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Solve the inequality x^2-x-12<0

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To solve the inequality x^2 - x - 12 < 0, we can use a method called factoring.

First, let's factor the quadratic expression x^2 - x - 12. We are looking for two numbers whose product is -12 and whose sum is -1 (the coefficient of the x term). The numbers -4 and 3 fit these criteria, so we can factor the expression as follows:

x^2 - x - 12 = (x - 4)(x + 3)

Now, we have the inequality in factored form: (x - 4)(x + 3) < 0.

To determine the solution, we need to consider the sign of the expression (x - 4)(x + 3) for different ranges of x.

1. When x < -3:
In this range, both factors (x - 4) and (x + 3) are negative, resulting in a positive product. Therefore, the inequality is not satisfied for x < -3.

2. When -3 < x < 4:
In this range, (x - 4) is negative, and (x + 3) is positive. A negative multiplied by a positive gives a negative product. Thus, the inequality is satisfied for -3 < x < 4.

3. When x > 4:
In this range, both factors (x - 4) and (x + 3) are positive, resulting in a positive product. Therefore, the inequality is not satisfied for x > 4.

Thus, the solution to the inequality x^2 - x - 12 < 0 is -3 < x < 4.
User Brian Topping
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