Answer
$\text{Set Builder Notation: }
\left\{ t|t\ge-1 \right\}
\\\text{Interval Notation: }
\left[ -1,\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
