The rules for solving absolute value inequalities depend on whether the absolute value expression is less than or greater than a constant. Here's a breakdown:
Understanding Absolute Value Inequalities
Absolute value inequalities involve expressions with absolute value symbols, such as |x|, compared to a numerical value using inequality signs (<, >, ≤, or ≥). Solving these inequalities means finding the range of values for the variable that satisfies the inequality.
Rules for Solving Absolute Value Inequalities
According to the provided reference, the following rules apply:
1. Absolute Value Inequality with "Less Than"
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Form: |X| ≤ p (or |X| < p)
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Rule: Replace this inequality with the compound inequality -p ≤ X ≤ p (or -p < X < p). Then, solve this compound inequality as usual.
- This means X must be greater than or equal to -p AND less than or equal to p.
- This is an "and" situation and requires the solution to satisfy both sides of the compound inequality.
2. Absolute Value Inequality with "Greater Than"
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Form: |X| ≥ p (or |X| > p)
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Rule: Replace this inequality with the compound inequality X ≤ -p or X ≥ p (or X < -p or X > p). Solve each of these inequalities separately.
- This means X must be less than or equal to -p OR greater than or equal to p.
- This is an "or" situation, where the solution must satisfy at least one of the two individual inequalities.
Summary Table of Rules
| Inequality Type | Equivalent Compound Inequality |
|---|---|
| X | |
| X | |
| X | |
| X |
Examples
Example 1: Less Than
Solve: |x - 2| < 3
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Rewrite as a compound inequality: -3 < x - 2 < 3
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Add 2 to all parts: -3 + 2 < x - 2 + 2 < 3 + 2
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Simplify: -1 < x < 5
- The solution is all x values between -1 and 5.
Example 2: Greater Than
Solve: |2x + 1| ≥ 5
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Rewrite as a compound inequality: 2x + 1 ≤ -5 OR 2x + 1 ≥ 5
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Solve the first inequality:
- 2x + 1 ≤ -5
- 2x ≤ -6
- x ≤ -3
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Solve the second inequality:
- 2x + 1 ≥ 5
- 2x ≥ 4
- x ≥ 2
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Combine solutions: x ≤ -3 or x ≥ 2
- The solution is all x values less than or equal to -3 or greater than or equal to 2.
Key Takeaways
- "Less than" inequalities translate to a single compound inequality with "and" condition.
- "Greater than" inequalities translate to a compound inequality with an "or" condition.
- Solving these inequalities involves applying basic algebraic manipulation rules after converting from absolute value form.
