# Chapter 2 - Equations, Inequalities, and Problem Solving - 2.6 Solving Inequalities - 2.6 Exercise Set: 54

$\text{Set Builder Notation: } \left\{ y|y\gt\dfrac{7}{12} \right\} \\\text{Interval Notation: } \left( \dfrac{7}{12},\infty \right)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the properties of inequality to solve the given inequality, $y-\dfrac{1}{3}\gt\dfrac{1}{4} .$ Write the answer in both set-builder notation and interval notation. 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, the given is equivalent to \begin{array}{l}\require{cancel} y-\dfrac{1}{3}\gt\dfrac{1}{4} \\\\ 12\left( y-\dfrac{1}{3} \right)\gt12\left(\dfrac{1}{4}\right) \\\\ 12y-4\gt3 \\\\ 12y\gt3+4 \\\\ 12y\gt7 \\\\ y\gt\dfrac{7}{12} .\end{array} Hence, the solution set is \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \left\{ y|y\gt\dfrac{7}{12} \right\} \\\text{Interval Notation: } \left( \dfrac{7}{12},\infty \right) .\end{array}

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