Question

graph the solution to the following inequality on the number line.
$(x + 4)(x - 2) < 0$

Explanation:

Step1: Find the critical points

Set \((x + 4)(x - 2)=0\). Using the zero - product property \(x+4 = 0\) gives \(x=-4\), and \(x - 2=0\) gives \(x = 2\). These are the critical points that divide the number line into three intervals: \((-\infty,-4)\), \((-4,2)\), and \((2,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,-4)\), let \(x=-5\). Then \((-5 + 4)(-5 - 2)=(-1)\times(-7)=7>0\).
• For the interval \((-4,2)\), let \(x = 0\). Then \((0 + 4)(0 - 2)=(4)\times(-2)=-8<0\).
• For the interval \((2,\infty)\), let \(x=3\). Then \((3 + 4)(3 - 2)=(7)\times(1)=7>0\).

Step3: Determine the solution

Since the inequality \((x + 4)(x - 2)<0\) is satisfied when \(x\in(-4,2)\). On the number line, we use open circles at \(x=-4\) and \(x = 2\) (because the inequality is strict, i.e., \(<\) not \(\leq\)) and shade the region between them.

Answer:

We place open circles at \(x=-4\) and \(x = 2\) on the number line and shade the region between \(-4\) and \(2\).