# Chapter 2 - Systems of Linear Equations and Inequalities - 2.5 Absolute Value Equations and Inequalities - 2.5 Exercises: 50

$m\le2 \text{ OR } m\ge4$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|m-3|\ge1 ,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} m-3\ge1 \\\\\text{OR}\\\\ m-3\le-1 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} m-3\ge1 \\\\ m\ge1+3 \\\\ m\ge4 \\\\\text{OR}\\\\ m-3\le-1 \\\\ m\le-1+3 \\\\ m\le2 .\end{array} Hence, the solution set is $m\le2 \text{ OR } m\ge4 .$ The graph above confirms the solution set of the inequality.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.