Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Solve compound inequalities.

Write and solve absolute-value equations and inequalities.

Objectives

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Recall that the absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because absolute value represents distance without regard to direction, the absolute value of any real number is nonnegative.

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Absolute-value equations and inequalities can be represented by compound statements. Consider the equation |x| = 3.

The solutions of |x| = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = –3 or x = 3.

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Solve the equation.

Example 1: Solving Absolute-Value Equations

Rewrite the absolute value as a disjunction.

This can be read as “the distance from k to –3 is 10.”

Add 3 to both sides of each equation.

|–3 + k| = 10

–3 + k = 10 or –3 + k = –10

k = 13 or k = –7

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Solve the equation.

Example 2: Solving Absolute-Value Equations

x = 16 or x = –16

Isolate the absolute-value expression.

Rewrite the absolute value as a disjunction.

Multiply both sides of each equation by 4.

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Example 3

|x + 9| = 13

Solve the equation.

Rewrite the absolute value as a disjunction.

This can be read as “the distance from x to +9 is 4.”

Subtract 9 from both sides of each equation.

x + 9 = 13 or x + 9 = –13

x = 4 or x = –22

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Example 4

|6x| – 8 = 22

Solve the equation.

Isolate the absolute-value expression.

Rewrite the absolute value as a disjunction. Divide both sides of each equation by 6.

|6x| = 30

6x = 30 or 6x = –30

x = 5 or x = –5

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Lesson Quiz: Part I

1.

2.

Solve each equation.

|2v + 5| = 9

4. |5b| – 7 = 13

2 or –7

+ 4

Holt McDougal Algebra 2

2-8Solving Absolute-Value Equations and Inequalities

Homework

Pg 154: 5-7, 16-19, 37, 38