Inequalities — A/L Combined Mathematics

How do I practise Inequalities?

1 question with a worked solution

1. What is the complete set of real values of x satisfying the rational inequality (x - 1) / (x + 2) ≤ 0?
   1. A) x ≤ -2 or x > 1
   2. B) -2 < x ≤ 1
   3. C) -2 ≤ x ≤ 1
   4. D) x < -2 or x ≥ 1
   Check my answer

   Correct answer: B) -2 < x ≤ 1

   The critical points are x = 1 (numerator zero) and x = -2 (denominator zero, where expression is undefined). Testing sign intervals: for x < -2, (-)/(-) = +; for -2 < x < 1, (-)/(+) = -; for x > 1, (+)/(+) = +. The inequality is non-positive for x in (-2, 1]. Note x = -2 is excluded because denominator cannot be zero.

   Show worked solution

   Step 1: Critical points: x = 1, x = -2 (undefined). Step 2: Sign analysis on intervals: (-inf, -2) positive, (-2, 1) negative, (1, inf) positive. Step 3: At x = 1, expression is 0 (valid); at x = -2, undefined (excluded). Answer: -2 < x ≤ 1.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Inequalities fit in the syllabus?

Year 1 · Term 1