# Chapter 2 - Systems of Linear Equations and Inequalities - 2.5 Absolute Value Equations and Inequalities - 2.5 Exercises: 67

all real numbers

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $-6|p-4|+3\le16 ,$ isolate first the absolute value expression. Then use the definition of absolute value to analyze the solution. $\bf{\text{Solution Details:}}$ Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} -6|p-4|\le16-3 \\\\ -6|p-4|\le13 \\\\ \dfrac{-6|p-4|}{-6}\ge\dfrac{13}{-6} \\\\ |p-4|\ge-\dfrac{13}{6} .\end{array} The absolute value of $x,$ written as $|x|,$ is the distance of $x$ from zero. Hence, it is always a nonnegative number. In the same way, the left side of the inequality above is a nonnegative number. This is always $\text{ greater than or equal to }$ the negative number at the right. Hence, the solution is the set of $\text{ all real numbers .}$

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