Problem 94 - College Algebra Essentials

Question

Solve each absolute value inequality. $$\left|2-\frac{x}{2}\right|-1 \leq 1$$

Step-by-Step Solution

Answer

The solution to the absolute value inequality \(\left|2-\frac{x}{2}\right|-1 \leq 1\) is \(x \geq 0\).

Step 1: Isolate the absolute value

The first step is to isolate the absolute value portion of the equation, which is \(\left|2-\frac{x}{2}\right|\). To do so, add after both sides of the equation, which will remove the -1. The modified equation is now \(\left|2-\frac{x}{2}\right| \leq 2\).

Step 2: Create a compound inequality

The inequality \(\left| Y \right| \leq a\) can also be written as \(-a \leq Y \leq a\). Replacing Y with \(2-\frac{x}{2}\), and a with 2, we can rewrite the inequality as \(-2 \leq 2-\frac{x}{2} \leq 2\). This turns our absolute value inequality into a compound inequality.

Step 3: Solve the compound inequality

Now we need to solve this compound inequality. Starting with the left-most inequality \(-2 \leq 2-\frac{x}{2}\), add \(\frac{x}{2}\) to both sides to get \(\frac{x}{2} \geq -2 + 2\), or \(\frac{x}{2} \geq 0\). Multiply both sides by 2 to solve for x, leaving us with \(x \geq 0\). Similarly, for the right-most inequality \(2-\frac{x}{2} \leq 2\), subtract 2 from both sides to get \(-\frac{x}{2} \leq 0\). Multiplying by -2 (and keeping in mind that multiplying or dividing by a negative number flips the inequality sign) results in \(x \geq 0\).

Step 4: Combine solutions

The solutions for both parts of the compound inequality are the same, \(x \geq 0\). Therefore, the final solution to the inequality is \(x \geq 0\)