Chapter 1.2, Problem 4QR

To determine

To find:The solution of the inequality $\left|x-2\right|\ge 5$ .

Answer to Problem 4QR

The solution of the inequality $\left|x-2\right|\ge 5$ is the interval $\left(-\infty ,-3\right]\cup \left[7,\infty \right)$ .

Explanation of Solution

Given information:The inequality is $\left|x-2\right|\ge 5$ .

Calculation:

Two cases arise for the inequality $\left|x-2\right|\ge 5$ . First case is $x-2\le -5$ and second case is $x-2\ge 5$ .

First take the inequality $x-2\le -5$ and simplify.

  $\begin{array}{c}x-2\le -5\\ x-2+2\le -5+2\\ x\le -3\end{array}$

So, the solution of the inequality $x-2\le -5$ is the interval $\left(-\infty ,-3\right]$ .

Now, take the inequality $x-2\ge 5$ and simplify.

  $\begin{array}{c}x-2\ge 5\\ x-2+2\ge 5+2\\ x\ge 7\end{array}$

So, the solution of the inequality $x-2\ge 5$ is the interval $\left[7,\infty \right)$ .

The solution of the inequality $\left|x-2\right|\ge 5$ is the union of boththe intervals.

Therefore, the solution of the inequality $\left|x-2\right|\ge 5$ is the interval $\left(-\infty ,-3\right]\cup \left[7,\infty \right)$ .