Question 59

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


$$f(x) = \frac{x^2 + 2x - 15}{x^2 - 7x - 18}$$<0


We have four inflection points -5, -2, 3, and 9.

For x<-5, all four terms (x+5), (x-3), (x-9), (x+2) will be negative. Hence, the overall expression will be positive. Similarly, when x>9, 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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