Answer
$x=-\dfrac{5}{32}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\dfrac{2}{3}\left( \dfrac{7}{8}-4x \right)-\dfrac{5}{8}=\dfrac{3}{8}
,$ remove first the fraction by multiplying both sides by the $LCD.$ Then use the properties of equality to isolate the variable. Do checking of the solution.
$\bf{\text{Solution Details:}}$
The $LCD$ of the denominators, $\{
3,8,8
\},$ is $
24
$ since this is the least number that can be evenly divided (no remainder) by all the denominators. Multiplying both sides by the $LCD,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}\left( \dfrac{7}{8}-4x \right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
24\left[\dfrac{2}{3}\left( \dfrac{7}{8}-4x \right)-\dfrac{5}{8}\right]=24\left[\dfrac{3}{8}\right]
\\\\
16\left( \dfrac{7}{8}-4x \right)-15=9
.\end{array}
Using the Distributive Property and the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
16\left( \dfrac{7}{8}-4x \right)-15=9
\\\\
14-64x-15=9
\\\\
-64x=9-14+15
\\\\
-64x=10
\\\\
x=\dfrac{10}{-64}
\\\\
x=-\dfrac{5}{32}
.\end{array}
Checking: If $x=-\dfrac{5}{32},$ then
\begin{array}{l}\require{cancel}
\dfrac{2}{3}\left( \dfrac{7}{8}-4x \right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{2}{3}\left( \dfrac{7}{8}-4\left(-\dfrac{5}{32}\right) \right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{2}{3}\left( \dfrac{7}{8}+\dfrac{5}{8} \right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{2}{3}\left( \dfrac{12}{8}\right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{2}{\cancel3}\left( \dfrac{\cancel3(4)}{8}\right)-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{8}{8}-\dfrac{5}{8}=\dfrac{3}{8}
\\\\
\dfrac{3}{8}=\dfrac{3}{8}
\text{ (TRUE) }
.\end{array}
Hence, the solution is $
x=-\dfrac{5}{32}
.$
