## Intermediate Algebra: Connecting Concepts through Application

$m=\left\{ -4,-2 \right\}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $-4|m+3|+7=3 ,$ isolate first the absolute value expression. Then use the definition of absolute value equality. Do checking of the solution/s. $\bf{\text{Solution Details:}}$ Using the properties of equality, the given equation is equivalent to \begin{array}{l}\require{cancel} -4|m+3|+7=3 \\\\ -4|m+3|=3-7 \\\\ -4|m+3|=-4 \\\\ \dfrac{-4|m+3|}{-4}=\dfrac{-4}{-4} \\\\ |m+3|=1 .\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} m+3=1 \\\\\text{OR}\\\\ m+3=-1 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} m+3=1 \\\\ m=1-3 \\\\ m=-2 \\\\\text{OR}\\\\ m+3=-1 \\\\ m=-1-3 \\\\ m=-4 .\end{array} If $m=-2,$ then \begin{array}{l}\require{cancel} -4|m+3|+7=3? \\\\ -4|-2+3|+7=3? \\\\ -4|1|+7=3? \\\\ -4(1)+7=3? \\\\ -4+7=3? \\\\ 3=3 \text{ (TRUE)} .\end{array} If $m=-4,$ then \begin{array}{l}\require{cancel} -4|m+3|+7=3? \\\\ -4|-4+3|+7=3? \\\\ -4|-1|+7=3? \\\\ -4(1)+7=3? \\\\ -4+7=3? \\\\ 3=3 \text{ (TRUE)} .\end{array} Hence, $m=\left\{ -4,-2 \right\} .$