## Intermediate Algebra: Connecting Concepts through Application

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $12+8|3d-8|\le10 ,$ isolate first the absolute value expression. Then use the definition of absolute value to analyze the solution. $\bf{\text{Solution Details:}}$ Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} 12+8|3d-8|\le10 \\\\ 8|3d-8|\le10-12 \\\\ 8|3d-8|\le-2 \\\\ |3d-8|\le-\dfrac{2}{8} \\\\ |3d-8|\le-\dfrac{1}{4} .\end{array} The absolute value of $x,$ written as $|x|,$ is the distance of $x$ from zero. Hence, it is always a nonnegative number. In the same way, the left side of the inequality above is always a nonnegative number. This is never $\text{ less than or equal to }$ the negative number at the right. Hence, there is $\text{ no solution .}$