Solving Rational Inequality

CAT 2024 Slot 2 · QA · Medium · Algebra

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All the values of xx satisfying the inequality 1x+5≤12x−3\frac{1}{x+5} \le \frac{1}{2x-3} are

  1. A.

    x < -5 \text{ or } x > \frac{3}{2}

  2. B.

    x < -5 \text{ or } \frac{3}{2} < x \le 8

  3. C.

    -5 < x < \frac{3}{2} \text{ or } \frac{3}{2} < x \le 8

  4. D.

    -5 < x < \frac{3}{2} \text{ or } x > \frac{3}{2}

Answer

B

Explanation

Rearranging the inequality: 1x+5−12x−3≤0\frac{1}{x+5} - \frac{1}{2x-3} \le 0 (2x−3)−(x+5)(x+5)(2x−3)≤0  ⟹  x−8(x+5)(2x−3)≤0\frac{(2x-3) - (x+5)}{(x+5)(2x-3)} \le 0 \implies \frac{x - 8}{(x+5)(2x-3)} \le 0

Critical points are x=−5,32,8x = -5, \frac{3}{2}, 8. Testing intervals:

  • x<−5x < -5: (−)(−)(−)=−≤0\frac{(-)}{(-)(-)} = - \le 0 (Satisfied)
  • −5<x<32-5 < x < \frac{3}{2}: (−)(+)(−)=+>0\frac{(-)}{(+)(-)} = + > 0 (Not satisfied)
  • 32<x≤8\frac{3}{2} < x \le 8: (−)(+)(+)=−≤0\frac{(-)}{(+)(+)} = - \le 0 (Satisfied)
  • x>8x > 8: (+)(+)(+)=+>0\frac{(+)}{(+)(+)} = + > 0 (Not satisfied)

Note that x≠−5x \ne -5 and x≠32x \ne \frac{3}{2} due to division by zero. Hence, x<−5x < -5 or 32<x≤8\frac{3}{2} < x \le 8.

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