All the values of x satisfying the inequality $\frac{1}{x+5} \leq \frac{1}{2 x-3}$ are

D) $x<-5$ or $\frac{3}{2}<x \leq 8$

CAT 2024 Slot 2 QA Question 16

Multiple choice (+3 / −1) · Algebra · Inequalities · Try it, then check the answer and solution below.

CAT 2024 Slot 2QAAlgebra • InequalitiesModerate

All the values of x satisfying the inequality

\frac{1}{x+5}

≤

\frac{1}{2 x-3}

$ are

Practice More Questions

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: D

Answer and solution

Answer: D) $x<-5$ or $\frac{3}{2}$ <x ≤ 8$

📌 Core Concept & Formula

To solve the inequality

\frac{1}{x+5}

≤

\frac{1}{2x-3}

, we use the method of combining fractions and analyzing the sign of the resulting expression. The critical points are found by setting the numerator and denominator to zero.

🔢 Step-by-Step Solution

Combine the fractions:

\frac{1}{x+5} - \frac{1}{2x-3} \leq 0

\frac{2x - 3 - (x + 5)}{(x+5)(2x-3)} \leq 0

\frac{x - 8}{(x+5)(2x-3)} \leq 0

Identify critical points:

Numerator zero:

x = 8

Denominator zero:

x = -5 , x = \frac{3}{2}

Test intervals:

x < -5

: Expression is negative.

- 5 < x <

\frac{3}{2}

$: Expression is positive.

\frac{3}{2} < x < 8

: Expression is negative.

x > 8

: Expression is positive.

Include critical points:

x = 8

is included.

x = -5

and \(x = \

CAT 2024 Slot 2 QA Question 16

Keep going

Related Inequalities questions