Question

solve for x and graph the solution.
(x - 1)(x + 5) ≥ 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: \(x - 1 = 0\) gives \(x = 1\); \(x + 5 = 0\) gives \(x = -5\). These are the critical points that divide the number line into intervals: \((-\infty, -5)\), \((-5, 1)\), and \((1, \infty)\).

Step2: Test intervals

• For \((-\infty, -5)\), let's pick \(x = -6\). Substitute into \((x - 1)(x + 5)\): \((-6 - 1)(-6 + 5)=(-7)(-1) = 7 \geq 0\), so this interval is part of the solution.
• For \((-5, 1)\), let's pick \(x = 0\). Substitute: \((0 - 1)(0 + 5)=(-1)(5) = -5 < 0\), so this interval is not part of the solution.
• For \((1, \infty)\), let's pick \(x = 2\). Substitute: \((2 - 1)(2 + 5)=(1)(7) = 7 \geq 0\), so this interval is part of the solution.

Step3: Include critical points

Since the inequality is \(\geq 0\), we include the critical points \(x = -5\) and \(x = 1\) (so the endpoints are closed).

Answer:

The solution to the inequality \((x - 1)(x + 5) \geq 0\) is \(x \leq -5\) or \(x \geq 1\). To graph this: plot closed circles at \(x = -5\) and \(x = 1\), draw a ray to the left of \(x = -5\) (including all points less than or equal to \(-5\)) and a ray to the right of \(x = 1\) (including all points greater than or equal to \(1\)).