Answer
$a=\dfrac{1}{4}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|7-4a|=|4a+5|
,$ use the definition of an absolute value equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since $|x|=|y|$ implies $x=y \text{ or } x=-y,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
7-4a=4a+5
\\\\\text{OR}\\\\
7-4a=-(4a+5)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
7-4a=4a+5
\\\\
-4a-4a=5-7
\\\\
-8a=-2
\\\\
a=\dfrac{-2}{-8}
\\\\
a=\dfrac{1}{4}
\\\\\text{OR}\\\\
7-4a=-(4a+5)
\\\\
7-4a=-4a-5
\\\\
-4a+4a=-5-7
\\\\
0=-12
\text{ (FALSE)}
.\end{array}
Hence, $
a=\dfrac{1}{4}
.$
