Question

graph the solution to the following inequality on the number line.

$x^{2}+2x\leq8$

note that you can use the aleks graphing calculator to help get your answer.

Explanation:

Step1: Rearrange the inequality

$$x^{2}+2x - 8\leq0$$

Step2: Factor the quadratic expression

$$(x + 4)(x - 2)\leq0$$

Step3: Find the critical points

Set \(x + 4=0\), then \(x=-4\); set \(x - 2=0\), then \(x = 2\)

Step4: Test intervals

• For \(x<-4\) (e.g., \(x=-5\)), \((-5 + 4)(-5 - 2)=(-1)\times(-7)=7>0\)
• For \(-4
• For \(x>2\) (e.g., \(x=3\)), \((3 + 4)(3 - 2)=7\times1 = 7>0\)

Answer:

The solution of the inequality \(x^{2}+2x\leq8\) is \(-4\leq x\leq2\). On the number - line, we plot a closed circle at \(x=-4\) and \(x = 2\) (because the inequality includes equality) and shade the region between them.