Question

Solve the absolute value inequalities. Other than 0, use interval notation to express the solution set and graph the solution set on a number line.

1. |x + 3| - 9 ≤ -3

2. |3x - 8| -4 > -2

Answer 1

a) The solution set in interval notation is (-9, 3]) and on a number.

b) The solution set in interval notation is (-∞, 2), (10/3, ∞)

How to determine values of absolute function

Given:

1. |x + 3| - 9 <= -3

Add 9 to both sides:

|x + 3| <= 6

This implies two cases:

x + 3 <=6 → x <= 3

-(x + 3) <=6 → x => -9

The solution set in interval notation is (-9, 3]) and on a number

2. For |3x - 8| - 4 > -2

Add 4 to both sides:

|3x - 8| > 2

This implies two cases:

3x - 8 > 2 → x > 10/3

-(3x - 8) > 2 → x < 2

The solution set in interval notation is (-∞, 2), (10/3, ∞)