## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x=\left\{ -11,11 \right\}$
$\bf{\text{Solution Outline:}}$ Given that $f(x)=|x|+7 ,$ to find $x$ for which $f(x)=18 ,$ use substitution. Then isolate the absolute value expression and use the definition of an absolute value equality. $\bf{\text{Solution Details:}}$ Replacing $f(x)$ with $18 ,$ then \begin{array}{l}\require{cancel} f(x)=|x|+7 \\\\ 18=|x|+7 \\\\ |x|+7=18 .\end{array} Using the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} |x|+7=18 \\\\ |x|=18-7 \\\\ |x|=11 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to \begin{array}{l}\require{cancel} x=11 \\\\\text{OR}\\\\ x=-11 .\end{array} Hence, $x=\left\{ -11,11 \right\} .$