What is the solution to (x - 5)(x + 2)(x - 7) <0?

For x < -2  f(x) < 0.

For   5 < x < 7 , f(x) < 0.

In interval notation it is:

(∞, -2) ∪ ( 5, 7).

Step-by-step explanation:

First find the critical values of x:

f(x) = (x - 5)(x + 2)(x - 7) = 0

x = 5, -2  and 7.

(x - 5)(x + 2)(x - 7) < 0

Consider x = -2:

When x < -2 the sign of the expression is - * - * - = -  

and it is 0 for x = -2

So for x < -2,  f(x) < 0.

For:

-2  < x < 5  the sign of f(x) is - * + * -  = +

So for  -2  < x < 5,  f(x) > 0

For:

5 < x < 7  the sign of f(x) is + * + * - = -

So for 5 < x < 7 , f(x) < 0.

For x > 7,   f(x) > 0.

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