Question

solve for x and graph the solution.

x² + 4x - 12 ≥ 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.

Explanation:

Step1: Factor the quadratic expression

$$x^{2}+4x - 12=(x + 6)(x - 2)$$
So the inequality becomes \((x + 6)(x - 2)\geq0\)

Step2: Find the roots

Set \((x + 6)(x - 2)=0\). Using the zero - product property \(x+6 = 0\) gives \(x=-6\), and \(x - 2=0\) gives \(x = 2\)

Step3: Test intervals

• Interval 1: \(x<-6\)

Let \(x=-7\). Then \((-7 + 6)(-7 - 2)=(-1)\times(-9)=9\geq0\)

• Interval 2: \(-6

Let \(x = 0\). Then \((0 + 6)(0 - 2)=(6)\times(-2)=-12<0\)

• Interval 3: \(x>2\)

Let \(x=3\). Then \((3 + 6)(3 - 2)=(9)\times(1)=9\geq0\)

Answer:

The solution of the inequality \(x^{2}+4x - 12\geq0\) is \(x\leq - 6\) or \(x\geq2\). On the number - line, we plot closed circles (since the inequality is \(\geq\)) at \(x=-6\) and \(x = 2\). Then we draw a ray to the left of \(x=-6\) (including \(x=-6\)) and a ray to the right of \(x = 2\) (including \(x = 2\))