Question

solve for x and graph the solution. (x + 2)(x + 3) ≥ 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

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x + 2 = 0\) gives \(x = -2\); \(x + 3 = 0\) gives \(x = -3\). These are the critical points that divide the number line into intervals.

Step2: Test intervals

• For \(x < -3\) (e.g., \(x = -4\)): \((-4 + 2)(-4 + 3)=(-2)(-1)=2\geq0\), so this interval is part of the solution.
• For \(-3 < x < -2\) (e.g., \(x = -2.5\)): \((-2.5 + 2)(-2.5 + 3)=(-0.5)(0.5)= -0.25<0\), not part of the solution.
• For \(x > -2\) (e.g., \(x = 0\)): \((0 + 2)(0 + 3)=(2)(3)=6\geq0\), so this interval is part of the solution.

Step3: Include critical points

Since the inequality is \(\geq0\), the critical points \(x = -3\) and \(x = -2\) are included.

Answer:

The solution to the inequality \((x + 2)(x + 3)\geq0\) is \(x\leq - 3\) or \(x\geq - 2\). To graph this: plot closed circles at \(x=-3\) and \(x = -2\), draw a ray to the left of \(x=-3\) (including \(x=-3\)) and a ray to the right of \(x=-2\) (including \(x=-2\)), and leave the segment between \(-3\) and \(-2\) unshaded (or deleted as per the graphing instructions).