Answer
$x \lt -3$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Multiply both sides of the given inequality $
-\dfrac{1}{2}x-\dfrac{1}{4} \gt \dfrac{1}{2}-\dfrac{1}{4}x
,$ by the $LCD$ to remove the denominators. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
The $LCD$ of the denominators, $\{
2,4,2,4
\}$ is $
4
$ since it is the least number that can be divided by all the denominators. Multiplying both sides by the $LCD,$ the given inequality is equivalent to
\begin{array}{l}\require{cancel}
-\dfrac{1}{2}x-\dfrac{1}{4} \gt \dfrac{1}{2}-\dfrac{1}{4}x
\\\\
4\left( -\dfrac{1}{2}x-\dfrac{1}{4} \right) \gt 4\left( \dfrac{1}{2}-\dfrac{1}{4}x \right)
\\\\
2(-1x)-1(1) \gt 2(1)-1(1x)
\\\\
-2x-1 \gt 2-x
.\end{array}
Using the properties of equality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-2x-1 \gt 2-x
\\\\
-2x+x \gt 2+1
\\\\
-x \gt 3
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-x \gt 3
\\\\
x \lt \dfrac{3}{-1}
\\\\
x \lt -3
.\end{array}
