In this explainer, we will learn how to solve one-variable inequalities that contain absolute values.
Before looking at absolute value inequalities, let us recap the definition of the absolute value function.
Definition: The Absolute Value
In words, the absolute value of a number can be described as the magnitude of the number without regard to its sign. For example, the number 2 has magnitude 2, and the number also has magnitude 2. More formally, we can define absolute value algebraically as follows:
Let us also recap how to plot a graph of the absolute value function. To do this, we can complete a table of value for :
Then, we can plot the coordinates on a pair of coordinate axes to draw the graph:
The ability to apply the definition of the absolute value and plot absolute value graphs is very useful when solving absolute value inequalities. So, practicing these skills is very important.
Let us now introduce the concept of absolute value inequalities. Consider the inequality
Before we look at this question algebraically, it can be useful to think about the problem graphically. So, on the same set of axes, let us plot the graphs and :
From this graph, we can see that the red graph, , is less than or equal to 3 when is greater than or equal to and less than or equal to 2. So, the solution to the inequality is
Now, we can also see from the graph how we might approach solving this inequality graphically. The red graph contains part of each of the graphs of and . So, we can consider solving the set of compound inequalities and to help us find the overall solution.
Starting with if we subtract 1 from each side, we find that
Then, looking at we start by expanding out the bracket to get
Then, we add to each side to get and, finally, subtract 3 from each side to find that
Combing these two inequalities, we can see that our overall solution is which agrees with our initial findings from inspecting the graphs.
Both methods are equally acceptable for solving absolute value inequalities, but it is worth practicing both, particularly solving graphically as this helps you to visualize the solution. It is also worth practicing giving your answer in various forms, including as simplified inequalities on number lines and as intervals.
Let us look at a couple more examples.
Example 1: Solving Absolute Value Inequalities
Find the solution set of the inequality .
We will solve this question first by using a graphical approach and then by using an algebraic approach. To solve the inequality graphically, we need to plot the graphs and on the same set of axes:
We can then see that the red graph is less than 9 when is greater than and less than 5. Hence, the solution to the inequality is
The question, however, asks for the solution set of the inequality, which would be written as .
If we want to solve the inequality algebraically, we consider the compound inequalities and . To solve we need to subtract four from each side of the inequality to get To solve we first expand the bracket to get
Then, add to each side to give us
Finally, we subtract 9 from each side to get
Our last step is to then combine the inequalities, which gives us as before.
Example 2: Solving Absolute Value Inequalities
Find algebraically the solution set of the inequality .
Notice here that the question explicitly asks us to calculate the solution set algebraically, however, for the benefit of explaining the solution, we will also present the graph. If we start by subtracting 3 from each side of the inequality, we get
Now, the left-hand side of the inequality is an absolute value which is always greater than or equal to zero, and the right-hand side is a negative number, so there is no solution as the left-hand side can never be less than or equal to the right-hand side. This can be seen clearly by drawing the graphs of and on the same set of axes:
The red graph here is obviously never less than the blue graph. Therefore, the solution set for the inequality is the empty set, .