Intermediate Algebra (12th Edition)

$x=\left\{ -\dfrac{5}{3},3 \right\}$
Using the properties of equality, the given equation, $|3x-2|+1=8 ,$ is equivalent to \begin{array}{l}\require{cancel} |3x-2|=8-1 \\\\ |3x-2|=7 .\end{array} Since for any $a\gt0$, $|x|=a$ implies $x=a$ or $x=-a$, then the equation above is equivalent to \begin{array}{l}\require{cancel} 3x-2=7 \text{ OR } 3x-2=-7 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 3x-2=7 \\\\ 3x=7+2 \\\\ 3x=9 \\\\ x=\dfrac{9}{3} \\\\ x=3 \\\\\text{ OR }\\\\ 3x-2=-7 \\\\ 3x=-7+2 \\\\ 3x=-5 \\\\ x=-\dfrac{5}{3} .\end{array} Hence, the solutions are $x=\left\{ -\dfrac{5}{3},3 \right\} .$
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