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CAT 2021 Slot 1 — QA Question 16

Domain & RangeEasy

f(x) = x2+2x-15x2-7x-18 is negative if and only if

Answer & solution

  • A

    X < -5 or 3 < x < 9

  • B

    -2 < x < 3 or x > 9

  • -5 < x < -2 or 3 < x <  9

  • D

    x < -5 or -2 < x < 3

Solution

Easy

Factor the numerator and denominator, find the four critical points where the expression is zero or undefined, and use a sign analysis across the number line. The fraction is negative where numerator and denominator have opposite signs.

1

Factor.

x2+2x−15=(x+5)(x−3)x2−7x−18=(x−9)(x+2)⇒ f(x)=(x+5)(x−3)(x−9)(x+2)\begin{aligned} &x^2+2x-15=(x+5)(x-3)\\ &x^2-7x-18=(x-9)(x+2)\\ &\Rightarrow\ f(x)=\frac{(x+5)(x-3)}{(x-9)(x+2)} \end{aligned}
2

Critical points (zeros of numerator and denominator), in order:

−5, −2, 3, 9\begin{aligned} &-5,\ -2,\ 3,\ 9 \end{aligned}
3

Sign analysis. All four factors are simple (odd multiplicity), so the sign of ff alternates between consecutive critical points. To the far right (x>9x>9) every factor is positive, so f>0f>0 there; alternate moving left.

\begin{aligned} &x>9:&&+\\ &3
4

Collect the negative intervals. f(x)<0f(x)<0 exactly on the two intervals marked −-.

\begin{aligned} &-5
-5
Need a hint?

Mark −5,−2,3,9-5,-2,3,9 on a line. Start positive on the rightmost region and flip the sign each time you cross a critical point (all are single roots). The "−-" regions are your answer.

CAT 2021 Slot 1 QA Q16: f(x) = x 2 + 2 x - 15 x 2 - 7 x - 18 is negative if and only if — Solution | TheCATExam