# Chapter 2 - Systems of Linear Equations and Inequalities - 2.5 Absolute Value Equations and Inequalities - 2.5 Exercises: 49

$p\le2 \text{ OR } p\ge6$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|p-4|\ge2 ,$ 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} p-4\ge2 \\\\\text{OR}\\\\ p-4\le-2 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} p-4\ge2 \\\\ p\ge2+4 \\\\ p\ge6 \\\\\text{OR}\\\\ p-4\le-2 \\\\ p\le-2+4 \\\\ p\le2 .\end{array} Hence, the solution set is $p\le2 \text{ OR } p\ge6 .$ The graph above confirms the solution set of the inequality.

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