Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set - Page 599: 62

$-2\lt x \lt4$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Solve the given inequality, $|x-1|\lt3 ,$ using the definition of a less than absolute value inequality. Then use the properties of inequality to isolate the variable. Finally, 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 $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the given inequality is equivalent to \begin{array}{l}\require{cancel} -3\lt x-1 \lt3 .\end{array} Using the properties of inequality to isolate the variable results to \begin{array}{l}\require{cancel} -3+1\lt x-1+1 \lt3+1 \\\\ -2\lt x \lt4 .\end{array} Hence, the solution set is $-2\lt x \lt4 .$

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