Question

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

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x + 3 = 0\) gives \(x = -3\); \(x - 5 = 0\) gives \(x = 5\). These are the critical points that divide the number line into intervals: \((-\infty, -3)\), \((-3, 5)\), and \((5, \infty)\).

Step2: Test intervals

• For interval \((-\infty, -3)\), pick \(x = -4\). Substitute into \((x + 3)(x - 5)\): \((-4 + 3)(-4 - 5)=(-1)(-9)=9>0\), so this interval is part of the solution.
• For interval \((-3, 5)\), pick \(x = 0\). Substitute: \((0 + 3)(0 - 5)=(3)(-5)= -15<0\), so this interval is not part of the solution.
• For interval \((5, \infty)\), pick \(x = 6\). Substitute: \((6 + 3)(6 - 5)=(9)(1)=9>0\), so this interval is part of the solution.

Step3: Determine solution

Since the inequality is strict (\(>\) not \(\geq\)), the endpoints \(x = -3\) and \(x = 5\) are not included. So the solution is \(x < -3\) or \(x > 5\).

Answer:

The solution for \(x\) is \(x < -3\) or \(x > 5\). To graph it, plot open circles at \(x = -3\) and \(x = 5\), then draw rays to the left of \(-3\) and to the right of \(5\) on the number line.