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Solve and Graph an Absolute Value Inequality

Nicole Conaway, Luke Winspur
  • Author
    Nicole Conaway

    Nicole Conaway has taught Secondary Math and Science [subjects] for over 20 years. They have a Master's Degree in Environmental Science from the University of Michigan and a Bachelor's degree in Biology form Wayne State University. They also have a Professional Teaching Certificate from the State of Michigan.

  • Instructor
    Luke Winspur

    Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Learn what absolute value inequalities are and how to solve absolute value inequalities. The lesson also covers the graphing of absolute value inequalities. Updated: 11/11/2021

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Absolute Value Inequalities

An absolute value inequality is a mathematical expression of two unequal quantities which includes at least one absolute value.

Absolute Value

The absolute value is the distance of a number from zero. Sometimes this is described as the magnitude. For example the absolute value of 5 is 5 and the absolute value of -5 is also 5. The symbol for absolute value is |n|, where n is a number. Absolute values may also be found for algebraic expressions. The number, variable, or expression within the absolute value symbol is called the argument. In general:

$$|n| = n $$

$$|-n| = n $$

Read: The absolute value of a number n is equal to n and the absolute value of -n is equal to n.

Inequalities

An inequality is a mathematical expression stating that one quantity is greater than, greater than or equal to, less than, or less than or equal to another quantity, indicated by the following symbols:

The point of the symbol points toward the lesser value and the open side is toward the greater value. Those that include equal to include a line under the less than or greater than symbol.

inequality symbols

A simple example of an absolute value inequality is $$|x| \geq 7 $$

Following the definition and example above, the absolute value of x is equal to 7 if x is 7 or -7. So The absolute value of x is equal to or greater than 7 if x is any number greater than 7 or less than -7. This is because any number less than -7 will have an absolute value greater than 7.

The solution is written:

$$x\geq 7 $$

or

$$x\leq - 7 $$

This solution is read: X is equal to or greater than 7 OR x is less than or equal to -7

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Absolute Value Inequality Rules

When solving absolute value inequalities there are some important rules to remember:

  • Absolute value inequalities give us compound inequalities, a combination of two inequalities.
  • Inequalities with a less than symbol have solutions that are intersections of two inequalities, described as "and"
  • Inequalities with greater than symbols have solutions that are unions of two inequalities, described as "or"
  • When solving inequalities, multiplication or division by a negative number reverses the direction of the inequality symbol
  • Since absolute value is positive, an absolute value inequality set less than a negative number will have no solution.
  • If the absolute value is greater than any negative number, then the solution is all real numbers

Use these steps to solve absolute value inequalities:

  • First use inverse operations to isolate the absolute value on the left side.
  • For 'greater than' inequalities, solve for the the inequality greater than the positive and flip the symbol and solve for the the negative value.

Solving 'Greater Than' Absolute Value Inequalities

For example:

{eq}3|2-x|-5> 7 {/eq}

First,use inverse operations to isolate the absolute value:

First, isolate the absolute value on the left.

Isolate the absolute value expression

Since this is a 'greater than' inequality, solve for the the inequality greater than the positive and flip the symbol and solve for the the negative value:

Solving for the positive option:

Remember the rule about multiplying or dividing by a negative number. When dividing by -1 to get x, flip the inequality symbol

Solving for the positive option

Flip the inequality symbol and solve for the negative option:

When solving for the negative option, remember to flip the inequality symbol first. In this case, it is flipped again when dividing by -1.

Solving for the negative option.

The solution is X<-2 or x>6

Solving a 'Less Than' Absolute Value Inequality

$$|3x+4|\leq25 $$

This is a 'less than' absolute inequality so the solution will be an 'and' complex inequality. The values that make the inequality true must satisfy both conditions: they must be greater than the negative value AND less than the positive value. This type of inequality is written with the argument of the absolute value in the middle of the negative and positive options like this:

$$-25\leq |3x+4|\leq 25 $$

Use inverse operations to solve for x:

Since the absolute value is already isolated, use inverse operations to solve this

Solving and &quot;and&quot; complex inequality

This solution can be read "-29/3 is less than or equal to x and x is less than or equal to 7" or simply "-29/3 is less than or equal to x is less than or equal to 7"

Here is one more example:

The first step is to isolate the absolute value expression.

Third Example which has no solution

In this case, when the absolute value expression is isolated, it is less than a negative value. Since absolute values are always positive, this inequality has no solution

Graphing Absolute Value Inequalities

Graphing the solution to an absolute value inequality helps to make sense of the solution.

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Frequently Asked Questions

How do you solve absolute value inequalities?

Solve one dimensional absolute value inequalities algebraically. First, isolate the absolute value on the left using inverse operations. Solve for both the negative and the positive values. For "greater than" inequalities, flip the inequality symbol before solving for the negative value.

Solve two variable absolute value inequalities by graphing them. Graph the corresponding absolute value equation. Use a solid line if the inequality includes 'or equal to' or a broken line if it does not. Shade in the appropriate area above or below the line.

How does an absolute value symbol affect an inequality?

Because each number and its opposite (both positive and negative) have the same absolute value, the absolute value symbol means the inequality will have a two part solution, or a complex inequality. Finding the solution involves solving two inequalities, one for the negative value and one for the positive value.

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