Solve the inequality x^2 + 7x + 12 < 0.
To solve the inequality x^2 + 7x + 12 < 0, we can use a method called factoring.
Step 1: Rewrite the inequality.
x^2 + 7x + 12 < 0
Step 2: Factor the quadratic expression on the left side.
(x + 3)(x + 4) < 0
The factors of the quadratic expression are (x + 3) and (x + 4).
Step 3: Determine the sign of each factor.
To determine the sign of each factor, we need to find the values of x that make each factor equal to zero.
(x + 3) = 0 --> x = -3
(x + 4) = 0 --> x = -4
Step 4: Create a sign chart.
We will create a sign chart using the critical values (-3 and -4).
-4 -3 0
+ - + +
Step 5: Analyze the sign chart.
From the sign chart, we can see that the expression (x + 3) is positive for x < -3 and negative for x > -3. Similarly, the expression (x + 4) is positive for x < -4 and negative for x > -4.
Step 6: Determine the solution.
To find the values of x that satisfy the inequality, we want to find the intervals where the expression (x + 3)(x + 4) is negative.
From the sign chart, we can see that the expression is negative between -4 and -3:
-4 < x < -3
Therefore, the solution to the inequality x^2 + 7x + 12 < 0 is -4 < x < -3.