1.2.4 Absolute Value Inequalities

Solve inequalities involving absolute values and interpret their geometric meaning on the number line.

定义

An absolute value inequality is an inequality that contains an absolute value expression. The absolute value of a number \(x\), denoted \(|x|\), represents the distance from \(x\) to zero on the number line, always non-negative. When solving absolute value inequalities, we must consider two cases based on whether the expression inside the absolute value is positive or negative. For an inequality of the form \(|A| < B\) (where \(B > 0\)), this is equivalent to the compound inequality \(-B < A < B\). For an inequality of the form \(|A| > B\) (where \(B > 0\)), this is equivalent to \(A < -B\) or \(A > B\). The geometric interpretation shows that \(|x - a| < r\) represents all points within distance \(r\) from point \(a\) on the number line, forming an open interval \((a-r, a+r)\), while \(|x - a| > r\) represents all points more than distance \(r\) away from point \(a\), forming the union of two rays \((-\infty, a-r) \cup (a+r, \infty)\).

核心公式

• \(|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\)
• \(|A| < B \iff -B < A < B\) (where \(B > 0\))
• \(|A| > B \iff A < -B \text{ or } A > B\) (where \(B > 0\))
• \(|x - a| < r \iff a - r < x < a + r\) (where \(r > 0\))
• \(|x - a| > r \iff x < a - r \text{ or } x > a + r\) (where \(r > 0\))

易错点

• ⚠️ Forgetting to flip the inequality sign or create a compound inequality: Students often treat \(|A| < B\) as a single inequality \(A < B\) instead of recognizing it as \(-B < A < B\), missing half of the solution set.
• ⚠️ Incorrectly handling the case when the absolute value expression is greater than a negative number: Students may fail to recognize that \(|A| > -5\) is always true for any value of \(A\) (since absolute values are always non-negative), leading to incorrect 'all real numbers' conclusions without proper justification.
• ⚠️ Misinterpreting the geometric meaning: Students struggle to connect the algebraic solution to the number line, failing to visualize that \(|x - 3| < 2\) represents an interval centered at 3 with radius 2, rather than two separate regions.
• ⚠️ Making errors with boundary points: Students often incorrectly include or exclude endpoints when dealing with strict versus non-strict inequalities (e.g., confusing \(|x| < 5\) with \(|x| \leq 5\)), resulting in incorrect interval notation with wrong bracket types.