CAT 2021

Slot 1

QA

Question & Solution

1. Concept Used

• Topic: Rational Inequalities — Wavy Curve / Sign Chart Method
• Formula: $$\frac{(x - a)(x - b)}{(x - c)(x - d)} < 0 \implies \text{Use sign chart across critical points } a, b, c, d$$

2. Calculation

We need to find where (f(x) < 0), i.e., where the expression is strictly negative.

First, factorize both the numerator and the denominator:

$$\text{Numerator: } x^2 + 2x - 15 = (x + 5)(x - 3)$$

$$\text{Denominator: } x^2 - 7x - 18 = (x - 9)(x + 2)$$

So the inequality becomes:

$$\frac{(x+5)(x-3)}{(x-9)(x+2)} < 0$$

The four critical points (where any factor equals zero) are: (x = -5,\ x = -2,\ x = 3,\ x = 9). Note that (x = -2) and (x = 9) are excluded since they make the denominator zero.

Now arrange the critical points on a number line and apply the Wavy Curve Method. In each interval, count the sign of each linear factor:

Interval (x < -5): All four factors ((x+5),\ (x-3),\ (x-9),\ (x+2)) are negative. Product of four negatives = positive. ❌

Interval (-5 < x < -2): ((x+5) > 0), while ((x-3),\ (x-9),\ (x+2)) are all negative. That gives ((+)(-)(-)(-) = \frac{(+)(-)}{(-)(-))} = \frac{(-)}{(+)} = ) negative. ✅

Interval (-2 < x < 3): ((x+5) > 0,\ (x+2) > 0), while ((x-3) < 0,\ (x-9) < 0). Result: (\frac{(+)(-)}{(-)(+)} = \frac{(-)}{(-)} = ) positive. ❌

Interval (3 < x < 9): ((x+5) > 0,\ (x+2) > 0,\ (x-3) > 0), while ((x-9) < 0). Result: (\frac{(+)(+)}{(-)(+)} = \frac{(+)}{(-)} = ) negative. ✅

Interval (x > 9): All four factors are positive. Result: positive. ❌

Thus, the expression is negative for (x \in (-5,\ -2) \cup (3,\ 9)), i.e., (-5 < x < -2) or (3 < x < 9).

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3. Solution

Answer = Option A ✅

The final answer is (-5 < x < -2) or (3 < x < 9).