Intermediate Algebra (12th Edition)

$x=\left\{ -\dfrac{5}{3},\dfrac{11}{3} \right\}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $|3(x-1)|=8 ,$ use the Distributive Property and the definition of absolute value equality. Then use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the Distributive Property, the given equation is equivalent to \begin{array}{l}\require{cancel} |3(x)+3(-1)|=8 \\\\ |3x-3|=8 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to \begin{array}{l}\require{cancel} 3x-3=8 \\\\\text{OR}\\\\ 3x-3=-8 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 3x-3=8 \\\\ 3x=8+3 \\\\ 3x=11 \\\\ x=\dfrac{11}{3} \\\\\text{OR}\\\\ 3x-3=-8 \\\\ 3x=-8+3 \\\\ 3x=-5 \\\\ x=-\dfrac{5}{3} .\end{array} Hence, $x=\left\{ -\dfrac{5}{3},\dfrac{11}{3} \right\} .$