# Does the inequation (x - 4)(x + 2)(x + 10) <= 0 have any integral solutions.

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The integral solutions of (x - 4)(x + 2)(x + 10) <= 0 need to be determined.

(x - 4)(x + 2)(x + 10) <= 0 if any one of the terms within the brackets is negative or if all the three terms within the brackets is negative.

(x - 4)<= 0, (x + 2)>= 0 and (x + 10) >= 0

=> x <= 4, x >= -2 and x >= -10

The values of x that satisfy this lie in [-2, 4]

(x - 4) >= 0, (x + 2) <= 0 and (x + 10) >= 0

=> x >= 4, x <= -2 and x >= -10

This does not have any solutions as x cannot be less than -2 and greater than 4 at the same time.

(x - 4) >= 0, (x + 2) >= 0 and (x + 10) <= 0

=> x >= 4, x >= -2 and x <= -10

This also does not have any solutions as x cannot be less than -10 and greater than 4 at the same time.

(x - 4) <= 0, (x + 2) <= 0 and (x + 10) <= 0

=> x <= 4, x <= -2 and x <= -10

All values that lie in (-inf., -10] satisfy these conditions

**It is not possible to write the integral values of x that satisfy the inequality individually but they should lie in (-inf., -10]U[-2, 4]**