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.