Problem

Solve. Graph the solution set. See Examples 9 through 16.Example 9 Solve |x| < 2 using...

Solve. Graph the solution set. See Examples 9 through 16.

Example 9 Solve |x| < 2 using a number line.
Solution: The solution set contains all numbers whose distance from 0 is less than 2 units on the number line.
The solution set is {x | − 2 < x < 2}, or (− 2, 2) in interval notation.
Example 10 Solve |x| ≥ 3 using a number line.
Solution: The solution set contains all numbers whose distance from 0 is 3 or more units. Thus 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.
This solution set Is {x | x ≤ − 3 or x ≥ 3}. In Interval notation, the solution set Is (− ∞, − 3] ∪ [3, ∞), since or means union.
The following box summarizes solving absolute value equations and inequalities.
Example 11 Solve: |x − 3| > 7
Solution: Since 7 Is positive, to solve |x − 3| > 7, we solve the compound inequality x − 3 − 7 or x − 3 > 7.
The solution set Is {x | x < − 4 or x > 10} or (∞, −4) U (10, ∞) in interval notation. Its graph Is shown.
Example 12 Solve: |x + l| = 6
Solution: This is an equation, so we solve
The solution set is {−7, 5}. Its graph Is shown.
Example 13: Solve: |x − 6| ≤ 2
Solution: To solve |x − 6| ≤ 2, we solve
The solution set Is {x|4 ≤ x < 8}, or [4, 8] in Interval notation. its graph is shown.
Example14 Solve: |5x + l| + 1 ≤ 10
Solution: First we get the absolute value expression alone by subtracting 1 from both sides.
Since 9 is positive, to solve |5x + l| ≤ 9, we solve
The solution set is .
Example 15 Solve: |x| ≤ − 3
Solution: The absolute value of a number is never negative. Thus It will then never be less than or equal to − 3. The solution set is { } or 0.
Example 16 Solve: |x − 1| > − 2
Solution: The absolute value of a number is always nonnegative. Thus it will always be greater than − 2,The solution set contains all real numbers, or (— ∞, ∞).