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

Explanation:

Step1: Find the critical points

Set \((x + 3)(x + 5)=0\).
Using the zero - product property \(a\times b = 0\Rightarrow a = 0\) or \(b = 0\).
If \(x+3 = 0\), then \(x=-3\). If \(x + 5=0\), then \(x=-5\).

Step2: Test intervals

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

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

Step3: Determine the solution set

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

Answer:

The solution of the inequality \((x + 3)(x + 5)\leq0\) is \(x\in[-5,-3]\). On the number line, we plot a closed circle at \(x=-5\) and \(x = - 3\) and shade the segment between them.