Question

solve for x.
(x + 1)(x - 1) ≥ 0
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

Explanation:

Step1: Analyze the critical points

First, find the values of \(x\) that make \((x + 1)(x - 1)=0\). Solving \(x + 1 = 0\) gives \(x=-1\), and solving \(x - 1 = 0\) gives \(x = 1\). These are the critical points that divide the number line into intervals.

Step2: Test intervals

The intervals to test are \((-\infty, - 1)\), \((-1,1)\), and \((1,\infty)\).

• For \(x<-1\) (e.g., \(x=-2\)): \((-2 + 1)(-2 - 1)=(-1)\times(-3)=3>0\), so the inequality holds here.
• For \(-1
• For \(x>1\) (e.g., \(x = 2\)): \((2 + 1)(2 - 1)=(3)\times(1)=3>0\), so the inequality holds here.

Step3: Include critical points

Since the inequality is \(\geq0\), we include the points where \((x + 1)(x - 1)=0\), which are \(x=-1\) and \(x = 1\).

Answer:

\(x\leq - 1\) or \(x\geq1\)