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Triangle Inequality Equality Condition

CAT 2017 Slot 2 · Quantitative Ability · Hard · Functions

This is a hard Quantitative Ability question from the CAT 2017 Slot 2 paper. It tests Functions. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Let f(x)=2x–5f(x) = 2x – 5 and g(x)=7–2xg(x) = 7 – 2x. Then ∣f(x)+g(x)∣=∣f(x)∣+∣g(x)∣|f(x) + g(x)| = |f(x)| + |g(x)| if and only if

  1. A.

    52<x<72\frac{5}{2} < x < \frac{7}{2}

  2. B.

    x≤52x \le \frac{5}{2} or x≥72x \ge \frac{7}{2}

  3. C.

    x<52x < \frac{5}{2} or x≥72x \ge \frac{7}{2}

  4. D.

    52≤x≤72\frac{5}{2} \le x \le \frac{7}{2}

Answer

D

Explanation

The property ∣A+B∣=∣A∣+∣B∣|A + B| = |A| + |B| holds if and only if AA and BB have the same sign, i.e., A⋅B≥0A \cdot B \ge 0.

Here A=f(x)=2x−5A = f(x) = 2x - 5 and B=g(x)=7−2xB = g(x) = 7 - 2x.

(2x−5)(7−2x)≥0(2x - 5)(7 - 2x) \ge 0 −(2x−5)(2x−7)≥0-(2x - 5)(2x - 7) \ge 0 (2x−5)(2x−7)≤0(2x - 5)(2x - 7) \le 0

Thus, 52≤x≤72\frac{5}{2} \le x \le \frac{7}{2}.

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