Question

solve for x and graph the solution.
(x + 2)(x + 6) ≤ 0
plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
submit

Explanation:

Step1: Find critical points

Set each factor equal to zero:
\(x + 2 = 0 \implies x = -2\)
\(x + 6 = 0 \implies x = -6\)

Step2: Test intervals

The critical points \(x = -6\) and \(x = -2\) divide the number line into three intervals:

1. \(x < -6\): Test \(x = -7\). \((-7 + 2)(-7 + 6) = (-5)(-1) = 5 > 0\) (not a solution).
2. \(-6 < x < -2\): Test \(x = -4\). \((-4 + 2)(-4 + 6) = (-2)(2) = -4 \leq 0\) (solution).
3. \(x > -2\): Test \(x = 0\). \((0 + 2)(0 + 6) = 12 > 0\) (not a solution).

Step3: Include endpoints

Since the inequality is \(\leq 0\), the endpoints \(x = -6\) and \(x = -2\) are included.

Answer:

The solution is \(-6 \leq x \leq -2\). To graph it, plot closed circles at \(-6\) and \(-2\), then draw a line segment connecting them.