2.15: Graphical Solutions to Absolute Value Inequalities
Solve the following inequality and graph the solution on a number line.
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Khan Academy Absolute Value Inequalities on a Number Line
Guidance
Recall that you can graph linear inequalities on number lines. For
Notice that there is only one solution set and therefore one section of the number line has the region shown in red.
What do you think would happen with absolute value linear inequalities? With absolute value linear inequalities, there are two inequalities to solve. Therefore there can be two sections of the number line showing solutions.
For
The solution is
For
The solution is
Graphing the solution set to an absolute value linear inequality gives you the same visual representation as you had when graphing the solution set to linear inequalities. The same rules apply when graphing absolute values of linear inequalities on a real number line. Once the solution is found, the open circle is used for absolute value inequalities containing the symbols > and <. The closed circle is used for absolute value inequalities containing the symbols
Example A
Represent the solution set to the following inequality on a number line:
Solution: First solve the inequality. Then, represent your solution on a number line.
The solution sets are
Example B
Solve the following inequality and graph the solution on a number line:
Solution: First solve the inequality. Then, represent your solution on a number line.
The solution sets are
Example C
Solve the following inequality and graph the solution on a number line:
Solution: First solve the inequality. Then, represent your solution on a number line.
The solution is
Concept Problem Revisited
Solve the following inequality and graph the solution on a number line.
First solve the inequality:
The solution is
Representing on a number line:
Vocabulary
 Absolute Value Linear Inequality

Absolute Value Linear inequalities can have one of four forms:
ax+b>c,ax+b<c,ax+b≥c , orax+b≤c . Absolute value linear inequalities have two related inequalities. For example forax+b>c , the two related inequalities areax+b>c andax+b<−c .
 Number Line
 A number line is a line that matches a set of points and a set of numbers one to one.
Guided Practice
1. Represent the solution set to the inequality
2. Represent the solution set to the inequality
3. Represent the solution set to the inequality
Answers:
1.
The solution sets are
2.
The solution sets are
3.
The solution sets are
Practice
Represent the solution sets to each absolute value inequality on a number line.

3−2x<3  \begin{align*}2\big\frac{2x}{3}+1\big\ge 4\end{align*}
 \begin{align*}\big\frac{2g9}{4}\big<1\end{align*}
 \begin{align*}\big\frac{4}{3}x5\big\ge 7\end{align*}
 \begin{align*}2x+5+4 \ge 7\end{align*}
 \begin{align*}p16>10\end{align*}
 \begin{align*}r+2<5\end{align*}
 \begin{align*}32k\ge 1\end{align*}
 \begin{align*}8y>5\end{align*}
 \begin{align*}8 \ge 5d2\end{align*}
 \begin{align*}s+25>8\end{align*}
 \begin{align*}10+8w2<16\end{align*}
 \begin{align*}2q+15 \le 7\end{align*}
 \begin{align*}\big \frac{1}{3}(g2) \big <4\end{align*}
 \begin{align*}2(e+4)>17\end{align*}
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Absolute Value Linear inequalities
Absolute value linear inequalities can have one of four forms , or . Absolute value linear inequalities have two related inequalities. For example for , the two related inequalities are and .number line
A number line is a line on which numbers are marked at intervals. Number lines are often used in mathematics to show mathematical computations.Image Attributions
Here you'll learn how to represent the solutions of an absolute value inequality on a number line.