden dimensions: A rectangular garden is 10 ft longer than wide, with area 875 ft$^2$. Find its dimensions.
42.Room width: A rectangular bedroom is 7 ft longer than wide, with area 228 ft$^2$. What is its width?
43.Garden dimensions: A rectangular garden plot is enclosed by 200 ft of fence and has area 2400 ft$^2$. Find its length and width. figure — add your own
44.Lot dimensions: A parcel is 6 ft longer than wide, and a corner-to-corner diagonal is 174 ft. Find its dimensions.
45.Lot dimensions: A rectangular parcel is 50 ft wide, and a diagonal is 10 ft longer than the length. Find the length.
46.Track dimensions: A running track has two straight sides and semicircular ends. If the total length is 440 yd and each straight is 110 yd, find the radius of the semicircular ends (to the nearest yard). figure — add your own
47.Length and area: Find the length $x$ in each L-shaped figure given its shaded area: (a) area = 144 cm$^2$; (b) area = 160 in$^2$. figure — add your own
48.Length and area: Find the length $y$ in each figure given its shaded area: (a) area = 120 in$^2$; (b) area = 1200 cm$^2$. figure — add your own
49.Framing a painting: A 20-in by 15-in watercolor is placed on a mat with a uniform border. If the mat's perimeter is 102 in, how wide is the border showing around the picture? figure — add your own
50.Poster dimensions: A poster has a 100-cm by 140-cm printed area with a uniform blank border. The poster's perimeter is $1\tfrac{1}{2}$ times the printed area's perimeter. How wide is the border? figure — add your own
51.Ladder reach: A $19\tfrac{1}{2}$-ft ladder leans against a building with its base $7\tfrac{1}{2}$ ft from the wall. How high up does it reach? figure — add your own
52.Flagpole height: A flagpole is held by two guy wires, each 5 ft longer than the pole. The two ground anchor points are separated by a distance equal to one wire's length. How tall is the pole (to the nearest inch)? figure — add your own
53.Shadow length: A 2-m-tall man walks away from a lamppost whose light is 6 m high. How long is his shadow when he is 10 m from the post? (Hint: use similar triangles. A figure accompanies this problem.)
54.Tree height: A woodcutter sights along the tops of a 20-ft tree (125 ft away) and a small tree 20 ft tall (25 ft away), with eye level 5 ft above ground. How tall is the taller tree? figure — add your own
55.Mixture: How much 60% acid solution must be mixed with 30% solution to make 300 mL of 50% solution?
56.Mixture: How much pure acid must be added to 300 mL of 50% acid solution to make a 60% solution?
57.Mixture: Five rings, each 18 g of a 10%-silver, 90%-gold alloy, are melted together. How much silver should be added to bring the gold content down to 75%?
58.Mixture: A pot holds 6 L of brine at 120 g/L. How much water should be boiled off to raise the concentration to 200 g/L?
59.Mixture: A 3.6-L radiator holds 60% antifreeze / 40% water, but the recommended mix is 50% antifreeze. How much coolant should be drained and replaced with pure water?
60.Mixture: A 100-gal sterilization tank holds 2% bleach in distilled water; the recommended strength is 5%. How much solution should be drained and replaced with pure bleach?
61.Mixture: A 750-mL bottle of fruit punch is 50% juice. Jill drinks 100 mL and refills with a cheaper punch; the juice concentration drops to 48%. What was the juice concentration of the punch Jill added?
62.Mixture: A merchant blends $3.00/oz tea with $2.75/oz tea to make 80 oz of a $2.90/oz blend. How many ounces of each are used?
63.Shared work: Candy delivers a paper route in 70 min; Tim takes 80 min. How long if they work together?
64.Shared work: Stan and Hilda together mow a lawn in 40 min. If Hilda works twice as fast as Stan, how long does Stan take alone?
65.Shared work: Working together, Betty and Karen paint a house in two-thirds the time Karen needs alone. Betty alone takes 6 h. How long does Karen take alone?
66.Shared work: Using both hoses, Bob and Jim fill a pool in 18 h. Bob's hose alone takes 20% less time than Jim's. How long does each hose take alone?
67.Shared work: Together, Henry and Irene wash all their windows in 1 h 48 min. Henry alone takes $1\tfrac{1}{2}$ h longer than Irene. How long does each take alone?
68.Shared work: Jack, Kay, and Lynn deliver flyers. Jack alone takes 4 h; Lynn takes 1 h longer than Kay. Together they finish in 40% of Kay's solo time. How long does Kay take alone?
69.Distance, speed, time: Wendy travels 300 mi from Davenport to Omaha, part by bus (averaging 40 mi/h) and the rest by train (averaging 60 mi/h); the total trip took $5\tfrac{1}{2}$ h. How long was she on the train?
70.Distance, speed, time: Two cyclists 90 mi apart ride toward each other at the same time; one rides twice as fast as the other. If they meet after 2 h, find each speed.
71.Distance, speed, time: A pilot flies 2500 mi from Montreal to Los Angeles, returning 20% faster. The round trip took 9 h 10 min. What was the outbound speed?
72.Distance, speed, time: A 14-ft car passes a 30-ft truck going 50 mi/h. How fast must the car go to completely pass the truck in 6 s? (Hint: work in feet and seconds. Figures accompany this problem.)
73.Distance, speed, time: A salesman drives 120 mi from Ajax to Barrington at a steady speed, then 150 mi from Barrington to Collins at 10 mi/h faster. The second leg took 6 min more. How fast was the first leg?
74.Distance, speed, time: Kiran drives 250 mi from Tortula to Cactus, then 360 mi from Cactus to Dry Junction at 10 mi/h faster. The total trip took 11 h. What was her Tortula-to-Cactus speed?
75.Distance, speed, time: A crew rows 6 km upstream and back in 2 h 40 min. If the current flows at 3 km/h, what is the crew's rowing speed in still water?
76.Boat speed: Two boats leave a harbor together, one heading east and one south; the eastbound boat is 3 mi/h faster. After 2 h they are 30 mi apart. Find the southbound boat's speed. figure — add your own
77.Law of the lever: A balanced lever satisfies $w_1 x_1 = w_2 x_2$ (weight times distance from the fulcrum, equal on both sides). A 100-lb boy sits 8 ft from the fulcrum of a seesaw; where should his 125-lb mother sit to balance it? figure — add your own
78.Law of the lever: A 30-ft plank rests on a rooftop edge with 5 ft projecting over the edge. A 240-lb worker sits on the far (on-roof) end. What is the largest weight that can hang from the projecting end while staying balanced? (Use $w_1 x_1 = w_2 x_2$. A figure accompanies this problem.)
79.Box dimensions: A plywood box has volume 180 ft$^3$; its length is 9 ft more than its height and its width is 4 ft less than its height. Find the dimensions. figure — add your own
80.Sphere radius: Three solid gold spheres of radii 2 mm, 3 mm, and 4 mm are melted into one sphere. What is the new radius?
81.Box dimensions: An open box with a square base is made from a square cardboard sheet by cutting 4-in squares from each corner and folding up the sides. The box must hold 100 in$^3$. What size cardboard is needed? figure — add your own
82.Can dimensions: A cylindrical can has volume $40\pi$ cm$^3$ and height 10 cm. What is its diameter? figure — add your own
83.Tank radius: A spherical tank holds 750 gallons. Using 1 gal $\approx$ 0.1337 ft$^3$, find its radius (to the nearest hundredth of a foot).
84.Lot dimensions: A right-triangular city lot has a hypotenuse 7 ft longer than one leg, and a perimeter of 392 ft. Find the length of each side.
85.Construction cost: Foxton lies 10 mi north of an east–west abandoned road through Grimley; the road point nearest Foxton is 40 mi from Grimley. Restoring old road costs $100,000/mi; new road costs $200,000/mi. How much abandoned road should be used to spend exactly $6.8 million, and would a direct new road cost less? figure — add your own
86.Distance, speed, time: A boardwalk runs parallel to and 210 ft inland from a straight shore. A man on the boardwalk is 750 ft across the sand from his umbrella at the shore. He walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before cutting across the sand to reach the umbrella in exactly 4 min 45 s? figure — add your own
87.Grain volume: Grain falls into a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it holds 1000 ft$^3$? figure — add your own
88.Monitors: Two monitors have the same screen height. One screen is 7 in wider than tall; the other is 1.8 times as wide as tall. The wider screen's diagonal is 3 in more than the smaller's. Find the screens' height (to the nearest 0.1 in). figure — add your own
89.Structure dimensions: A corn bin is a wire-mesh cylinder topped by a conical roof whose height is one-third the total structure height. If the total volume is $1400\pi$ ft$^3$ and the radius is 10 ft, find the height. figure — add your own
90.Comparing areas: A 360-in wire is cut into two pieces; one becomes a square, the other a circle. If both figures have equal area, find the two piece lengths (to the nearest tenth of an inch). figure — add your own
91.Ancient Chinese problem (from the Nine Chapters on the Mathematical Art, c. 250 B.C.): A 10-ft bamboo stem is broken so its tip touches the ground 3 ft from the base. What is the height of the break? (Hint: use the Pythagorean Theorem. A figure accompanies this problem.)

Discuss · Discover · Prove · Write

92.Write: Read the biographical notes on Pythagoras, Euclid, and Archimedes. Choose one, research him further, and write a short essay covering both his life and the mathematics he is known for.
93.Write: Invent a real-world story matching each equation, stating what the variables mean: (a) $C = 20{,}000 + 4.50x$; (b) $A = w(w + 10)$; (c) $C = 10.50x + 11.75y$.
94.Discuss (a Babylonian quadratic, c. 2000 B.C.): 'I have a reed of unknown length. I broke off one cubit, and it fit 60 times along my field's length. I restored the broken cubit, and it fit 30 times along the width. The field's area is 375 square ninda.' Find the reed's original length. (Use 1 ninda = 12 cubits.)

1.8 Inequalities 131 problems

Concepts

1.Fill each blank with the correct inequality sign: (a) if $x < 5$, then $x - 3 \;\square\; 2$; (b) if $x \le 5$, then $3x \;\square\; 15$; (c) if $x \ge 2$, then $-3x \;\square\; -6$; (d) if $x < -2$, then $-x \;\square\; 2$.
2.To solve $\dfrac{x + 1}{x - 2} \le 0$, first note that ______ and ______ are the zeros of the numerator and denominator; these split the line into three intervals. Complete a sign table for $x + 1$, $x - 2$, and the quotient, decide whether any endpoints fail, and state the solution.
3.(a) The solution of $|x| \le 3$ is the interval ______. (b) The solution of $|x| \ge 3$ is a union of two intervals ______ ∪ ______.
4.(a) The set of all points whose distance from 0 is less than 3 is described by $|x| \;\underline{\;\;}$. (b) The set whose distance from 0 is greater than 3 is described by $|x| \;\underline{\;\;}$.
5.Yes or No (give an example if No): (a) If $x(x + 1) > 0$, must $x$ be positive? (b) If $x(x + 1) > 5$, must $x > 5$?
6.State a logical first step for solving: (a) $3x \le 7$; (b) $5x - 2 \ge 1$; (c) $|3x + 2| \le 8$.

Skills

7.With $S = \{-5, -1, 0, \tfrac{2}{3}, \tfrac{5}{6}, 1, \sqrt{5}, 3, 5\}$, which elements satisfy $-2 + 3x \ge \tfrac{1}{3}$?
8.With $S = \{-5, -1, 0, \tfrac{2}{3}, \tfrac{5}{6}, 1, \sqrt{5}, 3, 5\}$, which elements satisfy $1 - 2x \ge 5x$?
9.With $S$ as given, which elements satisfy $1 < 2x - 4 \le 7$?
10.With $S$ as given, which elements satisfy $-2 \le 3 - x < 2$?
11.With $S$ as given, which elements satisfy $\dfrac{1}{x} \le \dfrac{1}{2}$?
12.With $S$ as given, which elements satisfy $x^{2} + 2 < 4$?
13.Solve the linear inequality; give the solution in interval notation and graph it: $2x \le 7$.
14.Solve and graph: $-4x \ge 10$.
15.Solve and graph: $2x - 5 > 3$.
16.Solve and graph: $3x + 11 < 5$.
17.Solve and graph: $7 - x \ge 5$.
18.Solve and graph: $5 - 3x \le -16$.
19.Solve and graph: $2x + 1 < 0$.
20.Solve and graph: $0 < 5 - 2x$.
21.Solve and graph: $4x - 7 < 8 + 9x$.
22.Solve and graph: $5 - 3x \ge 8x - 7$.
23.Solve and graph: $\tfrac{1}{2}x - \tfrac{2}{3} > 2$.
24.Solve and graph: $\tfrac{2}{5}x + 1 < \tfrac{1}{5} - 2x$.
25.Solve and graph: $\tfrac{1}{3}x + 2 < \tfrac{1}{6}x - 1$.
26.Solve and graph: $\tfrac{2}{3} - \tfrac{1}{2}x \ge \tfrac{1}{6} + x$.
27.Solve and graph: $4 - 3x \le -(1 + 8x)$.
28.Solve and graph: $2(7x - 3) \le 12x + 16$.
29.Solve and graph: $2 \le x + 5 < 4$.
30.Solve and graph: $5 \le 3x - 4 \le 14$.
31.Solve and graph: $-1 < 2x - 5 < 7$.
32.Solve and graph: $1 < 3x + 4 \le 16$.
33.Solve and graph: $-2 < 8 - 2x \le -1$.
34.Solve and graph: $-3 \le 3x + 7 \le \tfrac{1}{2}$.
35.Solve and graph: $\tfrac{1}{6} < \dfrac{2x - 13}{12} \le \tfrac{2}{3}$.
36.Solve and graph: $-\tfrac{1}{2} \le \dfrac{4 - 3x}{5} \le \tfrac{1}{4}$.
37.Solve the nonlinear inequality; give interval notation and graph: $(x + 2)(x - 3) < 0$.
38.Solve and graph: $(x - 5)(x + 4) \ge 0$.
39.Solve and graph: $x(2x + 7) \ge 0$.
40.Solve and graph: $x(2 - 3x) \le 0$.
41.Solve and graph: $x^{2} - 3x - 18 \le 0$.
42.Solve and graph: $x^{2} + 5x + 6 > 0$.
43.Solve and graph: $2x^{2} + x \ge 1$.
44.Solve and graph: $x^{2} < x + 2$.
45.Solve and graph: $3x^{2} - 3x < 2x^{2} + 4$.
46.Solve and graph: $5x^{2} + 3x \ge 3x^{2} + 2$.
47.Solve and graph: $x^{2} > 3(x + 6)$.
48.Solve and graph: $x^{2} + 2x > 3$.
49.Solve and graph: $x^{2} < 4$.
50.Solve and graph: $x^{2} \ge 9$.
51.Solve and graph: $(x + 2)(x - 1)(x - 3) \le 0$.
52.Solve and graph: $(x - 5)(x - 2)(x + 1) > 0$.
53.Solve and graph: $(x - 4)(x + 2)^{2} < 0$.
54.Solve and graph: $(x + 3)^{2}(x + 1) > 0$.
55.Solve and graph: $(x + 3)^{2}(x - 2)(x + 5) \ge 0$.
56.Solve and graph: $4x^{2}(x^{2} - 9) \le 0$.
57.Solve and graph: $x^{3} - 4x > 0$.
58.Solve and graph: $16x \le x^{3}$.
59.Solve the quotient inequality; give interval notation and graph: $\dfrac{x - 3}{x + 1} \ge 0$.
60.Solve and graph: $\dfrac{2x + 6}{x - 2} < 0$.
61.Solve and graph: $\dfrac{x}{x + 1} > 3$.
62.Solve and graph: $\dfrac{x - 4}{2x + 1} < 5$.
63.Solve and graph: $\dfrac{2x + 1}{x - 5} \le 3$.
64.Solve and graph: $\dfrac{3 + x}{3 - x} \ge 1$.
65.Solve and graph: $\dfrac{4}{x} < x$.
66.Solve and graph: $\dfrac{x}{x + 1} > 3x$.
67.Solve and graph: $1 + \dfrac{2}{x + 1} \le \dfrac{2}{x}$.
68.Solve and graph: $\dfrac{3}{x - 1} - \dfrac{4}{x} \ge 1$.
69.Solve and graph: $\dfrac{6}{x - 1} - \dfrac{6}{x} \ge 1$.
70.Solve and graph: $\dfrac{x}{2} \ge \dfrac{5}{x + 1} + 4$.
71.Solve and graph: $\dfrac{x + 2}{x + 3} < \dfrac{x - 1}{x - 2}$.
72.Solve and graph: $\dfrac{1}{x + 1} + \dfrac{1}{x + 2} \le 0$.
73.Solve and graph: $x^{4} > x^{2}$.
74.Solve and graph: $x^{5} > x^{2}$.
75.Solve the absolute-value inequality; give interval notation and graph: $|5x| < 20$.
76.Solve and graph: $|16x| \le 8$.
77.Solve and graph: $|2x| > 7$.
78.Solve and graph: $\tfrac{1}{2}|x| \ge 1$.
79.Solve and graph: $|x - 5| \le 3$.
80.Solve and graph: $|x + 1| \ge 1$.
81.Solve and graph: $|3x + 2| < 4$.
82.Solve and graph: $|5x - 2| < 8$.
83.Solve and graph: $|3x - 2| \ge 5$.
84.Solve and graph: $|8x + 3| > 12$.
85.Solve and graph: $\left|\dfrac{x - 2}{3}\right| < 2$.
86.Solve and graph: $\left|\dfrac{x + 1}{2}\right| \ge 4$.
87.Solve and graph: $|x + 6| < 0.001$.
88.Solve and graph: $3 - |2x + 4| \le 1$.
89.Solve and graph: $8 - |2x - 1| \ge 6$.
90.Solve and graph: $7|x + 2| + 5 > 4$.
91.Express as an absolute-value inequality: all real $x$ less than 3 units from 0.
92.Express as an absolute-value inequality: all real $x$ more than 2 units from 0.
93.Express as an absolute-value inequality: all real $x$ at least 5 units from 7.
94.Express as an absolute-value inequality: all real $x$ at most 4 units from 2.
95.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
96.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
97.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
98.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
99.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
100.Write an absolute-value inequality describing the graphed set on the number line. figure — add your own
101.Find the values of the variable for which the expression is a real number: $\sqrt{x^{2} - 9}$.
102.Find the values for which the expression is real: $\sqrt{x^{2} - 5x - 50}$.
103.Find the values for which the expression is real: $\left(\dfrac{1}{x^{2} - 3x - 10}\right)^{1/2}$.
104.Find the values for which the expression is real: $\sqrt[4]{\dfrac{1 - x}{2 + x}}$.

Skills Plus

105.Solve for $x$ (constants $a, b, c$ positive): $a(bx - c) \ge bc$.
106.Solve for $x$: $a \le bx + c < 2a$.
107.Solve for $x$: $a|bx - c| + d \ge 4a$.
108.Solve for $x$: $\left|\dfrac{bx + c}{a}\right| > 5a$.

Applications

109.Temperature scales: Using $C = \tfrac{5}{9}(F - 32)$, find the Fahrenheit interval corresponding to $20 \le C \le 30$.
110.Temperature scales: What Celsius interval corresponds to $50 \le F \le 95$?
111.Car rental: Plan A is $30/day plus $0.10/mile; Plan B is $50/day with unlimited mileage. For what range of daily miles does Plan B save money?
112.Phone plans: Plan A is $25/month plus $0.05/min; Plan B is $5/month plus $0.12/min. For what range of international minutes is Plan B cheaper?
113.Driving cost: Annual driving cost is $C = 0.35m + 2200$ dollars for $m$ miles. Jane budgets between $6400 and $7100. What mileage range fits her budget?
114.Air temperature: Rising dry air cools about 1°C per 100-m gain (up to ~12 km). (a) If the ground temperature is 20°C, write a formula for the temperature at height $h$. (b) What temperature range occurs for a plane reaching a maximum height of 5 km?
115.Ticket price: A charter flight's 120 seats all sell at a $200 ticket price; each $3 increase loses one sold seat. (a) Write a formula for seats sold at price $P$. (b) If between 90 and 115 seats sold, what was the price range?
116.Scale accuracy: A merchant sells 3 lb of Kona coffee at $6.50/lb; the scale is accurate to within ±0.03 lb. By how much could the customer be over- or under-charged?
117.Gravity: The Earth's gravitational force on a 100-kg object is $F = \dfrac{4{,}000{,}000}{d^{2}}$ newtons, with $d$ the distance (km) from Earth's center. For what distances is $F$ between 0.0004 N and 0.01 N?
118.Bonfire temperature: Near a fire, the temperature at distance $x$ meters is $T = \dfrac{600{,}000}{x^{2} + 300}$ °C. At what range of distances is the temperature below 500°C? figure — add your own
119.Falling ball: A ball thrown up at 16 ft/s from a 128-ft building has height $h = 128 + 16t - 16t^{2}$ after $t$ seconds. During what time interval is it at least 32 ft above ground? figure — add your own
120.Gas mileage: A vehicle's mileage at speed $v$ (mi/h) is $g = 10 + 0.9v - 0.01v^{2}$, valid for $10 \le v \le 75$. For what speeds is the mileage 30 mi/gal or better?
121.Stopping distance: A car traveling $v$ mi/h needs stopping distance $d = v + \dfrac{v^{2}}{20}$ feet. Kerry wants $d$ no more than 240 ft. What speed range is allowed? figure — add your own
122.Profit: For $x$ units, revenue is $R = 20x$ and cost is $C = 2000 + 8x + 0.0025x^{2}$. Using profit = revenue − cost, how many units must be sold for a profit of at least $2400?
123.Fencing a garden: A gardener has 120 ft of fence for a rectangular plot and wants enclosed area of at least 800 ft$^2$. What range of lengths is possible?
124.Laminate thickness: A laminate is made 0.020 in thick with a tolerance of 0.003 in. (a) Write an absolute-value inequality for the allowed thickness. (b) Solve it. figure — add your own
125.Height range: Adult-male heights average 68.2 in, and 95% satisfy $\left|\dfrac{h - 68.2}{2.9}\right| \le 2$. Solve for the height range.

Discuss · Discover · Prove · Write

126.Discuss / Discover: If $a < b$, is $a^{2} < b^{2}$? (Test positive and negative values.) If $a < b$, is $a^{3} < b^{3}$? Based on your tests, state a general rule for when $a^{n} < b^{n}$ given $a < b$ and positive integer $n$.
127.Discuss / Discover: Solving $1 < 3/x$ by multiplying both sides by $x$ gives $x < 3$, suggesting $(-\infty, 3)$ — but $x = -1$ fails the original. Explain why this method breaks down (consider the sign of $x$), then solve the inequality correctly.
128.Discuss / Discover: Since $|a - b|$ is the distance between $a$ and $b$, interpret $|x - 1|$ and $|x - 3|$, and use this to solve $|x - 1| < |x - 3|$ geometrically. In general, for $a < b$, what is the solution of $|x - a| < |x - b|$?
129.Prove (Rule 6 for inequalities): If $a < b$ and $c < d$, then $a + c < b + d$. (Hint: use Rule 1 to get $a + c < b + c$ and $b + c < b + d$, then Rule 7.)
130.Prove: If $a, b, c, d$ are positive with $\dfrac{a}{b} < \dfrac{c}{d}$, then $\dfrac{a}{b} < \dfrac{a + c}{b + d} < \dfrac{c}{d}$. (Hint: show $\dfrac{ad}{b} + a < c + a$ and $a + c < \dfrac{cb}{d} + c$.)
131.Prove (Arithmetic–Geometric Mean Inequality, two-number case): For nonnegative $x$ and $y$: (a) show that if $x \le y$ then $x^{2} \le y^{2}$ (Hint: use Rule 3 to get $x^{2} \le xy$ and $xy \le y^{2}$); (b) prove that $\sqrt{xy} \le \dfrac{x + y}{2}$.

1.9 The Coordinate Plane; Graphs of Equations; Circles 127 problems

Concepts

1.The point 3 units right of the $y$-axis and 5 units below the $x$-axis has coordinates ( ____ , ____ ).
2.The distance between $(a, b)$ and $(c, d)$ is ______. So the distance between $(1, 2)$ and $(7, 10)$ is ______.
3.The midpoint of $(a, b)$ and $(c, d)$ is ______. So the midpoint of $(1, 2)$ and $(7, 10)$ is ______.
4.If $(2, 3)$ lies on the graph of an equation in $x$ and $y$, then the equation holds when $x$ is replaced by ______ and $y$ by ______. Is $(2, 3)$ on the graph of $2y = x + 1$? Complete a table and sketch the graph.
5.(a) To find $x$-intercepts, set ______ equal to 0 and solve for ______; the $x$-intercept of $2y = x + 1$ is ______. (b) To find $y$-intercepts, set ______ equal to 0 and solve for ______; the $y$-intercept of $2y = x + 1$ is ______.
6.The graph of $(x - 1)^{2} + (y - 2)^{2} = 9$ is a circle with center ( ____ , ____ ) and radius ______.
7.(a) If a graph is symmetric about the $x$-axis and $(a, b)$ is on it, then ( ____ , ____ ) is also on it. (b) If symmetric about the $y$-axis and $(a, b)$ is on it, then ( ____ , ____ ) is also on it. (c) If symmetric about the origin and $(a, b)$ is on it, then ( ____ , ____ ) is also on it.
8.For the displayed graph: (a) the $x$-intercept(s) are ______ and the $y$-intercept(s) are ______; (b) the graph is symmetric about the ______ ($x$-axis / $y$-axis / origin). figure — add your own
9.Yes or No (give a reason if No): If a graph is symmetric about both the $x$- and $y$-axes, must it be symmetric about the origin?
10.Yes or No (give a reason if No): If a graph is symmetric about the origin, must it be symmetric about the $x$- or $y$-axis?

Skills

11.Find the coordinates of the labeled points shown in the figure. figure — add your own
12.List which labeled points lie in Quadrants I and III. figure — add your own
13.Plot the points: $(0, 5)$, $(-1, 0)$, $(-1, -2)$, $\left(\tfrac{1}{2}, \tfrac{2}{3}\right)$.
14.Plot the points: $(-5, 0)$, $(2, 0)$, $(2.6, -1.3)$, $(-2.5, 3.5)$.
15.Sketch the region: (a) $\{(x, y) \mid x \ge 2\}$; (b) $\{(x, y) \mid y = 2\}$.
16.Sketch the region: (a) $\{(x, y) \mid y < 3\}$; (b) $\{(x, y) \mid x = -4\}$.
17.Sketch the region: (a) $\{(x, y) \mid -3 < x < 3\}$; (b) $\{(x, y) \mid |x| \le 2\}$.
18.Sketch the region: (a) $\{(x, y) \mid 0 \le y \le 2\}$; (b) $\{(x, y) \mid |y| > 2\}$.
19.Sketch the region: (a) $\{(x, y) \mid -2 < x < 2 \text{ and } y \ge 1\}$; (b) $\{(x, y) \mid xy < 0\}$.
20.Sketch the region: (a) $\{(x, y) \mid |x| \le 1 \text{ and } |y| \le 3\}$; (b) $\{(x, y) \mid xy > 0\}$.
21.For the graphed pair of points: (a) find the distance between them; (b) find the midpoint of the joining segment. figure — add your own
22.For the graphed pair of points: (a) find the distance; (b) find the midpoint. figure — add your own
23.For the graphed pair of points: (a) find the distance; (b) find the midpoint. figure — add your own
24.For the graphed pair of points: (a) find the distance; (b) find the midpoint. figure — add your own
25.For $(0, 8)$ and $(6, 16)$: (a) plot the points; (b) find the distance between them; (c) find the midpoint of the joining segment.
26.For $(-2, 5)$ and $(10, 0)$: (a) plot; (b) find the distance; (c) find the midpoint.
27.For $(3, -2)$ and $(-4, 5)$: (a) plot; (b) find the distance; (c) find the midpoint.
28.For $(-1, 1)$ and $(-6, -3)$: (a) plot; (b) find the distance; (c) find the midpoint.
29.For $(6, -2)$ and $(-6, 2)$: (a) plot; (b) find the distance; (c) find the midpoint.
30.For $(0, -6)$ and $(5, 0)$: (a) plot; (b) find the distance; (c) find the midpoint.
31.Draw the rectangle with vertices $A(1, 3)$, $B(5, 3)$, $C(1, -3)$, $D(5, -3)$ and find its area.
32.Draw the parallelogram with vertices $A(1, 2)$, $B(5, 2)$, $C(3, 6)$, $D(7, 6)$ and find its area.
33.Plot $A(1, 0)$, $B(5, 0)$, $C(4, 3)$, $D(2, 3)$; draw segments $AB$, $BC$, $CD$, $DA$. What kind of quadrilateral is $ABCD$, and what is its area?
34.Plot $P(5, 1)$, $Q(0, 6)$, $R(-5, 1)$; locate $S$ so that $PQRS$ is a square, and find its area.
35.Which point is closer to the origin: $A(6, 7)$ or $B(-5, 8)$?
36.Which point is closer to $E(-2, 1)$: $C(-6, 3)$ or $D(3, 0)$?
37.Which point is closer to $R(-1, -1)$: $P(3, 1)$ or $Q(-1, 3)$?
38.(a) Show that $(7, 3)$ and $(3, 7)$ are equidistant from the origin. (b) Show that $(a, b)$ and $(b, a)$ are equidistant from the origin.
39.Show that the triangle with vertices $A(0, 2)$, $B(-3, -1)$, $C(-4, 3)$ is isosceles.
40.Find the area of the triangle shown in the figure. figure — add your own
41.For triangle $ABC$ in the figure: (a) use the converse of the Pythagorean Theorem to show it is a right triangle; (b) find its area. figure — add your own
42.Show that the triangle with vertices $A(6, -7)$, $B(11, -3)$, $C(2, -2)$ is a right triangle (converse of the Pythagorean Theorem) and find its area.
43.Show that $A(-2, 9)$, $B(4, 6)$, $C(1, 0)$, $D(-5, 3)$ are the vertices of a square.
44.Show that $A(-1, 3)$, $B(3, 11)$, $C(5, 15)$ are collinear by checking that $d(A, B) + d(B, C) = d(A, C)$.
45.Find a point on the $y$-axis equidistant from $(5, -5)$ and $(1, 1)$.
46.Find the lengths of the medians of the triangle with vertices $A(1, 0)$, $B(3, 6)$, $C(8, 2)$. (A median joins a vertex to the midpoint of the opposite side.)
47.Plot $P(-1, -4)$, $Q(1, 1)$, $R(4, 2)$; locate $S$ so that $PQRS$ is a parallelogram.
48.If $M(6, 8)$ is the midpoint of segment $AB$ and $A = (2, 3)$, find the coordinates of $B$.
49.(a) Sketch the parallelogram with vertices $A(-2, -1)$, $B(4, 2)$, $C(7, 7)$, $D(1, 4)$. (b) Find the midpoints of its diagonals. (c) Conclude from (b) that the diagonals bisect each other.
50.Point $M$ in the figure is the midpoint of segment $AB$. Show that $M$ is equidistant from the vertices of triangle $ABC$. figure — add your own
51.Determine which of $(0, 0)$, $(1, 0)$, $(-1, -1)$ lie on the graph of $x - 2y - 1 = 0$.
52.Determine which of $(1, 1)$, $\left(1, \tfrac{1}{2}\right)$, $\left(-1, \tfrac{1}{2}\right)$ lie on the graph of $y(x^{2} + 1) = 1$.
53.Determine which of $(0, -2)$, $(1, -2)$, $(2, -2)$ lie on the graph of $x^{2} + xy + y^{2} = 4$.
54.Determine which of $(0, 1)$, $\left(\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}}\right)$, $\left(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right)$ lie on the graph of $x^{2} + y^{2} = 1$.
55.Make a table of values and sketch the graph of $4x + 5y = 40$.
56.Make a table of values and sketch: $3x - 5y = 30$.
57.Make a table of values and sketch: $y = x^{2} + 4$.
58.Make a table of values and sketch: $y = 3 - x^{2}$.
59.Make a table of values and sketch: $y = |x| - 1$.
60.Make a table of values and sketch: $y = |x + 1|$.
61.Use a graphing calculator to graph $y = 0.01x^{3} - x^{2} + 5$ in the viewing rectangle $[-100, 150]$ by $[-2000, 2000]$.
62.Graph $y = \sqrt{12x - 17}$ in $[0, 10]$ by $[0, 20]$.
63.Graph $y = \dfrac{x}{x^{2} + 25}$ in $[-50, 50]$ by $[-0.2, 0.2]$.
64.Graph $y = x^{4} - 4x^{3}$ in $[-4, 6]$ by $[-50, 100]$.
65.Make a table, sketch, find the intercepts, and test for symmetry: (a) $2x - y = 6$; (b) $y = -(x + 1)^{2}$.
66.Make a table, sketch, find intercepts, test symmetry: (a) $x - 4y = 8$; (b) $y = -x^{2} + 3$.
67.Make a table, sketch, find intercepts, test symmetry: (a) $y = \sqrt{x + 1}$; (b) $y = -|x|$.
68.Make a table, sketch, find intercepts, test symmetry: (a) $y = 3 - \sqrt{x}$; (b) $x = |y|$.
69.Make a table, sketch, find intercepts, test symmetry: (a) $y = \sqrt{4 - x^{2}}$; (b) $x = y^{3} + 2y$.
70.Make a table, sketch, find intercepts, test symmetry: (a) $y = -\sqrt{4 - x^{2}}$; (b) $x = y^{3}$.
71.Find the $x$- and $y$-intercepts of: (a) $y = x + 6$; (b) $y = x^{2} - 5$.
72.Find the $x$- and $y$-intercepts of: (a) $4x^{2} + 25y^{2} = 100$; (b) $x^{2} - xy + 3y = 1$.
73.Find the $x$- and $y$-intercepts of: (a) $9x^{2} - 4y^{2} = 36$; (b) $y - 2xy + 4x = 1$.
74.Find the $x$- and $y$-intercepts of: (a) $y = \sqrt{x^{2} - 16}$; (b) $y = \sqrt{64 - x^{3}}$.
75.An equation and its graph are given; find the $x$- and $y$-intercepts: $y = 4x - x^{2}$. figure — add your own
76.Find the intercepts from the given equation and graph: $\dfrac{x^{2}}{9} + \dfrac{y^{2}}{4} = 1$. figure — add your own
77.Find the intercepts from the given equation and graph: $x^{4} + y^{2} - xy = 16$. figure — add your own
78.Find the intercepts from the given equation and graph: $x^{2} + y^{3} - x^{2}y^{2} = 64$. figure — add your own
79.For $y = x^{3} - x^{2}$ in $[-2, 2]$ by $[-1, 1]$: (a) graph it; (b) read the intercepts from the graph; (c) verify them algebraically.
80.For $y = x^{4} - 2x^{3}$ in $[-2, 3]$ by $[-3, 3]$: (a) graph; (b) read intercepts; (c) verify algebraically.
81.For $y = -\dfrac{2}{x^{2} + 1}$ in $[-5, 5]$ by $[-3, 1]$: (a) graph; (b) read intercepts; (c) verify algebraically.
82.For $y = \sqrt[3]{1 - x^{2}}$ in $[-5, 5]$ by $[-5, 3]$: (a) graph; (b) read intercepts; (c) verify algebraically.
83.Find the center and radius of the circle and sketch it: $x^{2} + y^{2} = 9$.
84.Find center and radius and sketch: $x^{2} + y^{2} = 5$.
85.Find center and radius and sketch: $x^{2} + (y - 4)^{2} = 1$.
86.Find center and radius and sketch: $(x + 1)^{2} + y^{2} = 9$.
87.Find center and radius and sketch: $(x + 3)^{2} + (y - 4)^{2} = 25$.
88.Find center and radius and sketch: $(x + 1)^{2} + (y + 2)^{2} = 36$.
89.Find an equation of the circle with center $(2, -1)$ and radius 3.
90.Find an equation of the circle with center $(-1, -4)$ and radius 8.
91.Find an equation of the circle centered at the origin passing through $(4, 7)$.
92.Find an equation of the circle with center $(-1, 5)$ passing through $(-4, -6)$.
93.Find an equation of the circle whose diameter has endpoints $P(-1, 1)$ and $Q(5, 9)$.
94.Find an equation of the circle whose diameter has endpoints $P(-1, 3)$ and $Q(7, -5)$.
95.Find an equation of the circle with center $(7, -3)$ tangent to the $x$-axis.
96.Find an equation of the circle of radius 5 lying in the first quadrant and tangent to both the $x$- and $y$-axes.
97.Find the equation of the circle shown in the figure. figure — add your own
98.Find the equation of the circle shown in the figure. figure — add your own
99.Show that the equation represents a circle, and find its center and radius: $x^{2} + y^{2} + 4x - 6y + 12 = 0$.
100.Show it is a circle and find center and radius: $x^{2} + y^{2} + 6y + 2 = 0$.
101.Show it is a circle and find center and radius: $x^{2} + y^{2} - \tfrac{1}{2}x + \tfrac{1}{2}y = \tfrac{1}{8}$.
102.Show it is a circle and find center and radius: $x^{2} + y^{2} + \tfrac{1}{2}x + 2y + \tfrac{1}{16} = 0$.
103.Show it is a circle and find center and radius: $2x^{2} + 2y^{2} - 3x = 0$.
104.Show it is a circle and find center and radius: $3x^{2} + 3y^{2} + 6x - y = 0$.
105.Test the equation for symmetry: $y = x^{4} + x^{2}$.
106.Test the equation for symmetry: $x = y^{4} - y^{2}$.
107.Test the equation for symmetry: $x^{2}y^{2} + xy = 1$.
108.Test the equation for symmetry: $x^{4}y^{4} + x^{2}y^{2} = 1$.
109.Test the equation for symmetry: $y = x^{3} + 10x$.
110.Test the equation for symmetry: $y = x^{2} + |x|$.
111.Complete the partial graph of $y = \dfrac{1}{1 + x^{2}}$ using symmetry about the $y$-axis. figure — add your own
112.Complete the partial graph of $y^{2} - x^{2} = 1$ using symmetry about the $x$-axis. figure — add your own
113.Complete the partial graph of $y = \dfrac{x}{1 + x^{2}}$ using symmetry about the origin. figure — add your own
114.Complete the partial graph of $y = \dfrac{1}{x^{3}}$ using symmetry about the origin. figure — add your own

Skills Plus

115.Sketch the region: $\{(x, y) \mid x^{2} + y^{2} \le 1\}$.
116.Sketch the region: $\{(x, y) \mid x^{2} + y^{2} > 4\}$.
117.Find the area of the region outside the circle $x^{2} + y^{2} = 4$ but inside the circle $x^{2} + y^{2} - 4y - 12 = 0$.
118.Sketch the region satisfying both $x^{2} + y^{2} \le 9$ and $y \ge |x|$. What is its area?
119.Shifting the plane: Each point is shifted 3 units right and 2 units up. (a) Where does $(5, 3)$ go? (b) Where does $(a, b)$ go? (c) Which point maps to $(3, 4)$? (d) Triangle $ABC$ shifts to $A'B'C'$; find $A'$, $B'$, $C'$. figure — add your own
120.Reflecting in the plane: The $y$-axis reflects each right-side point to the left. (a) Where does $(3, 7)$ reflect? (b) Where does $(a, b)$ reflect? (c) Which point reflects to $(-4, -1)$? (d) Triangle $ABC$ reflects to $A'B'C'$; find $A'$, $B'$, $C'$. figure — add your own
121.Making a graph symmetric: The graph shown is not symmetric about any axis or the origin. Add as few segments as possible to make it symmetric about: (a) the $x$-axis; (b) the $y$-axis; (c) the origin. figure — add your own

Applications

122.City distances: Streets run N–S and avenues E–W, evenly spaced. The walking distance from $A$ to $B$ is 7 blocks (3 east, 4 north); straight-line distance uses the Distance Formula. (a) Find the straight-line distance (in blocks) between $A$ and $B$. (b) Find walking and straight-line distances between 4th St. & 2nd Ave. and 11th St. & 26th Ave. (c) When does walking distance equal straight-line distance? figure — add your own
123.Halfway point (same city grid): One friend lives at 3rd St. & 7th Ave., the other at 27th St. & 17th Ave. They meet at a coffee shop halfway between. (a) At which intersection is the shop? (b) How far must each walk to reach it?
124.Satellite orbit: A satellite orbits the moon (center at origin, distances in megameters) along $\dfrac{(x - 3)^{2}}{25} + \dfrac{y^{2}}{16} = 1$. (a) From the graph, find the closest and farthest distances to the moon's center. (b) Two orbit points have $y$-coordinate 2; find their $x$-coordinates and distances to the center. figure — add your own

Discuss · Discover · Prove · Write

125.Write: Plot $M(6, 8)$ and $A(2, 3)$. If $M$ is the midpoint of segment $AB$, find $B$, and write a short description of the steps and reasoning you used.
126.Write: Plot $P(0, 3)$, $Q(2, 2)$, $R(5, 3)$. Where should $S$ go so that $PQRS$ is a parallelogram? Describe the steps and reasoning.
127.Discover: Complete the square in the general equation $x^{2} + ax + y^{2} + by + c = 0$ and simplify. Under what conditions on $a$, $b$, $c$ does it represent a circle? A single point? The empty set? When it is a circle, find its center and radius.

1.10 Lines 96 problems

Concepts

1.The slope of a line through two points is the difference in their ______-coordinates divided by the difference in their ______-coordinates. So the line through $(0, 1)$ and $(2, 5)$ has slope ______.
2.For the line $y = 3x + 2$: (a) its slope is ______; (b) any parallel line has slope ______; (c) any perpendicular line has slope ______.
3.The point-slope equation of the line with slope 3 through $(1, 2)$ is ______.
4.For $2x + 3y - 12 = 0$: the $x$-intercept is ______ and the $y$-intercept is ______; in slope-intercept form $y = \underline{\;\;}$; the slope is ______.
5.The slope of a horizontal line is ______; the equation of the horizontal line through $(2, 3)$ is ______.
6.The slope of a vertical line is ______; the equation of the vertical line through $(2, 3)$ is ______.
7.Yes or No (give a reason if No): (a) Is $y = -3$ a horizontal line? (b) Is $x = -3$ a vertical line? (c) Does a line perpendicular to a horizontal line have slope 0? (d) Does a line perpendicular to a vertical line have slope 0?
8.Sketch the graphs of $y = -3$ and $x = -3$. Are the lines perpendicular?

Skills

9.Find the slope of the line through $P$ and $Q$: $P(-1, 2)$, $Q(0, 0)$.
10.Find the slope through $P(0, 0)$, $Q(3, -1)$.
11.Find the slope through $P(2, -2)$, $Q(7, -1)$.
12.Find the slope through $P(-5, 1)$, $Q(3, -2)$.
13.Find the slope through $P(5, 4)$, $Q(0, 4)$.
14.Find the slope through $P(4, 3)$, $Q(1, -1)$.
15.Find the slope through $P(10, -2)$, $Q(6, -5)$.
16.Find the slope through $P(3, -2)$, $Q(6, -2)$.
17.Find the slopes of the lines $l_1, l_2, l_3, l_4$ shown in the figure. figure — add your own
18.Slope: (a) Sketch lines through $(0, 0)$ with slopes 1, $\tfrac{1}{2}$, 2, and $-1$. (b) Sketch lines through $(0, 0)$ with slopes $\tfrac{1}{3}$, $\tfrac{1}{2}$, $-\tfrac{1}{3}$, and 3.
19.Find an equation for the line whose graph is sketched. figure — add your own
20.Find an equation for the line whose graph is sketched. figure — add your own
21.Find an equation for the line whose graph is sketched. figure — add your own
22.Find an equation for the line whose graph is sketched. figure — add your own
23.Find an equation of the line with slope 3 and $y$-intercept $-2$.
24.Find an equation of the line with slope $\tfrac{2}{5}$ and $y$-intercept 4.
25.Find an equation of the line through $(2, 3)$ with slope 5.
26.Find an equation of the line through $(-2, 4)$ with slope $-1$.
27.Find an equation of the line through $(1, 7)$ with slope $\tfrac{2}{3}$.
28.Find an equation of the line through $(-3, -5)$ with slope $-\tfrac{7}{2}$.
29.Find an equation of the line through $(2, 1)$ and $(1, 6)$.
30.Find an equation of the line through $(-1, -2)$ and $(4, 3)$.
31.Find an equation of the line through $(-2, 5)$ and $(-1, -3)$.
32.Find an equation of the line through $(1, 7)$ and $(4, 7)$.
33.Find an equation of the line with $x$-intercept 1 and $y$-intercept $-3$.
34.Find an equation of the line with $x$-intercept $-8$ and $y$-intercept 6.
35.Find an equation of the line through $(1, 3)$ with slope 0.
36.Find an equation of the line through $(-1, 4)$ with undefined slope.
37.Find an equation of the line through $(2, -1)$ with undefined slope.
38.Find an equation of the line through $(5, 1)$ with slope 0.
39.Find an equation of the line through $(1, 2)$ parallel to $y = 3x - 5$.
40.Find an equation of the line through $(-3, 2)$ perpendicular to $y = -\tfrac{1}{2}x + 7$.
41.Find an equation of the line through $(4, 5)$ parallel to the $x$-axis.
42.Find an equation of the line through $(4, 5)$ parallel to the $y$-axis.
43.Find an equation of the line through $(1, -6)$ parallel to $x + 2y = 6$.
44.Find an equation of the line with $y$-intercept 6 parallel to $2x + 3y + 4 = 0$.
45.Find an equation of the line through $(-1, 2)$ parallel to $x = 5$.
46.Find an equation of the line through $(2, 6)$ perpendicular to $y = 1$.
47.Find an equation of the line through $(-1, -2)$ perpendicular to $2x + 5y + 8 = 0$.
48.Find an equation of the line through $\left(\tfrac{1}{2}, -\tfrac{2}{3}\right)$ perpendicular to $4x - 8y = 1$.
49.Find an equation of the line through $(1, 7)$ parallel to the line through $(2, 5)$ and $(-2, 1)$.
50.Find an equation of the line through $(-2, -11)$ perpendicular to the line through $(1, 1)$ and $(5, -1)$.
51.(a) Sketch the line with slope $\tfrac{3}{2}$ through $(-2, 1)$. (b) Find an equation for this line.
52.(a) Sketch the line with slope $-2$ through $(4, -1)$. (b) Find an equation for this line.
53.Use a graphing device to graph the family $y = -2x + b$ for $b = 0, \pm 1, \pm 3, \pm 6$ in one viewing rectangle. What do the lines share?
54.Graph the family $y = mx - 3$ for $m = 0, \pm 0.25, \pm 0.75, \pm 1.5$. What do the lines share?
55.Graph the family $y = m(x - 3)$ for $m = 0, \pm 0.25, \pm 0.75, \pm 1.5$. What do the lines share?
56.Graph the family $y = 2 + m(x + 3)$ for $m = 0, \pm 0.5, \pm 1, \pm 2, \pm 6$. What do the lines share?
57.Find the slope and $y$-intercept and draw the graph: $y = 3 - x$.
58.Find slope and $y$-intercept and graph: $y = \tfrac{2}{3}x - 2$.
59.Find slope and $y$-intercept and graph: $-2x + y = 7$.
60.Find slope and $y$-intercept and graph: $2x - 5y = 0$.
61.Find slope and $y$-intercept and graph: $4x + 5y = 10$.
62.Find slope and $y$-intercept and graph: $3x - 4y = 12$.
63.Find slope and $y$-intercept and graph: $y = 4$.
64.Find slope and $y$-intercept and graph: $x = -5$.
65.Find slope and $y$-intercept and graph: $x = 3$.
66.Find slope and $y$-intercept and graph: $y = -2$.
67.Find the $x$- and $y$-intercepts and draw the graph: $5x + 2y - 10 = 0$.
68.Find intercepts and graph: $6x - 7y - 42 = 0$.
69.Find intercepts and graph: $\tfrac{1}{2}x - \tfrac{1}{3}y + 1 = 0$.
70.Find intercepts and graph: $\tfrac{1}{3}x - \tfrac{1}{5}y - 2 = 0$.
71.Find intercepts and graph: $y = 6x + 4$.
72.Find intercepts and graph: $y = -4x - 10$.
73.Determine whether the lines are parallel, perpendicular, or neither: $y = 2x + 3$; $2y - 4x - 5 = 0$.
74.Parallel, perpendicular, or neither: $y = \tfrac{1}{2}x + 4$; $2x + 4y = 1$.
75.Parallel, perpendicular, or neither: $-3x + 4y = 4$; $4x + 3y = 5$.
76.Parallel, perpendicular, or neither: $2x - 3y = 10$; $3y - 2x - 7 = 0$.
77.Parallel, perpendicular, or neither: $7x - 3y = 2$; $9y + 21x = 1$.
78.Parallel, perpendicular, or neither: $6y - 2x = 5$; $2y + 6x = 1$.

Skills Plus

79.Use slopes to show that $A(1, 1)$, $B(7, 4)$, $C(5, 10)$, $D(-1, 7)$ are vertices of a parallelogram.
80.Use slopes to show that $A(-3, -1)$, $B(3, 3)$, $C(-9, 8)$ are vertices of a right triangle.
81.Use slopes to show that $A(1, 1)$, $B(11, 3)$, $C(10, 8)$, $D(0, 6)$ are vertices of a rectangle.
82.Use slopes to decide whether the points are collinear: (a) $(1, 1)$, $(3, 9)$, $(6, 21)$; (b) $(-1, 3)$, $(1, 7)$, $(4, 15)$.
83.Find an equation of the perpendicular bisector of the segment joining $A(1, 4)$ and $B(7, -2)$.
84.Find the area of the triangle formed by the coordinate axes and the line $2y + 3x - 6 = 0$.
85.Two-intercept form: (a) Show that if a line has nonzero $x$- and $y$-intercepts $a$ and $b$, its equation can be written $\dfrac{x}{a} + \dfrac{y}{b} = 1$. (b) Use this to find the equation of the line with $x$-intercept 6 and $y$-intercept $-8$.
86.Tangent to a circle: (a) Find an equation of the line tangent to $x^{2} + y^{2} = 25$ at $(3, -4)$. (b) At what other point on the circle is the tangent line parallel to this one? figure — add your own

Applications

87.Global warming: Average global surface temperature is modeled by $T = 0.02t + 15.0$, with $T$ in °C and $t$ years since 1950. (a) What do the slope and $T$-intercept represent? (b) Predict the average temperature in 2050.
88.Drug dosage: For an adult dose $D$ (mg), a child of age $a$ gets $c = 0.0417D(a + 1)$. For an adult dose of 200 mg: (a) find the slope and explain its meaning; (b) find the dose for a newborn.
89.Flea market: A manager knows that charging $x$ dollars per rental space rents $y = 200 - 4x$ spaces. (a) Sketch this line (rental charge and number of spaces must be nonnegative). (b) Interpret the slope, $y$-intercept, and $x$-intercept.
90.Production cost: A manufacturer's monthly cost for $x$ toaster ovens is $y = 6x + 3000$ dollars. (a) Sketch the line. (b) Interpret the slope and $y$-intercept.
91.Temperature scales: The scales relate by $F = \tfrac{9}{5}C + 32$. (a) Complete a table comparing the scales at the given values. (b) Find the temperature at which the two scales agree (set $F = a$ and $C = a$, then solve).
92.Crickets and temperature: Chirping rate is nearly linear in temperature: 120 chirps/min at 70°F and 168 chirps/min at 80°F. (a) Find the linear equation relating temperature $t$ and chirps/min $n$. (b) Estimate the temperature when chirping at 150 chirps/min.
93.Depreciation: A $4000 computer is expected to be worth $200 after 4 years; the business uses linear depreciation for value $V$ at time $t$. (a) Find a linear equation relating $V$ and $t$. (b) Sketch it. (c) Interpret the slope and $V$-intercept. (d) Find the value 3 years after purchase.
94.Pressure and depth: At the surface, ocean pressure equals air pressure, 15 lb/in$^2$; it rises 4.34 lb/in$^2$ for each 10 ft of depth. (a) Find an equation relating pressure and depth. (b) Sketch it. (c) Interpret the slope and $y$-intercept. (d) At what depth is the pressure 100 lb/in$^2$?

Discuss · Discover · Prove · Write

95.Discuss: Suppose outdoor temperature over time graphs as a line. How is the weather changing if the slope is positive? Negative? Zero?
96.Discuss: Given three points, how can you test whether they are collinear using slopes? Using the Distance Formula? Can you think of another method?

1.11 Solving Equations and Inequalities Graphically 50 problems

Concepts

1.The solutions of $x^{2} - 2x - 3 = 0$ are the ______-intercepts of the graph of $y = x^{2} - 2x - 3$.
2.The solutions of $x^{2} - 2x - 3 > 0$ are the $x$-coordinates of the points on the graph of $y = x^{2} - 2x - 3$ lying ______ the $x$-axis.
3.The figure shows $y = x^{4} - 3x^{3} - x^{2} + 3x$. Use it to: (a) find the solutions of $x^{4} - 3x^{3} - x^{2} + 3x = 0$; (b) find the solutions of $x^{4} - 3x^{3} - x^{2} + 3x \le 0$. figure — add your own
4.The figure shows the graphs of $y = 5x - x^{2}$ and $y = 4$. Use them to: (a) solve $5x - x^{2} = 4$; (b) solve $5x - x^{2} > 4$. figure — add your own

Skills

5.Solve both algebraically and graphically: $x - 4 = 5x + 12$.
6.Solve algebraically and graphically: $\tfrac{1}{2}x - 3 = 6 + 2x$.
7.Solve algebraically and graphically: $\dfrac{2}{x} + \dfrac{1}{2x} = 7$.
8.Solve algebraically and graphically: $\dfrac{4}{x + 2} - \dfrac{6}{2x} = \dfrac{5}{2x + 4}$.
9.Solve algebraically and graphically: $x^{2} - 32 = 0$.
10.Solve algebraically and graphically: $x^{3} + 16 = 0$.
11.Solve algebraically and graphically: $x^{2} + 9 = 0$.
12.Solve algebraically and graphically: $x^{2} + 3 = 2x$.
13.Solve algebraically and graphically: $16x^{4} = 625$.
14.Solve algebraically and graphically: $2x^{5} - 243 = 0$.
15.Solve algebraically and graphically: $(x - 5)^{4} - 80 = 0$.
16.Solve algebraically and graphically: $6(x + 2)^{5} = 64$.
17.Solve graphically in the given interval, rounding to two decimals: $x^{2} - 7x + 12 = 0$; $[0, 6]$.
18.Solve graphically (round to two decimals): $x^{2} - 0.75x + 0.125 = 0$; $[-2, 2]$.
19.Solve graphically (round to two decimals): $x^{3} - 6x^{2} + 11x - 6 = 0$; $[-1, 4]$.
20.Solve graphically (round to two decimals): $16x^{3} + 16x^{2} = x + 1$; $[-2, 2]$.
21.Solve graphically (round to two decimals): $x - \sqrt{x + 1} = 0$; $[-1, 5]$.
22.Solve graphically (round to two decimals): $1 + \sqrt{x} = \sqrt{1 + x^{2}}$; $[-1, 5]$.
23.Solve graphically (round to two decimals): $x^{1/3} - x = 0$; $[-3, 3]$.
24.Solve graphically (round to two decimals): $x^{1/2} + x^{1/3} - x = 0$; $[-1, 5]$.
25.Use the graphical method to solve the equation in Exercise 97 of Section 1.5.
26.Use the graphical method to solve the equation in Exercise 98 of Section 1.5.
27.Use the graphical method to solve the equation in Exercise 105 of Section 1.5.
28.Use the graphical method to solve the equation in Exercise 106 of Section 1.5.
29.Find all real solutions, rounded to two decimals: $x^{3} - 2x^{2} - x - 1 = 0$.
30.Find all real solutions (round to two decimals): $x^{4} - 8x^{2} + 2 = 0$.
31.Find all real solutions (round to two decimals): $x(x - 1)(x + 2) = \tfrac{1}{6}x$.
32.Find all real solutions (round to two decimals): $x^{4} = 16 - x^{3}$.
33.Solve the inequality by drawing appropriate graphs (round to two decimals): $x^{2} \le 3x + 10$.
34.Solve graphically (round to two decimals): $0.5x^{2} + 0.875x \le 0.25$.
35.Solve graphically (round to two decimals): $x^{3} + 11x \le 6x^{2} + 6$.
36.Solve graphically (round to two decimals): $16x^{3} + 24x^{2} > -9x - 1$.
37.Solve graphically (round to two decimals): $x^{1/3} < x$.
38.Solve graphically (round to two decimals): $\sqrt{0.5x^{2} + 1} \le 2|x|$.
39.Solve graphically (round to two decimals): $(x + 1)^{2} < (x - 1)^{2}$.
40.Solve graphically (round to two decimals): $(x + 1)^{2} \le x^{3}$.
41.Use the graphical method to solve the inequality in Exercise 45 of Section 1.8.
42.Use the graphical method to solve the inequality in Exercise 46 of Section 1.8.
43.Use the graphical method to solve the inequality in Exercise 55 of Section 1.8.
44.Use the graphical method to solve the inequality in Exercise 56 of Section 1.8.

Skills Plus

45.Another graphical method: In Example 2 the equation $5 - 3x = 8x - 20$ was solved by graphing two equations. Solve it instead by graphing a single equation, and compare your answer to Example 2's.
46.Finding more solutions: In Example 3 two solutions of $x^{3} - 6x^{2} + 9x = \sqrt{x}$ were found in $[1, 6]$. Find two more, rounded to two decimals.

Applications

47.Estimating profit: A manufacturer's monthly profit from $x$ cooktops is $y = 10x + 0.5x^{2} - 0.001x^{3} - 5000$, for $0 \le x \le 450$. (a) Graph it. (b) How many cooktops must be produced to start making a profit? (c) For what range of $x$ is the profit above $15,000?
48.How far can you see: From height $x$ (ft) above sea level on a ship in calm sea, the farthest visible distance $y$ (mi) is $y = \sqrt{1.5x + \left(\dfrac{x}{5280}\right)^{2}}$. (a) Graph it for $0 \le x \le 100$. (b) How high must you be to see 10 mi? figure — add your own

Discuss · Discover · Prove · Write

49.Write: Compose a short essay comparing the algebraic and graphical methods for solving equations, with your own examples illustrating each method's advantages and disadvantages.
50.Discuss: A student wants to graph $y = x^{1/3}$ and $y = \dfrac{x}{x + 4}$ but enters Y1 = X^1/3 and Y2 = X/X+4, and the calculator draws two lines instead. Explain what went wrong.

1.12 Modeling Variation 57 problems

Concepts

1.If $x$ and $y$ satisfy $y = 3x$, we say $y$ is ______ ______ to $x$, and the constant of ______ is 3.
2.If $x$ and $y$ satisfy $y = \dfrac{3}{x}$, we say $y$ is ______ ______ to $x$, and the constant of ______ is 3.
3.If $x$, $y$, $z$ satisfy $z = 3\dfrac{x}{y}$, we say $z$ is ______ ______ to $x$ and ______ ______ to $y$.
4.If $z$ is directly proportional to the product of $x$ and $y$, and $z = 10$ when $x = 4$, $y = 5$, then $x$, $y$, $z$ are related by the equation $z = \underline{\;\;}$.
5.For each, state whether $y$ is directly proportional, inversely proportional, or not proportional to $x$: (a) $y = 3x$; (b) $y = 3x + 1$.
6.For each, state whether $y$ is directly proportional, inversely proportional, or not proportional to $x$: (a) $y = \dfrac{3}{x + 1}$; (b) $y = \dfrac{3}{x}$.

Skills

7.Write an equation expressing the statement: $T$ varies directly as $x$.
8.Write an equation: $P$ is directly proportional to $w$.
9.Write an equation: $v$ is inversely proportional to $z$.
10.Write an equation: $w$ is proportional to the product of $m$ and $n$.
11.Write an equation: $y$ is proportional to $s$ and inversely proportional to $t$.
12.Write an equation: $P$ varies inversely as $T$.
13.Write an equation: $z$ is proportional to the square root of $y$.
14.Write an equation: $A$ is proportional to the square of $x$ and inversely proportional to the cube of $t$.
15.Write an equation: $V$ is proportional to the product of $l$, $w$, and $h$.
16.Write an equation: $S$ is proportional to the product of the squares of $r$ and $\theta$.
17.Write an equation: $R$ is proportional to the product of the squares of $P$ and $t$ and inversely proportional to the cube of $b$.
18.Write an equation: $A$ is jointly proportional to the square roots of $x$ and $y$.
19.Express as an equation and find the constant of proportionality: $y$ is directly proportional to $x$; $y = 42$ when $x = 6$.
20.Express as an equation and find the constant: $w$ is inversely proportional to $t$; $w = 3$ when $t = 8$.
21.Express as an equation and find the constant: $A$ varies inversely as $r$; $A = 7$ when $r = 3$.
22.Express as an equation and find the constant: $P$ is directly proportional to $T$; $P = 20$ when $T = 300$.
23.Express as an equation and find the constant: $A$ is directly proportional to $x$ and inversely proportional to $t$; $A = 42$ when $x = 7$, $t = 3$.
24.Express as an equation and find the constant: $S$ is proportional to the product of $p$ and $q$; $S = 180$ when $p = 4$, $q = 5$.
25.Express as an equation and find the constant: $W$ is inversely proportional to the square of $r$; $W = 10$ when $r = 6$.
26.Express as an equation and find the constant: $t$ is proportional to the product of $x$ and $y$ and inversely proportional to $r$; $t = 25$ when $x = 2$, $y = 3$, $r = 12$.
27.Express as an equation and find the constant: $C$ is jointly proportional to $l$, $w$, and $h$; $C = 128$ when $l = w = h = 2$.
28.Express as an equation and find the constant: $H$ is jointly proportional to the squares of $l$ and $w$; $H = 36$ when $l = 2$, $w = \tfrac{1}{3}$.
29.Express as an equation and find the constant: $R$ is inversely proportional to the square root of $x$; $R = 2.5$ when $x = 121$.
30.Express as an equation and find the constant: $M$ is jointly proportional to $a$, $b$, $c$ and inversely proportional to $d$; $M = 128$ when $a$ and $d$ are equal and $b = c = 2$.
31.Express as a proportionality equation, then determine the factor by which $z$ changes when $x$ is tripled and $y$ is doubled: $z$ varies directly as the cube of $x$ and inversely as the square of $y$.
32.Express as an equation, then find the factor change in $z$ when $x$ is tripled and $y$ doubled: $z$ is directly proportional to the square of $x$ and inversely proportional to the fourth power of $y$.
33.Express as an equation, then find the factor change in $z$ when $x$ is tripled and $y$ doubled: $z$ is jointly proportional to the cube of $x$ and the fifth power of $y$.
34.Express as an equation, then find the factor change in $z$ when $x$ is tripled and $y$ doubled: $z$ is inversely proportional to the square of $x$ and the cube of $y$.

Applications

35.Hooke's Law: The force to stretch a spring $x$ units past its natural length is proportional to $x$ (the constant is the spring constant). (a) Write Hooke's Law as an equation. (b) A spring of natural length 5 cm needs 30 N to stretch it to 9 cm; find the spring constant. (c) What force keeps it stretched to 11 cm? figure — add your own
36.Printing cost: The cost $C$ of printing a magazine is jointly proportional to the page count $p$ and the number of copies $m$. (a) Write the joint-variation equation. (b) Find the constant if printing 4000 copies of a 120-page magazine costs $60,000. (c) Find the cost to print 5000 copies of a 92-page magazine.
37.Windmill power: The power $P$ from a windmill is directly proportional to the cube of the wind speed $s$. (a) Write the equation. (b) Find the constant for a windmill producing 96 W at 20 mi/h wind. (c) How much power at 30 mi/h?
38.Boat power: The power $P$ (hp) to propel a boat is directly proportional to the cube of its speed $s$. (a) Write the equation. (b) Find the constant for a boat needing an 80-hp engine at 10 knots. (c) How much power is needed at 15 knots? figure — add your own
39.Stopping distance: A car's stopping distance $D$ varies directly as the square of its speed $s$. A car at 40 mi/h stops in 150 ft. What is the maximum speed if it must stop within 200 ft?
40.Aerodynamic lift: The lift $L$ on a wing varies jointly as the square of the plane's speed $s$ and the wing area $A$. A 500-ft$^2$ wing at 50 mi/h gives 1700 lb of lift. How much lift for a 600-ft$^2$ wing at 40 mi/h? figure — add your own
41.Boat drag: The drag force $F$ on a boat is jointly proportional to the wetted hull area $A$ and the square of the boat's speed $s$. A boat at 5 mi/h with 40 ft$^2$ wetted area has 220 lb of drag. How fast is a boat with 28 ft$^2$ wetted area experiencing 175 lb of drag?
42.Kepler's Third Law: The square of a planet's orbital period $T$ is directly proportional to the cube of its average distance $d$ from the sun. (a) Write the equation. (b) Find the constant using Earth's period of ~365 days and distance ~93 million mi. (c) Neptune is about $2.79\times 10^{9}$ mi from the sun; find its period.
43.Ideal Gas Law: The pressure $P$ of a gas is directly proportional to temperature $T$ and inversely proportional to volume $V$. (a) Write the equation. (b) Find the constant if 100 L of gas exerts 33.2 kPa at 400 K. (c) If $T$ rises to 500 K and $V$ drops to 80 L, find the pressure.
44.Skidding in a curve: The force $F$ needed to keep a car from skidding on a circular curve is jointly proportional to the car's weight $w$ and the square of its speed $s$, and inversely proportional to the curve radius $r$. (a) Write the equation. (b) A 1600-lb car rounds the curve at 60 mi/h; a 2500-lb car needs the same force. How fast is the second car going? figure — add your own
45.Loudness of sound: Loudness $L$ (dB) is inversely proportional to the square of distance $d$ from the source. (a) Write the equation. (b) Find the constant if a person 10 ft from a lawn mower hears 70 dB. (c) If that distance doubles, by what factor does loudness change? (d) If it is halved, by what factor?
46.Jet of water: The power $P$ of a water jet is jointly proportional to the jet's cross-sectional area $A$ and the cube of its velocity $v$. (a) Write the equation. (b) If velocity doubles and area halves, by what factor does power change? (c) If velocity halves and area triples, by what factor? figure — add your own
47.Electrical resistance: A wire's resistance $R$ varies directly as its length $L$ and inversely as the square of its diameter $d$. (a) Write the joint-variation equation. (b) Find the constant if a wire 1.2 m long and 0.005 m in diameter has resistance 140 ohms. (c) Find the resistance of a same-material wire 3 m long with diameter 0.008 m. (d) If diameter doubles and length triples, by what factor does $R$ change?
48.Growing cabbages: A cabbage's final size is proportional to the nutrients it gets and inversely proportional to the number of neighboring cabbages. A cabbage receiving 20 oz of nutrients with 12 neighbors grew to 30 lb. How big would it grow with 10 oz of nutrients and 5 neighbors?
49.Radiation energy: The radiation energy $E$ per unit area from a heated surface varies as the fourth power of its absolute temperature $T$. The sun's surface is 6000 K, Earth's is 300 K. (a) How many times more radiation per unit area does the sun emit than Earth? (b) Earth's radius is 3960 mi and the sun's is 435,000 mi; how many times more total radiation does the sun emit than Earth?
50.Lot value: A building lot's value is jointly proportional to its area and the water output of its well. A 200-ft by 300-ft lot with a 10-gal/min well is worth $48,000. What is a 400-ft by 400-ft lot worth if its well produces 4 gal/min?
51.Law of the pendulum: A pendulum's period varies directly as the square root of its length. (a) Write the equation. (b) To double the period, how must the length $l$ change? figure — add your own
52.Campfire heat: The heat a hiker feels is proportional to the amount of burning wood and inversely proportional to the cube of his distance from the fire. At 20 ft from the fire, if the wood burning doubles, how far must he stand to feel the same heat as before? figure — add your own
53.String frequency: A violin string's vibration frequency $f$ is inversely proportional to its length $L$ (the positive constant $k$ depends on tension and density). (a) Write the equation. (b) What effect does doubling the string length have on the frequency?
54.Disease spread: The spread rate $r$ in a population of size $P$ is jointly proportional to the number infected $x$ and the number not infected $P - x$. For a town of $P = 5000$: (a) write $r$ as a function of $x$; (b) compare the spread rate when 10 are infected to when 1000 are infected — which is larger and by what factor? (c) Find the rate when the whole population is infected, and explain why this makes sense.
55.Star brightness: Using brightness $B$, luminosity $L$, distance $d$, and proportionality constant $k = 0.080$: a star has $L = 2.5\times 10^{26}$ W and $d = 2.4\times 10^{19}$ m. How bright does it appear from Earth?
56.Distance to a star: With the same brightness–luminosity–distance relationship ($k = 0.080$): a star has $L = 5.8\times 10^{30}$ W and appears with brightness $B = 8.2\times 10^{-16}$ W/m$^2$ from Earth. Find its distance from Earth.

Discuss · Discover · Prove · Write

57.Discuss: Many laws of physics and chemistry are proportionalities. Give at least one example of a function arising in the sciences that is not a proportionality.