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1. Learning Objectives
- Understand the geometric definition of absolute value as distance.
- Solve absolute value inequalities using compound inequalities.
- Represent solutions on a number line and in interval notation.
2. Fundamental Rules
To solve inequalities of the form |ax + b| < c or |ax + b| > c (where c > 0):
1. The "LESS THAN" Case (< or ≤):
If |x| < a, then -a < x < a. (Intersection/Between)
2. The "GREATER THAN" Case (> or ≥):
If |x| > a, then x < -a OR x > a. (Union/Opposite directions)
Click the button below each question to reveal the explanation.
1. Solve: |x| < 5
Using the "Less Than" rule: -5 < x < 5. Ans: (-5, 5)
2. Solve: |x| ≥ 3
Using the "Greater Than" rule: x ≤ -3 OR x ≥ 3. Ans: (-∞, -3] ∪
6. Solve: |x| < -2
Absolute value represents distance and cannot be negative. Ans: No Solution
7. Solve: |x| > -1
Absolute value is always ≥ 0, so it is always > -1. Ans: All Real Numbers
8. Solve: |x - 5| ≤ 0
Absolute value cannot be < 0, only = 0. So x - 5 = 0. Ans: x = 5
9. Solve: |3x + 3| < 9
-9 < 3x + 3 < 9 → -12 < 3x < 6 → -4 < x < 2. Ans: (-4, 2)
10. Solve: |x/2| ≥ 1
x/2 ≤ -1 OR x/2 ≥ 1 → x ≤ -2 OR x ≥ 2. Ans: (-∞, -2] ∪
17. Solve: |x - 7| > -5
An absolute value is always ≥ 0, which is always > -5. Ans: R (All Reals)
18. Solve: |5x| < 0
Absolute value cannot be negative. Ans: No Solution
19. Solve: |x + 1| > 1
x + 1 < -1 OR x + 1 > 1 → x < -2 OR x > 0. Ans: (-∞, -2) ∪ (0, ∞)
20. Solve: |x - 2| ≤ 3
-3 ≤ x - 2 ≤ 3 → -1 ≤ x ≤ 5. Ans: [-1, 5]