Precalculus · Chapter 1: Fundamentals

1.1Real Numbers

Real Numbers · Properties · Addition & Subtraction · Multiplication & Division · The Real Line · Sets & Intervals · Absolute Value & Distance

The Real Number System

The natural numbers are \(1,2,3,\dots\) The integers add zero and negatives: \(\dots,-2,-1,0,1,2,\dots\) A rational number is any number expressible as a ratio \(\tfrac{a}{b}\) of integers with \(b\neq 0\). Numbers that cannot be written this way, such as \(\sqrt{2}\) and \(\pi\), are irrational. Together the rationals and irrationals form the real numbers.

Rational numbers have decimal expansions that either terminate or eventually repeat, while irrational numbers have nonrepeating, nonterminating decimals, for example \(\sqrt{2}=1.41421356\ldots\) and \(\pi=3.14159265\ldots\) Truncating a decimal gives an approximation, written with the symbol \(\approx\).

Properties of Real Numbers

Property	Statement	Example
Commutative	\(a+b=b+a\), \(\;ab=ba\)	\(7+3=3+7\)
Associative	\((a+b)+c=a+(b+c)\), \(\;(ab)c=a(bc)\)	\((2+4)+7=2+(4+7)\)
Distributive	\(a(b+c)=ab+ac\)	\(2(3+5)=2\cdot3+2\cdot5\)

EXAMPLE 1Using the Distributive Property

Expand \(2(x+3)\) and \((a+b)(x+y)\).

Solution

\(2(x+3)=2\cdot x+2\cdot 3=2x+6\)

\((a+b)(x+y)=(a+b)x+(a+b)y=ax+bx+ay+by\)

Negatives and Fractions

Additive & Multiplicative Identities

The number \(0\) is the additive identity since \(a+0=a\); every \(a\) has a negative with \(a+(-a)=0\). The number \(1\) is the multiplicative identity since \(a\cdot1=a\); every nonzero \(a\) has an inverse \(1/a\) with \(a\cdot(1/a)=1\). Subtraction is \(a-b=a+(-b)\) and division is \(a\div b=a\cdot\tfrac1b\).

Properties of Fractions

\[ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd},\qquad \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c},\qquad \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c},\qquad \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd} \]

EXAMPLE 3Using the LCD to Add Fractions

Evaluate \(\dfrac{5}{36}+\dfrac{7}{120}\).

Solution

Factor denominators: \(36=2^2\cdot3^2\) and \(120=2^3\cdot3\cdot5\).

LCD uses the highest power of each prime: \(2^3\cdot3^2\cdot5=360\).

\(\dfrac{5}{36}+\dfrac{7}{120}=\dfrac{50}{360}+\dfrac{21}{360}=\dfrac{71}{360}\)

The Real Line, Sets, and Intervals

The real numbers correspond to points on a line. We write \(aset is a collection of elements; \(a\in S\) means \(a\) belongs to \(S\). The union \(S\cup T\) collects elements in either set, the intersection \(S\cap T\) collects elements in both, and \(\varnothing\) is the empty set.

Interval Notation

\[ (a,b)=\{x\mid a

EXAMPLE 6Unions and Intersections of Intervals

Graph \((1,3)\cap[2,7]\) and \((1,3)\cup[2,7]\).

Solution

Intersection — numbers in both: \((1,3)\cap[2,7]=\{x\mid 2\le x<3\}=[2,3)\).

Union — numbers in either: \((1,3)\cup[2,7]=\{x\mid 1

Absolute Value and Distance

Definition of Absolute Value

\[ |a|=\begin{cases} a & \text{if } a\ge 0\\ -a & \text{if } a<0 \end{cases} \]

Distance Between Points on the Real Line

\[ d(a,b)=|b-a|=|a-b| \]

EXAMPLE 8Distance Between Two Points

Find the distance between \(-8\) and \(2\).

Solution

\(d(-8,2)=|2-(-8)|=|10|=10\)

Key IdeaAbsolute value measures distance from \(0\); it is never negative. The Triangle Inequality \(|a+b|\le|a|+|b|\) holds for all real numbers.

Instructor Note

Reserved for instructor notes on Section 1.1. (Add your guidance here.)

1.2Exponents and Radicals

Integer Exponents · Laws of Exponents · Scientific Notation · Radicals · Rational Exponents · Rationalizing the Denominator

Exponential Notation

If \(a\) is a real number and \(n\) a positive integer, the \(n\)th power of \(a\) is \(a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}}\). Here \(a\) is the base and \(n\) the exponent. For \(a\neq0\): \(a^0=1\) and \(a^{-n}=\dfrac{1}{a^n}\).

Laws of Exponents

\[ a^m a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad (ab)^n=a^n b^n,\quad \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \]

EXAMPLE 3Using the Laws of Exponents

Solution

\(x^4 x^7 = x^{4+7}=x^{11}\)

\(\dfrac{c^9}{c^5}=c^{9-5}=c^4\)

\((b^4)^5=b^{20}\), and \((3x)^3=27x^3\)

Scientific Notation

A number is in scientific notation when written as \(a\times10^{n}\) with \(1\le|a|<10\) and \(n\) an integer. Positive \(n\) shifts the decimal right (large numbers); negative \(n\) shifts it left (small numbers).

Radicals and Rational Exponents

nth Roots and Rational Exponents

\[ \sqrt[n]{a}=b \iff b^n=a,\qquad a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^{m} \]

Laws of Radicals

\[ \sqrt[n]{ab}=\sqrt[n]{a}\,\sqrt[n]{b},\qquad \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\qquad \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a} \]

EXAMPLE 9Rational Exponents

Solution

\(4^{3/2}=(\sqrt{4})^3=2^3=8\)

\(8^{-2/3}=\dfrac{1}{8^{2/3}}=\dfrac{1}{(\sqrt[3]{8})^2}=\dfrac{1}{4}\)

Key IdeaTo rationalize a denominator \(\sqrt[n]{a^m}\), multiply numerator and denominator by \(\sqrt[n]{a^{\,n-m}}\) so the radicand becomes a perfect \(n\)th power.

Instructor Note

Reserved for instructor notes on Section 1.2. (Add your guidance here.)

1.3Algebraic Expressions

Polynomials · Adding & Subtracting · Multiplying · Special Product Formulas · Factoring · Special Factoring Formulas · Factoring by Grouping

Polynomials

A polynomial in \(x\) has the form \(a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1 x+a_0\), where the \(a_i\) are real coefficients and \(n\) is a nonnegative integer. If \(a_n\neq0\), the degree is \(n\). A one-term polynomial is a monomial, two terms a binomial, three a trinomial.

We add and subtract polynomials by combining like terms. To multiply, we apply the Distributive Property repeatedly; for two binomials the FOIL pattern (First, Outer, Inner, Last) is a convenient shortcut.

Special Product Formulas

\[ (A+B)(A-B)=A^2-B^2 \]\[ (A\pm B)^2=A^2\pm 2AB+B^2 \]\[ (A\pm B)^3=A^3\pm3A^2B+3AB^2\pm B^3 \]

EXAMPLE 2Multiplying Binomials Using FOIL

Expand \((2x+1)(3x-5)\).

Solution

\((2x+1)(3x-5)=6x^2-10x+3x-5\)

\(=6x^2-7x-5\)

Factoring

Special Factoring Formulas

\[ A^2-B^2=(A-B)(A+B) \]\[ A^2\pm 2AB+B^2=(A\pm B)^2 \]\[ A^3\pm B^3=(A\pm B)(A^2\mp AB+B^2) \]

EXAMPLE 7Factoring by Grouping

Factor \(x^3+x^2+4x+4\).

Solution

Group: \((x^3+x^2)+(4x+4)\)

Factor each group: \(x^2(x+1)+4(x+1)\)

Common factor: \((x+1)(x^2+4)\)

Instructor Note

Reserved for instructor notes on Section 1.3. (Add your guidance here.)

1.4Rational Expressions

Domain of an Expression · Simplifying · Multiplying & Dividing · Adding & Subtracting · Compound Fractions · Rationalizing

Rational Expression & Domain

A rational expression is a fraction in which numerator and denominator are polynomials. The domain is the set of real numbers the variable may take — we must exclude any value that makes a denominator zero, and (for even roots) any value that makes a radicand negative.

Cancellation Property of Fractions

\[ \frac{AC}{BC}=\frac{A}{B}\qquad (B,C\neq 0) \]

EXAMPLE 2Simplifying by Cancellation

Simplify \(\dfrac{x^2-1}{x^2+x-2}\).

Solution

Factor: \(\dfrac{(x-1)(x+1)}{(x-1)(x+2)}\)

Cancel the common factor \((x-1)\): \(\dfrac{x+1}{x+2}\)

To multiply, multiply numerators and denominators then simplify. To divide, invert the divisor and multiply. To add or subtract, rewrite over a common denominator (preferably the LCD) and combine numerators.

EXAMPLE 5Adding Rational Expressions

Combine \(\dfrac{3}{x-1}+\dfrac{x}{x+2}\).

Solution

LCD is \((x-1)(x+2)\): \(\dfrac{3(x+2)+x(x-1)}{(x-1)(x+2)}\)

\(=\dfrac{3x+6+x^2-x}{(x-1)(x+2)}=\dfrac{x^2+2x+6}{(x-1)(x+2)}\)

CautionYou may cancel only common factors, never individual terms. In \(\dfrac{x^2+2}{x^2}\) the \(x^2\) terms cannot be cancelled because \(x^2\) is not a factor of the numerator.

Instructor Note

Reserved for instructor notes on Section 1.4. (Add your guidance here.)

1.5Equations

Solving Linear Equations · Solving for One Variable · Solving Quadratic Equations · Other Types of Equations

Equations and Equivalence

An equation asserts that two expressions are equal; a solution is a value of the variable that makes it true. Adding the same quantity to both sides, or multiplying both sides by the same nonzero quantity, yields an equivalent equation with the same solutions.

A linear (first-degree) equation has the form \(ax+b=0\) with \(a\neq0\). We solve it by isolating the variable.

EXAMPLE 1Solving a Linear Equation

Solve \(7x-4=3x+8\).

Solution

Collect variable terms: \(7x-3x=8+4\)

\(4x=12\)

\(x=3\)

Quadratic Equations

The Quadratic Formula

\[ ax^2+bx+c=0 \;\Longrightarrow\; x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]

The DiscriminantThe quantity \(D=b^2-4ac\) determines the nature of the roots: \(D>0\) gives two real solutions, \(D=0\) gives one (repeated) real solution, and \(D<0\) gives two complex solutions.

Zero-Product Property

\[ AB=0 \iff A=0 \ \text{ or } \ B=0 \]

EXAMPLE 4Solving by Factoring

Solve \(x^2+5x-24=0\).

Solution

Factor: \((x+8)(x-3)=0\)

Zero-Product Property: \(x=-8\) or \(x=3\)

Instructor Note

Reserved for instructor notes on Section 1.5. (Add your guidance here.)

1.6Complex Numbers

Arithmetic on Complex Numbers · Square Roots of Negative Numbers · Complex Solutions of Quadratic Equations

The Imaginary Unit and Complex Numbers

The imaginary unit \(i\) is defined by \(i^2=-1\). A complex number has the form \(a+bi\), where \(a\) is the real part and \(b\) the imaginary part. Two complex numbers are equal exactly when their real parts and imaginary parts match.

Arithmetic of Complex Numbers

\[ (a+bi)+(c+di)=(a+c)+(b+d)i \]\[ (a+bi)(c+di)=(ac-bd)+(ad+bc)i \]

EXAMPLE 2Multiplying Complex Numbers

Express \((3+5i)(4-2i)\) in the form \(a+bi\).

Solution

\(12-6i+20i-10i^2\)

Since \(i^2=-1\): \(12+14i+10=22+14i\)

Complex Conjugate & Division

The conjugate of \(z=a+bi\) is \(\bar z=a-bi\), and \(z\bar z=a^2+b^2\) is real. To divide, multiply numerator and denominator by the conjugate of the denominator.

Square Roots of Negative Numbers

\[ \text{If } c>0,\quad \sqrt{-c}=i\sqrt{c} \]

EXAMPLE 5Complex Solutions of a Quadratic

Solve \(x^2+4=0\).

Solution

\(x^2=-4\)

\(x=\pm\sqrt{-4}=\pm 2i\)

Instructor Note

Reserved for instructor notes on Section 1.6. (Add your guidance here.)

1.7Modeling with Equations

Making & Using Models · Interest · Area & Length · Mixtures · Time to Do a Job · Distance–Rate–Time

Guidelines for Modeling with Equations

1. Identify the variable — the quantity the problem asks for, and name it. 2. Express all other quantities in terms of that variable. 3. Set up the model by translating the words into an equation. 4. Solve the equation and check that the answer makes sense.

Simple Interest

\[ I=Prt \]

Here \(I\) is the interest, \(P\) the principal, \(r\) the annual rate (as a decimal), and \(t\) the time in years.

Distance, Rate, and Time

\[ \text{distance}=\text{rate}\times\text{time},\qquad d=rt \]

EXAMPLE 3Dimensions of a Garden

A rectangular garden has area \(1200\ \text{ft}^2\); its length is \(20\) ft more than its width. Find the dimensions.

Solution

Let width \(=w\); then length \(=w+20\).

Model the area: \(w(w+20)=1200\).

\(w^2+20w-1200=0\Rightarrow(w+ 60)(w-20)=0\).

Reject \(w=-60\); so \(w=20\) ft and length \(=40\) ft.

Modeling TipFor "work-rate" problems, if a job takes \(H\) hours then the rate is \(1/H\) of the job per hour, and rates of workers acting together add.

Instructor Note

Reserved for instructor notes on Section 1.7. (Add your guidance here.)

1.8Inequalities

Linear Inequalities · Nonlinear Inequalities · Absolute Value Inequalities · Modeling with Inequalities

Rules for Inequalities

Adding the same quantity to both sides preserves the inequality. Multiplying both sides by a positive number preserves it; multiplying by a negative number reverses the inequality sign.

EXAMPLE 1Solving a Linear Inequality

Solve \(3x<9x+4\).

Solution

\(3x-9x<4 \Rightarrow -6x<4\)

Divide by \(-6\) and reverse: \(x>-\tfrac{2}{3}\)

Guidelines for Nonlinear Inequalities

Move all terms to one side and factor. Find the values that make each factor zero (the endpoints). These divide the line into intervals; test a point in each interval to determine the sign of the product, then read off the solution.

Absolute Value Inequalities

\[ |x|c \iff x<-c \ \text{ or } \ x>c \]

EXAMPLE 7An Absolute Value Inequality

Solve \(|3x-2|\ge 4\).

Solution

\(3x-2\le-4\) or \(3x-2\ge4\)

\(3x\le-2\) or \(3x\ge6\)

\(x\le-\tfrac{2}{3}\) or \(x\ge2\)

Instructor Note

Reserved for instructor notes on Section 1.8. (Add your guidance here.)

1.9The Coordinate Plane; Graphs of Equations; Circles

The Coordinate Plane · Distance & Midpoint Formulas · Graphs of Equations · Intercepts · Circles · Symmetry

The Coordinate Plane

Two perpendicular axes meeting at the origin assign to each point an ordered pair \((x,y)\). The axes divide the plane into four quadrants. The graph of an equation in \(x\) and \(y\) is the set of all points whose coordinates satisfy the equation.

Distance Formula

\[ d(A,B)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

Midpoint Formula

\[ M=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right) \]

EXAMPLE 2Applying the Distance Formula

Find the distance between \(A(2,5)\) and \(B(8,13)\).

Solution

\(d=\sqrt{(8-2)^2+(13-5)^2}=\sqrt{36+64}\)

\(=\sqrt{100}=10\)

Circles

Equation of a Circle

\[ (x-h)^2+(y-k)^2=r^2 \]

This is a circle of radius \(r\) centered at \((h,k)\). When the center is the origin it reduces to \(x^2+y^2=r^2\).

Tests for SymmetryA graph is symmetric about the \(x\)-axis if replacing \(y\) by \(-y\) leaves the equation unchanged, about the \(y\)-axis if replacing \(x\) by \(-x\) does, and about the origin if replacing both \(x,y\) by \(-x,-y\) does.

Instructor Note

Reserved for instructor notes on Section 1.9. (Add your guidance here.)

1.10Lines

Slope of a Line · Point-Slope Form · Slope-Intercept Form · Vertical & Horizontal Lines · General Equation · Parallel & Perpendicular Lines

Slope

The slope of a line measures its steepness as the ratio of vertical change (rise) to horizontal change (run). A positive slope rises to the right, a negative slope falls; a horizontal line has slope \(0\), and a vertical line has undefined slope.

Slope of a Line

\[ m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}\qquad (x_1\neq x_2) \]

EXAMPLE 1Slope Through Two Points

Find the slope of the line through \(P(2,1)\) and \(Q(8,5)\).

Solution

\(m=\dfrac{5-1}{8-2}=\dfrac{4}{6}=\dfrac{2}{3}\)

Equations of a Line

\[ \text{Point-slope: } y-y_1=m(x-x_1) \]\[ \text{Slope-intercept: } y=mx+b \]\[ \text{General: } Ax+By+C=0 \]

Parallel and Perpendicular Lines

Two nonvertical lines are parallel when they have equal slopes, \(m_1=m_2\). They are perpendicular when the product of their slopes is \(-1\), that is \(m_1 m_2=-1\), so each slope is the negative reciprocal of the other.

EXAMPLE 2Equation from Point and Slope

Find an equation of the line through \((1,-3)\) with slope \(-\tfrac{1}{2}\).

Solution

Point-slope: \(y-(-3)=-\tfrac{1}{2}(x-1)\)

\(y+3=-\tfrac{1}{2}x+\tfrac{1}{2}\)

\(y=-\tfrac{1}{2}x-\tfrac{5}{2}\)

Instructor Note

Reserved for instructor notes on Section 1.10. (Add your guidance here.)

1.11Solving Equations and Inequalities Graphically

Solving Equations Graphically · Solving Inequalities Graphically

The Graphical Method

To solve an equation graphically, move all terms to one side and graph \(y=f(x)\); the real solutions are the \(x\)-intercepts (zeros), where the graph crosses the \(x\)-axis. A graphing calculator finds these intercepts to any desired accuracy, which is useful when an equation cannot be solved algebraically.

Algebraic vs. GraphicalThe algebraic method gives exact solutions but works only for certain equations. The graphical method works for virtually any equation but gives approximate solutions. The two methods complement each other.

EXAMPLE 1Solving a Quadratic Two Ways

Solve \(x^2-4x+2=0\).

Solution

Quadratic Formula: \(x=\dfrac{4\pm\sqrt{16-8}}{2}=\dfrac{4\pm2\sqrt2}{2}=2\pm\sqrt2\).

Graphically, \(y=x^2-4x+2\) crosses the \(x\)-axis at \(x\approx0.59\) and \(x\approx3.41\) — matching \(2\pm\sqrt2\).

Solving Inequalities Graphically

To solve \(f(x)\ge 0\), graph \(y=f(x)\) and read off the intervals where the graph lies on or above the \(x\)-axis. For \(f(x)\le g(x)\), find where the graph of \(f\) lies below that of \(g\).

Instructor Note

Reserved for instructor notes on Section 1.11. (Add your guidance here.)

1.12Modeling Variation

Direct Variation · Inverse Variation · Combining Different Types of Variation

Direct Variation

\[ y=kx \]

\(y\) is directly proportional to \(x\); the constant \(k\neq0\) is the constant of proportionality. The graph is a line through the origin with slope \(k\).

Inverse Variation

\[ y=\frac{k}{x} \]

\(y\) is inversely proportional to \(x\). Boyle's Law — pressure inversely proportional to volume at constant temperature — is a classic example.

Joint & Combined Variation

\[ z=k\,\frac{x}{y} \quad\text{(directly as } x\text{, inversely as } y\text{)} \]

EXAMPLE 4Newton's Law of Gravity

Two objects of masses \(m_1\) and \(m_2\) attract with a force jointly proportional to their masses and inversely proportional to the square of the distance \(r\) between them. Write the law.

Solution

Combine joint and inverse variation: \(F=G\,\dfrac{m_1 m_2}{r^2}\), where \(G\) is the gravitational constant.

Engineering LinkVariation models appear throughout mechanical engineering — stress proportional to load, deflection inversely proportional to the moment of inertia, and the inverse-square laws of gravity and radiation.

Instructor Note

Reserved for instructor notes on Section 1.12. (Add your guidance here.)