Absolute Value Inequalities & Absolute Value Graphs

A series of free Basic Algebra Lessons with worked examples and solutions.

In these lessons, we will learn

• how to solve and graph absolute value inequalities
• how to graph absolute value inequalities in two variables
• how to shift or transform absolute value graphs
• how to find equations of shifted absolute value graphs

We also have an absolute value equation calculator that can solve and graph absolute value equtaions. Use it to check your answers.

Absolute Value Inequalities

To solve an absolute value inequality, knowledge of absolute values and solving inequalities are necessary. Absolute value inequalities can have one or two variables. Solving and graphing absolute value inequalities with two variables is also a skill that math teachers require students to master in Algebra I.
Absolute Value Inequalities.
Absolute Value Inequalities 2

Absolute value inequalities 3
This video shows how to solve absolute value inequalities. It shows a short cut. More is or is for greater than absolute value inequalities and has arrows going opposite directions on a number line graph. Less is nest is for less than absolute value inequalities and has the line filled in between two boundary points.

Absolute Value Inequality Graphs in Two Variables

Graphing inequalities with two variables can be tricky and is made even more tricky when we graph inequalities with two variables and absolute value. With absolute value graphing, if the inequality is similar to the equation of a line, (for example y > m|x| + b), then we get a V shape, and we shade above or below the V. This is very similar to graphing inequalities with two variables. The difference is that we are graphing inequalities with absolute values which makes the V-shape.
The following video examines how to graph linear and absolute value inequalities.

Shifts in Absolute Value Graphs

Once we learn about absolute value graphing, we can use shifts in absolute value graphs to more easily graph new absolute value equations. We can use specific rules for the addition of constants within the equation to shift the graph horizontally or vertically. These techniques can also be used when graphing parabolas.
All absolute value graphs have a V shape
Shifts:
|x| + a means move up a
|x| − a means move down a
|x - a| means move a units right
|x + a| means move a units left
Dilations:
a|x| skinny if |a| >1 and wide if |a| < 1
Upside down if a is negative
Graphs of absolute value functions using vertical and horizontal shifts

Graphing absolute value functions.
Stretching and Shrinking Functions.

How to graph absolute value graphs beginning with the basic parent graph and then transforming it and looking at shifts, stretches, and reflections. It begins with a review of transformations and how it was done with parabolas (quadratics).

Find the equations of shifted absolute value graphs

This video explains how to determine the equation of an absolute value function that has been shifted, up/down, left/right.

This video explains how to determine the equation of an absolute value function that has been shifted, up/down, left/right.
This video explains how to determine the equation of an absolute value function that has been shifted, up/down, left/right.

Absolute Value Equation Calculator

This absolute equation calculator will help you solve and graph absolute value equations.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.