# Chapter 1 - Section 1.5 - Linear Inequalities in One Variable - 1.5 Exercises - Page 100: 31

$\left( -\infty, 4 \right)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $-3(x-6)\gt2x-2 ,$ use the Distributive Property and the properties of inequality. For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$ For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the Distributive Property and the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -3(x)-3(-6)\gt2x-2 \\\\ -3x+18\gt2x-2 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -3x-2x\gt-2-18 \\\\ -5x\gt-20 .\end{array} Dividing both sides by a negative number (and consequently reversing the sign), the inequality above is equivalent to \begin{array}{l}\require{cancel} x\lt\dfrac{-20}{-5} \\\\ x\lt4 .\end{array} The red graph is the graph of the solution set. In interval notation, the solution set is $\left( -\infty, 4 \right) .$

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