## Ask Professor Puzzler

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How do you solve an inequality which looks like the following?

So we first need to find the points at which the different factors *change *sign, because those are the points at which the sign of the entire expression changes. We find these points of interest by setting each factor equal to zero (after all, the point at which the factor is zero is the point at which the factor is changing sign).

x + 2 = 0

x = -2

x - 3 = 0

x = 3

x = 0

x - 1 = 0

x = 1

So our points of interest are x = -2, x = 0, x = 1, and x = 3.

I listed those in ascending order; which will make it a little easier to finish up the problem.

If x < -2, then all the factors are negative, and the result is positive.

If -2 < x < 0, then x, (x - 1) and (x - 3) are negative, giving a negative result.

If 0 < x < 1, then (x - 1) and (x - 3) are negative, giving a positive result.

If 1 < x < 3, only (x - 3) is negative, giving a negative result.

If x > 3, none of the factors are negative, and the result is positive.

Thus, the solution is x < -2 or 0 < x < 1 or x > 3

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