Question

graph the solution to the following inequality on the number lin
$(x - 5)(x + 3) \leq 0$

Explanation:

Step1: Find the critical points

Set \((x - 5)(x + 3)=0\).
Using the zero - product property \(x-5 = 0\) gives \(x = 5\), and \(x+3=0\) gives \(x=-3\).

Step2: Test intervals

We have three intervals: \((-\infty,-3)\), \((-3,5)\), and \((5,\infty)\).

• For \(x=-4\) (in the interval \((-\infty,-3)\)): \((-4 - 5)(-4+3)=(-9)\times(-1)=9>0\).
• For \(x = 0\) (in the interval \((-3,5)\)): \((0 - 5)(0 + 3)=(-5)\times(3)=-15<0\).
• For \(x=6\) (in the interval \((5,\infty)\)): \((6 - 5)(6 + 3)=(1)\times(9)=9>0\).

Since the inequality is \((x - 5)(x + 3)\leq0\), the solution includes the interval where the product is non - positive and the critical points.

Answer:

The solution of the inequality \((x - 5)(x + 3)\leq0\) is \([-3,5]\). On the number line, we draw a solid circle at \(x=-3\) and \(x = 5\) (because the inequality includes equality) and shade the region between them.