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# CAT 2021 Quant Question [Slot 1] with Solution 15

Question
>$f(x) = \frac{{{x^2} + 2x - 15}}{{{x^2} - 7x - 18}}$ is negative if and only if
1. 5 < x < -2 or 3 < x < 9
2. x < -5 or -2 < x < 3
3. 2 < x < 3 or x > 9
4. x < -5 or 3 < x < 9
Option: 1
Solution:
$f(x) = \frac{{{x^2} + 2x - 15}}{{{x^2} - 7x - 18}}$<0
$\frac{{\left( {x + 5} \right)\left( {x - 3} \right)}}{{\left( {x - 9} \right)\left( {x + 2} \right)}} < 0$
We have four inflection points -5, -2, 3, and 9.
For x9, all four terms will be positive.
When x belongs to (-2,3), two terms are negative and two are positive. Hence, the overall expression is positive again.
We are left with the range (-5,-2) and (3,9) where the expression will be negative.
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