Sign of Rational Function

CAT 2021 Slot 2 · QA · Easy · Functions

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f(x)=x2+2x−15x2−7x−18f(x) = \frac{x^2 + 2x - 15}{x^2 - 7x - 18} is negative if and only if

  1. A.

    -5 < x < -2 or 3 < x < 9

  2. B.

    -2 < x < 3 or x > 9

  3. C.

    x < -5 or 3 < x < 9

  4. D.

    x < -5 or -2 < x < 3

Answer

A

Explanation

Factoring numerator and denominator: f(x)=(x+5)(x−3)(x−9)(x+2)f(x) = \frac{(x+5)(x-3)}{(x-9)(x+2)}

Critical points in increasing order are −5,−2,3,9-5, -2, 3, 9. Checking signs on intervals:

  • For x>9x > 9: Positive
  • For 3<x<93 < x < 9: Negative
  • For −2<x<3-2 < x < 3: Positive
  • For −5<x<−2-5 < x < -2: Negative
  • For x<−5x < -5: Positive

Thus, f(x)<0f(x) < 0 when −5<x<−2-5 < x < -2 or 3<x<93 < x < 9.

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