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Solving Fractional Exponent Inequality

CAT 2006 Slot 1 · Quantitative Ability · Medium · Quantitative Aptitude

This is a medium Quantitative Ability question from the CAT 2006 Slot 1 paper. It tests Quantitative Aptitude. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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What values of xx satisfy x2/3+x1/3−2≤0x^{2/3} + x^{1/3} - 2 \le 0 (xx is a real number)?

  1. A.

    -8 <= x <= 1

  2. B.

    -1 <= x <= 8

  3. C.

    1 < x < 8

  4. D.

    1 <= x <= 8

  5. E.

    -8 <= x <= 8

Answer

-8 <= x <= 1

Explanation

Let y=x1/3y = x^{1/3}. The inequality becomes: y2+y−2≤0  ⟹  (y+2)(y−1)≤0  ⟹  −2≤y≤1y^2 + y - 2 \le 0 \implies (y + 2)(y - 1) \le 0 \implies -2 \le y \le 1. Substituting y=x1/3y = x^{1/3}: −2≤x1/3≤1  ⟹  (−2)3≤x≤13  ⟹  −8≤x≤1-2 \le x^{1/3} \le 1 \implies (-2)^3 \le x \le 1^3 \implies -8 \le x \le 1.

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