Chapter A.9, Problem 103AYU

To determine

To express: The fact that $x$ differs from $-3$ by more than 2 as an inequality involving an absolute value. Solve for x.

Answer to Problem 103AYU

$\left\{x|x < -5 or x > -1\right\}\text{ or }\left(-\infty , -5\right) \cup \left(-1, \infty \right)$

Explanation of Solution

$\begin{array}{l}|x - \left(-3\right)| > 2\hfill \\ |x + 3| > 2\hfill \\ x + 3 < -2\text{ or }x + 3 > 2\hfill \\ x + 3 - 3 < -2 -3\text{ or }x + 3 - 3 > 2 - 3\hfill \\ x < -5\text{ or }x > -1\hfill \end{array}$

The solution set is $\left\{x|x < -5 or x > -1\right\}$ or using interval notation, all numbers in the interval $\left(-\infty , -5\right) \cup \left(-1, \infty \right)$.