How to Graph Absolute Value Inequalities?
To graph an absolute value inequality, first, you need to solve the absolute value inequality.
![How to Graph Absolute Value Inequalities?](https://www.effortlessmath.com/wp-content/uploads/2022/02/blog-How-to-Graph-Absolute-Value-Inequalities-512x240.jpg)
The absolute value of a number describes its distance from zero. To graph absolute value inequality you must first solve the inequality and then represent absolute value inequalities on a number line.
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A step-by-step guide to graph absolute value inequalities
- Isolate the absolute value expression.
- Write the equivalent compound inequality.
- Find two values.
- Draw a number line and Place dots or open dots on the two points corresponding to the solutions. if the symbol is (\(≥\) or \(≤\)) then you fill in the dot if the symbol is (\(>\) or \(<\)) then you do not fill in the dot.
- Sketch a line and show all numbers the variable can be.
Graphing Absolute Value Inequalities – Example 1:
Solve and graph \(|3x-3|<9\).
Solution:
Use absolute properties: if \(|a|<b\), \(b > 0\) then: \( -b<a<b → -9 < 3x-3 < 9\)
Add \(3\) to each side of inequality: \(-9+3 < 3x-3+3 < 9+3 → -6 < 3x < 12\)
Divide each side of inequality by \(3\): \( -2 < x < 4\)
Graph it on a number line.
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/Picture3-1024x138.png)
Graphing Absolute Value Inequalities – Example 2:
Solve and graph \(|x+2| ≥ 3\).
Solution:
Split into two inequalities: \(x+2 ≥ 3\) or \(x+2 ≤ -3\).
Subtract \(2\) from each side of each inequality:
\(x+2-2 ≥ 3-2\) → \(x ≥ 1\)
or
\(x+2-2 ≤ -3-2\) → \(x ≤ -5\)
\(x ≥ 1\) or \(x ≤ -5\).
Graph the numbers that satisfy both conditions:
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/4-1024x137.png)
Exercises for Graph Absolute Value Inequalities
Solve and graph the following inequalities.
- \(\color{blue}{4|x| ≥ 12}\)
![This image has an empty alt attribute; its file name is 33-10.png](https://www.effortlessmath.com/wp-content/uploads/2019/09/33-10.png)
- \(\color{blue}{|x-2| < 1}\)
![This image has an empty alt attribute; its file name is 33-10.png](https://www.effortlessmath.com/wp-content/uploads/2019/09/33-10.png)
- \(\color{blue}{|x+2| ≥ 2}\)
![This image has an empty alt attribute; its file name is 33-10.png](https://www.effortlessmath.com/wp-content/uploads/2019/09/33-10.png)
- \(\color{blue}{5|x+2| ≥ 15}\)
![This image has an empty alt attribute; its file name is 33-10.png](https://www.effortlessmath.com/wp-content/uploads/2019/09/33-10.png)
![This image has an empty alt attribute; its file name is answers.png](https://www.effortlessmath.com/wp-content/uploads/2019/12/answers.png)
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/j1-1024x137.png)
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/j2-1024x137.png)
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/j3-1-1024x137.png)
![](https://www.effortlessmath.com/wp-content/uploads/2022/02/j4-1-1024x137.png)
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