Posted by **Dee Williams** on Tuesday, November 7, 2006 at 9:08pm.

Solve Rational Inequalities

x^2-x-2/x^2+5x+6<0

I cant figure this out. Please help

First try to factor whenever you can to determine critical solutions:

(x - 2)(x + 1) / (x + 3)(x + 2) < 0

Critical solutions:

x = 2

x = -1

x = -3

x = -2

Test 5 intervals using a number line:

<==== -3 == -2 == -1 === 0 ====== 2 ===>

x < -3 ...let's choose -4. If we choose -4 for x, the result will be positive and not less than 0.

-3 < x < -2 ...let's choose -5/2. If we choose -5/2 for x, the result will be negative and less than 0.

-2 < x < -1 ...let's choose -3/2. If we choose -3/2 for x, the result will be negative and less than 0.

-1 < x < 2 ...let's choose 0. If we choose 0 for x, the result will be negative and less than 0.

x > 2 ...let's choose 3. If we choose 3 for x, the result will be positive and not less than 0.

Three of the tests work. If you are familiar with interval notation, the solution set can be written this way:

(-3,-2)U(-2,-1)U(-1,2) -->U = union of sets.

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