Objectives Solve compound inequalities. Write and solve 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.

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.

Example 1: Solving Absolute-Value Equations Solve the equation. This can be read as “the distance from k to –3 is 10.” |–3 + k| = 10 Rewrite the absolute value as a disjunction. –3 + k = 10 or –3 + k = –10 Add 3 to both sides of each equation. k = 13 or k = –7

Example 2: Solving Absolute-Value Equations Solve the equation. Isolate the absolute-value expression. Rewrite the absolute value as a disjunction. Multiply both sides of each equation by 4. x = 16 or x = –16

Example 3 Solve the equation. This can be read as “the distance from x to +9 is 4.” |x + 9| = 13 Rewrite the absolute value as a disjunction. x + 9 = 13 or x + 9 = –13 Subtract 9 from both sides of each equation. x = 4 or x = –22

Example 4 Solve the equation. |6x| – 8 = 22 Isolate the absolute-value expression. |6x| = 30 Rewrite the absolute value as a disjunction. 6x = 30 or 6x = –30 Divide both sides of each equation by 6. x = 5 or x = –5

Lesson Quiz: Part I Solve each equation. 1. |2v + 5| = 9 2 or –7 2. 4. |5b| – 7 = 13 + 4

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