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

$\text{Set Builder Notation: } \left\{ t|t\ge-1 \right\} \\\text{Interval Notation: } \left[ -1,\infty \right)$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $\dfrac{1}{2}t-\dfrac{1}{4}\le\dfrac{3}{4}t ,$ remove first the fraction by multiplying both sides by the $LCD.$ Then use the properties of inequality to isolate the variable. $\bf{\text{Solution Details:}}$ The $LCD$ of the denominators, $\{ 2,4,4 \},$ is $4$ since this is the least number that can be evenly divided (no remainder) by all the denominators. Multiplying both sides by the $LCD,$ the given inequality is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{2}t-\dfrac{1}{4}\le\dfrac{3}{4}t \\\\ 4\left( \dfrac{1}{2}t-\dfrac{1}{4} \right) \le4\left( \dfrac{3}{4}t \right) \\\\ 2t-1\le3t .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} 2t-1\le3t \\\\ 2t-3t\le1 \\\\ -t\le1 .\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} -t\le1 \\\\ \dfrac{-t}{-1}\le\dfrac{1}{-1} \\\\ t\ge-1 .\end{array} Hence, the solution set is \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \left\{ t|t\ge-1 \right\} \\\text{Interval Notation: } \left[ -1,\infty \right) .\end{array}