绝对值不等式juéduìzhí bùděngshì

Core Concept

An Absolute Value Inequality contains absolute value symbols. Solving requires case analysis based on the definition or using geometric meaning of absolute value.

Definition

$|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

Geometric Meaning

$|x|$ represents distance from point $x$ to origin on number line. $|x - a|$ represents distance from point $x$ to point $a$.

Basic Types

Type 1: $|x| < a$

$|x| < a \Leftrightarrow -a < x < a \quad (a > 0)$

Example: Solve $|x - 2| < 3$

Solution: $-3 < x - 2 < 3 \Rightarrow -1 < x < 5$

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Type 2: $|x| > a$

$|x| > a \Leftrightarrow x < -a \text{ or } x > a \quad (a > 0)$

Example: Solve $|2x + 1| > 5$

Solution: $2x + 1 < -5 \text{ or } 2x + 1 > 5$ $x < -3 \text{ or } x > 2$

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Type 3: Sum of Distances

$|x - a| + |x - b| \geq |a - b|$

Equality holds when $x$ is between $a$ and $b$.

Triangle Inequality

$|a + b| \leq |a| + |b|$

Equality holds when $ab \geq 0$.

Common Misconceptions

❌ Misconception 1: Wrong solution for $|x| < a$

Wrong: $|x| < 2 \Rightarrow x < -2$ or $x < 2$

Correct: $|x| < 2 \Rightarrow -2 < x < 2$

❌ Misconception 2: Union vs intersection

Wrong: $|x| > 2 \Rightarrow -2 < x < 2$

Correct: $|x| > 2 \Rightarrow x < -2$ or $x > 2$

Study Tips

1. ✅ Master basic formulas: $|x| < a$ and $|x| > a$
2. ✅ Understand geometry: Distance concept
3. ✅ Practice case analysis: Zero-point method
4. ✅ Remember triangle inequality: $|a + b| \leq |a| + |b|$

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💡 Exam Tip: Absolute value inequalities are high-frequency in CSCA exams!