Question

solve for x and graph the solution.
4x² - 16x - 48 ≥ 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: Simplify the inequality

Divide both sides of \(4x^{2}-16x - 48\geq0\) by \(4\) to get \(x^{2}-4x - 12\geq0\).
Factor the quadratic: \(x^{2}-4x - 12=(x - 6)(x+ 2)\geq0\).

Step2: Find the critical points

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

• For \(x\lt - 2\), let \(x=-3\). Then \((-3 - 6)(-3 + 2)=(-9)\times(-1)=9\geq0\).
• For \(-2\lt x\lt6\), let \(x = 0\). Then \((0 - 6)(0 + 2)=(-6)\times2=-12\lt0\).
• For \(x\gt6\), let \(x = 7\). Then \((7 - 6)(7 + 2)=(1)\times9 = 9\geq0\).

Answer:

The solution of the inequality \(4x^{2}-16x - 48\geq0\) is \(x\leq - 2\) or \(x\geq6\). On the number - line, we plot closed circles (since the inequality includes equality) at \(x=-2\) and \(x = 6\). Then we draw rays: one to the left from \(x=-2\) (towards \(-\infty\)) and one to the right from \(x = 6\) (towards \(+\infty\)).