Solve Inequalities Containing Absolute Value

Let us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us start with a simple inequality.

[latex]\left|x\right|\leq 4[/latex]

This inequality is read, “the absolute value of x is less than or equal to [latex]4[/latex].” If you are asked to solve for x, you want to find out what values of x are [latex]4[/latex] units or less away from [latex]0[/latex] on a number line. You could start by thinking about the number line and what values of x would satisfy this equation.

[latex]4[/latex] and [latex]−4[/latex] are both four units away from [latex]0[/latex], so they are solutions. [latex]3[/latex] and [latex]−3[/latex] are also solutions because each of these values is less than [latex]4[/latex] units away from [latex]0[/latex]. So are [latex]1[/latex] and [latex]−1[/latex],[latex]0.5[/latex] and [latex]−0.5[/latex], and so on—there are an infinite number of values for x that will satisfy this inequality.

The graph of this inequality will have two closed circles, at [latex]4[/latex] and [latex]−4[/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.

The solution can be written this way:

Inequality notation: [latex]-4\leq x\leq4[/latex]

Interval notation: [latex]\left[-4,4\right][/latex]

The situation is a little different when the inequality sign is “greater than” or “greater than or equal to.” Consider the simple inequality [latex]\left|x\right|>3[/latex]. Again, you could think of the number line and what values of x are greater than [latex]3[/latex] units away from zero. This time, [latex]3[/latex] and [latex]−3[/latex] are not included in the solution, so there are open circles on both of these values. [latex]2[/latex] and [latex]−2[/latex] would not be solutions because they are not more than [latex]3[/latex] units away from [latex]0[/latex]. But [latex]5[/latex] and [latex]−5[/latex] would work and so would all of the values extending to the left of [latex]−3[/latex] and to the right of [latex]3[/latex]. The graph would look like the one below.

The solution to this inequality can be written this way:

Inequality notation: [latex]x<−3[/latex] or [latex]x>3[/latex].

Interval notation: [latex]\left(-\infty, -3\right)\cup\left(3,\infty\right)[/latex]

In the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both and and or.

Writing Solutions to Absolute Value Inequalities

For any positive value of a and x, a single variable, or any algebraic expression:

Absolute Value Inequality	Equivalent Inequality	Interval Notation
[latex]\left|{ x }\right|\le{ a}[/latex]	[latex]{ -a}\le{x}\le{ a} \\ \mbox{Equivalently: } x\le a \mbox{ and } x\ge -a [/latex]	[latex]\left[-a, a\right][/latex]
[latex]\left| x \right|\lt{a}[/latex]	[latex]{ -a}\lt{x}\lt{ a} \\ \mbox{Equivalently: } x\lt a \mbox{ and } x\gt -a [/latex]	[latex]\left(-a, a\right)[/latex]
[latex]\left| x \right|\ge{ a}[/latex]	[latex]{x}\le\text{−a}[/latex] or [latex]{x}\ge{ a}[/latex]	[latex]\left(-\infty,-a\right]\cup\left[a,\infty\right)[/latex]
[latex]\left| x \right|\gt\text{a}[/latex]	[latex]\displaystyle{x}\lt\text{−a}[/latex] or [latex]{x}\gt{ a}[/latex]	[latex]\left(-\infty,-a\right)\cup\left(a,\infty\right)[/latex]

Let us look at a few more examples of inequalities containing absolute value.

In section 2.4, we saw that tripartite inequalities could also be solved by splitting the problem into a compound inequality adjoined by and.  This strategy can also be utilized here, providing an approach more similar to the “greater than” inequalities.  Below, you will find the previous example redone with this approach.

 

In the following video, you will see an example of solving multi-step absolute value inequalities involving an and situation.

The next video gives another example of an and situation.

In the last video that follows, you will see an example of an or situation where you need to isolate the absolute value first.

Identify Cases of Inequalities Containing Absolute Value That Have No Solutions

As with equations, there may be instances where there is no solution to an inequality.  This occurs if we obtain a statement that implies an absolute value is less than a negative number.  No value for the variable can ever make this true since absolute values are always nonnegative.

Identify Cases of Inequalities Containing Absolute Value That Have All Real Numbers as the Solution

Since absolute values are always nonnegative, if we instead arrive at a statement that implies an absolute value is greater than a negative number, this will hold true for every value of the variable.

 

Summary

Inequalities containing absolute value can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an or inequality or an and inequality. If the inequality is greater than a number, we will use or. If the inequality is less than a number, we will use and. Remember that if we end up with an absolute value less than, or less than or equal to a negative number, there is no solution. If we end with an absolute value greater than a negative number, the solution is all real numbers.