##### Inequalities of form \(\abs{ax+b} \lt c\) or \(\abs{ax+b} \le c\)

A graph of the function \(y=\abs{x}\) is shown in Figure 1.3.1. The line \(y=3\) is also. Note that the points where the absolute function has \(y\)-coordinates less than \(3\) all have \(x\)-coordinates between \(-3\) and \(3\text{.}\) This indicates that the solution set to the inequality \(\abs{x} \lt 3\) is the interval \((3,3)\text{.}\)

In general, if \(k \gt 0\text{,}\) then the solution set to an inequality of the form \(\abs{ax+b} \lt k\) will be an open interval that is determined by solving a compound inequality. Specifically:

is equivalent to the compound inequality

###### Example1.3.2

Determine the solution set to the inequality \(\abs{3x+7} \le 14\text{.}\) State the solution set using interval notation (if possible).

We begin by writing an equivalent compound inequality that does not include an absolute value expression. We then solve that compound inequality.

The solution set to the give inequality is \(\left[-7,\frac{7}{3}\right]\text{.}\)

###### Example1.3.3

Determine the solution set to the inequality \(\abs{5x+19} \lt -10\text{.}\) State the solution set using interval notation (if possible).

Let's pause for a moment and consider what's actually being asked. We are asked to determine what values of \(x\) will cause an absolute value to be less than \(-10\text{.}\) Since no absolute value is negative, no absolute value is ever less than \(-10\text{.}\) So no value of \(x\) makes the inequality \(\abs{5x+19} \lt -10\) true, and the solution set to that inequality is \(\emptyset\text{.}\)

###### Example1.3.4

Determine the solution set to the inequality \(\abs{-\frac{3}{7}x+6} \lt 9\text{.}\) State the solution set using interval notation (if possible).

We need to write and solve an equivalent compound inequality that does not include an absolute value expression. While solving, we need to remember that the direction of the inequality sign reverses anytime we multiply or divide both sides of the equation by a negative number,

The solution set to the given inequality is \((-7,35)\text{.}\)