Chapter 2.7 – Absolute Value Inequalities

Objectives Solve absolute value inequalities of the form /x/ < a Solve absolute value inequalities of the form /x/ > a

Example 1 Solve: ￨ x ￨ 3 [ ] The solution set is {-3, 3}

Solving Absolute Value expression of the form ￨ x ￨ < a If a is a positive number, then ￨ x ￨ < a is equivalent to – a < x< a

Example 2: Solve for m: ￨ m – 6 ￨ < 2 Step 1: -a < x < a Replace x with m - 6 and a with 2 -2 < m – 6 < 2

Example 2 continued Solve the compound inequality -2 < m – 6 < < m – < < m < 8 The solution set is (4, 8) and its graph is,

Give it a try! Solve ￨ x-2 ￨ 1

HINT MUST ISOLATE the absolute value expression Before using an absolute value inequality property, you MUST ISOLATE the absolute value expression on one side of the inequality!

Example 3: Solve for x: ￨ 5x + 1 ￨ + 110

Give it a try! Solve: ￨ 2x - 5 ￨ + 29

Example 4 Solve for x: The absolute value of a number is always nonnegative and can never be less than – 13. This inequality has NO solution! The solution set is { } or 0

Give it a try! Solve:

Let ￨ x ￨ 3 The solution set includes all numbers who distance from 0 is 3 or more units. The graph of the solution set contains 3 and all points to the right of 3 on the number line or – 3 and all points to the left of – 3 on the number line.. The solution is ( -, -3] U [3, ) Form of ￨ x ￨ > a

Solving Absolute Value Inequalities of the form ￨ x ￨ > a If a is a positive number, the ￨ x ￨ > a is equivalent to x a

Example 5 Solve for y: Step 1: Rewrite the inequalities without absolute value bars y – 3 7 Step 2: Solve the compound inequality y 10

Example 5 Continued Step 3: Graph the inequalities Step 4: Write the solution set (-∞, -4) U (10, ∞)

Give it a try! Solve:

Example 6: Isolate the Absolute Value expression! Solve: Step 1: Isolate the Absolute Value Expression

Example 6 - Continued ***Remember*** The absolute value of any number is always a nonnegative and thus is always great than -2. The inequality and the original inequality are true for all values of x. The solution set is {x/ x is a real number} or (-∞,∞)

Example 7: Isolate the Absolute Value Expression! Solve: Step 1: Isolate the Absolute Value Expression

Example 7 - Continued Step 2: Write the absolute value inequality as an equivalent compound inequality.

Example 7 continued Step 3: Solve each inequality

Example 7 continued Step 4: Write the solution set and graph (-∞, -3] U [9, ∞)

Give it a try! Solve:

Example 8 - Zero Solve for x: equal **Remember – the absolute value of any expression will never be less than 0, but it may be equal to 0. Thus solve the equation by setting it equal to zero!

Example 8 Continued

You give try! Solve:

Chart on Page 113 Copy the chart from 113 into your notes!