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1.2.2 Compound Inequalities

Solve compound inequalities using 'and' and 'or' connectives, including intersection and union of solution sets.

定义

A compound inequality is a statement that combines two or more inequalities using logical connectives 'and' or 'or'. When two inequalities are connected by 'and', the solution set is the intersection of the individual solution sets, meaning a value must satisfy both inequalities simultaneously. When connected by 'or', the solution set is the union of the individual solution sets, meaning a value must satisfy at least one of the inequalities. For example, \(a < x < b\) (read as "\(x\) is between \(a\) and \(b\)") is a compound inequality with 'and', equivalent to \(x > a\) AND \(x < b\). The solution to a compound inequality can be expressed in interval notation, set-builder notation, or graphed on a number line.

核心公式

  • \(a < x < b \text{ is equivalent to } (x > a) \text{ AND } (x < b)\)
  • \(x < a \text{ OR } x > b \text{ represents the union of two disjoint intervals: } (-\infty, a) \cup (b, \infty)\)
  • \(a \leq x \leq b \text{ in interval notation is } [a, b]\)
  • \(a < x < b \text{ in interval notation is } (a, b)\)
  • \(\text{For 'and' inequalities: solution set} = S_1 \cap S_2; \text{ For 'or' inequalities: solution set} = S_1 \cup S_2\)

易错点

  • ⚠️ Confusing 'and' with 'or': Students often write \(x > 3\) OR \(x < 5\) when they mean \(x > 3\) AND \(x < 5\). Remember that 'and' means both conditions must be true simultaneously (intersection), while 'or' means at least one condition is true (union).
  • ⚠️ Incorrectly flipping inequality signs when multiplying/dividing by negative numbers in compound inequalities: When solving \(-2x > 6\) AND \(-2x < 10\), students must flip both inequality signs to get \(x < -3\) AND \(x > -5\), which is equivalent to \(-5 < x < -3\).
  • ⚠️ Misinterpreting the solution set for 'or' inequalities: For \(x < 2\) OR \(x > 5\), the solution is NOT a single interval but two separate intervals \((-\infty, 2) \cup (5, \infty)\). Students often incorrectly try to write this as a single compound inequality.
  • ⚠️ Forgetting to apply operations to all parts of a compound inequality: When solving \(2 < 3x + 1 < 7\), students must subtract 1 from all three parts to get \(1 < 3x < 6\), then divide all three parts by 3 to get \(\frac{1}{3} < x < 2\). Applying operations to only some parts leads to incorrect solutions.