CAT 2024

Slot 2

QA

Question & Solution

There are two critical points for the inequality to consider: $x=-5$ and $x=\frac{3}{2}$

Region I: $x > \frac{3}{2}$
In this scenario, both the terms would be positive; cross-multiplying, we get the relation $2x-3 \le x+5$
Giving the boundary $x \le 8$, hence giving us the valid range as $\frac{3}{2} < x \le 8$

Region II: $-5 < x < \frac{3}{2}$
In this case, the right-hand side will be a negative value, and hence, the sign would change when multiplying, giving the inequality
$2x-3 \ge x+5$
Which will give $x > 8$, which is out of bounds for this region

Another way is to put a value in the region to check for the validity of the inequality; by putting $x=0$, we could see that the inequality does not hold in this region

Region III: $x < -5$
In this scenario, both the terms are negative, essentially giving us the same boundary as region 1; we take the lower bounds, giving us that x has to be less than 5

Therefore, for the given inequality to hold true $x < -5$ or $\frac{3}{2} < x \le 8$

Hence, Option A is the correct answer.