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 \( x = -5 \).

Step2: Test intervals

The critical points \( x = -5 \) and \( x = 3 \) divide the number line into three intervals:

1. \( (-\infty, -5) \): Test \( x = -6 \). \( (-6 - 3)(-6 + 5) = (-9)(-1) = 9 > 0 \), so inequality fails.
2. \( (-5, 3) \): Test \( x = 0 \). \( (0 - 3)(0 + 5) = (-3)(5) = -15 \leq 0 \), so inequality holds.
3. \( (3, \infty) \): Test \( x = 4 \). \( (4 - 3)(4 + 5) = (1)(9) = 9 > 0 \), so inequality fails.

Step3: Include endpoints

Since the inequality is \( \leq 0 \), include \( x = -5 \) and \( x = 3 \) (closed circles).

Answer:

The solution is \( -5 \leq x \leq 3 \). To graph: Plot closed circles at \( x = -5 \) and \( x = 3 \), then shade the segment between them on the number line.