# Chapter 9 - Inequalities and Problem Solving - 9.1 Inequalities and Applications - 9.1 Exercise Set - Page 580: 24

$x\le-\dfrac{23}{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $8x-3(3x+2)-5\ge3(x+4)-2x ,$ use the Distributive Property and the properties of inequality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} 8x-3(3x+2)-5\ge3(x+4)-2x \\\\ 8x-3(3x)-3(2)-5\ge3(x)+3(4)-2x \\\\ 8x-9x-6-5\ge3x+12-2x .\end{array} Using the properties of inequality to isolate the variable, then \begin{array}{l}\require{cancel} 8x-9x-6-5\ge3x+12-2x \\\\ 8x-9x-3x+2x\ge12+6+5 \\\\ -2x\ge23 .\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} -2x\ge23 \\\\ x\le\dfrac{23}{-2} \\\\ x\le-\dfrac{23}{2} .\end{array}

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