Question

1. please solve the absolute value inequality shown below and then show your solution on the number line provided.

$left|\frac{x}{2}-1
ight|\leq 2$

Explanation:

Step1: Apply absolute value inequality rule

For \(|A| \leq B\) (where \(B \geq 0\)), we have \(-B \leq A \leq B\). So for \(|\frac{x}{2}-1| \leq 2\), we get \(-2 \leq \frac{x}{2}-1 \leq 2\).

Step2: Solve left - hand inequality

Add 1 to all parts of the compound inequality. First, solve \(-2 \leq \frac{x}{2}-1\):
Add 1 to both sides: \(-2 + 1\leq\frac{x}{2}-1 + 1\), which simplifies to \(-1\leq\frac{x}{2}\).
Multiply both sides by 2: \(-1\times2\leq\frac{x}{2}\times2\), so \(-2\leq x\).

Step3: Solve right - hand inequality

Solve \(\frac{x}{2}-1\leq2\):
Add 1 to both sides: \(\frac{x}{2}-1 + 1\leq2 + 1\), which simplifies to \(\frac{x}{2}\leq3\).
Multiply both sides by 2: \(\frac{x}{2}\times2\leq3\times2\), so \(x\leq6\).

Answer:

The solution to the inequality \(|\frac{x}{2}-1|\leq2\) is \(-2\leq x\leq6\)