Question

solve for x and graph the solution.
$(x + 1)(x + 6) \geq 0$
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.
(number line from -6 to 6 with ticks at each integer)
submit

Explanation:

Step1: Find the critical points

Set \((x + 1)(x+6)=0\). Using the zero - product property \(x+1 = 0\) gives \(x=-1\), and \(x + 6=0\) gives \(x=-6\).

Step2: Test intervals

• Interval 1: \(x\lt - 6\)

Let \(x=-7\). Then \((-7 + 1)(-7+6)=(-6)\times(-1)=6\geq0\).

• Interval 2: \(-6\lt x\lt - 1\)

Let \(x=-3\). Then \((-3 + 1)(-3+6)=(-2)\times3=-6\lt0\).

• Interval 3: \(x\gt - 1\)

Let \(x=0\). Then \((0 + 1)(0+6)=1\times6 = 6\geq0\).

Step3: Consider the endpoints

Since the inequality is \(\geq\), when \(x=-6\), \((-6 + 1)(-6+6)=0\); when \(x=-1\), \((-1 + 1)(-1+6)=0\). Both endpoints satisfy the inequality.

Answer:

The solution of the inequality \((x + 1)(x + 6)\geq0\) is \(x\leq-6\) or \(x\geq-1\). On the number - line, we plot closed circles at \(x=-6\) and \(x=-1\), and shade the regions to the left of \(x =-6\) and to the right of \(x=-1\).