Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x \lt -3$
$\bf{\text{Solution Outline:}}$ 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}