Answer
$\left( -\infty,-\dfrac{1}{7}
\right)\cup\left( 1,\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|7x-3|\gt4
,$ use the definition of absolute value inequalities.
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
7x-3\gt4
\\\\\text{OR}\\\\
7x-3\lt-4
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
7x-3\gt4
\\\\
7x\gt4+3
\\\\
7x\gt7
\\\\
x\gt\dfrac{7}{7}
\\\\
x\gt1
\\\\\text{OR}\\\\
7x-3\lt-4
\\\\
7x\lt-4+3
\\\\
7x\lt-1
\\\\
x\lt-\dfrac{1}{7}
.\end{array}
Hence, the solution set is the interval $
\left( -\infty,-\dfrac{1}{7}
\right)\cup\left( 1,\infty \right)
.$
