2.1.2 Absolute Value Inequalities¶

Solving and graphing inequalities with absolute value, including compound inequalities and interval notation.

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 yielding a non-negative result. 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 solution set can be expressed using interval notation, set-builder notation, or graphically on a number line. When \(B \leq 0\), special cases apply: \(|A| < B\) has no solution if \(B \leq 0\), while \(|A| > B\) is true for all real numbers if \(B < 0\), or all \(A \neq 0\) if \(B = 0\).

核心公式¶

\(|A| < B \iff -B < A < B\) (where \(B > 0\))
\(|A| > B \iff A < -B \text{ or } A > B\) (where \(B > 0\))
\(|A| \leq B \iff -B \leq A \leq B\) (where \(B \geq 0\))
\(|A| \geq B \iff A \leq -B \text{ or } A \geq B\) (where \(B \geq 0\))
\(|A - B| < C \iff B - C < A < B + C\) (distance interpretation: \(A\) is within distance \(C\) from \(B\))

易错点¶

⚠️ Forgetting to reverse the inequality sign or create a compound inequality: Students often write \(|x - 3| > 5\) as \(x - 3 > 5\) only, missing the second case \(x - 3 < -5\), resulting in an incomplete solution set.
⚠️ Incorrectly handling negative bounds: When solving \(|x| < -2\), students may not recognize this has no solution, or when solving \(|x| > -3\), they fail to recognize this is true for all real numbers.
⚠️ Misinterpreting the 'and' vs 'or' logic: For \(|A| < B\), the solution uses 'and' (compound inequality), but for \(|A| > B\), the solution uses 'or'. Students frequently confuse these logical connectors.
⚠️ Errors in converting to interval notation: Students may write the solution to \(|x + 2| \leq 3\) as \([-5, 1]\) correctly but then make mistakes when the inequality involves fractions or when endpoints should be excluded (using parentheses vs brackets).