Answer
$x\lt-\dfrac{1}{2} \text{ or } x\gt\dfrac{7}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inqeuality, $
6+|3-2x|\gt10
,$ use the properties of inequality to isolate the absolute value expression. Then use the definition of a greater than absolute value inequality and solve each resulting inequality. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality to isolate the absolute value expression, the given is equivalent to
\begin{array}{l}\require{cancel}
6+|3-2x|\gt10
\\\\
|3-2x|\gt10-6
\\\\
|3-2x|\gt4
.\end{array}
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}
3-2x\gt4
\\\\\text{OR}\\\\
3-2x\lt-4
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
3-2x\gt4
\\\\
-2x\gt4-3
\\\\
-2x\gt1
\\\\\text{OR}\\\\
-2x\lt-4-3
\\\\
-2x\lt-7
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to
\begin{array}{l}\require{cancel}
-2x\gt1
\\\\
x\lt\dfrac{1}{-2}
\\\\
x\lt-\dfrac{1}{2}
\\\\\text{OR}\\\\
-2x\lt-7
\\\\
x\gt\dfrac{-7}{-2}
\\\\
x\gt\dfrac{7}{2}
.\end{array}
Hence, the solution set is $
x\lt-\dfrac{1}{2} \text{ or } x\gt\dfrac{7}{2}
.$
