# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 55

$\left( -\infty, -5 \right) \cup \left( 13,\infty \right)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|-4+x| \gt 9 ,$ use the definition of absolute value inequalities. Then use the properties of inequality to isolate the variable. For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$ For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ 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} -4+x \gt 9 \\\\\text{OR}\\\\ -4+x \lt -9 .\end{array} Using the properties of inequality to isolate the variable results to \begin{array}{l}\require{cancel} -4+x \gt 9 \\\\ x \gt 9+4 \\\\ x \gt 13 \\\\\text{OR}\\\\ -4+x \lt -9 \\\\ x \lt -9+4 \\\\ x \lt -5 .\end{array} In interval notation, the solution set is $\left( -\infty, -5 \right) \cup \left( 13,\infty \right) .$ The colored graph is the graph of the solution set.

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