Question

solve for x and graph the solution.
(x + 2)(x + 4) > 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 \((x + 2)(x + 4)=0\). Then \(x+2 = 0\) gives \(x=-2\), and \(x + 4=0\) gives \(x=-4\). These are the critical points that divide the number line into intervals: \((-\infty,-4)\), \((-4,-2)\), and \((-2,\infty)\).

Step2: Test intervals

• For \(x < -4\) (e.g., \(x=-5\)): \((-5 + 2)(-5 + 4)=(-3)(-1)=3>0\), so this interval is part of the solution.
• For \(-4 < x < -2\) (e.g., \(x=-3\)): \((-3 + 2)(-3 + 4)=(-1)(1)=-1<0\), so this interval is not part of the solution.
• For \(x > -2\) (e.g., \(x = 0\)): \((0 + 2)(0 + 4)=(2)(4)=8>0\), so this interval is part of the solution.

Since the inequality is strict (\(>\)), the endpoints \(x=-4\) and \(x=-2\) are not included.

Answer:

The solution for \(x\) is \(x < -4\) or \(x > -2\). To graph it, plot open circles at \(x=-4\) and \(x=-2\), then draw rays to the left of \(x=-4\) and to the right of \(x=-2\) on the number line.