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Related Lessons: Graphing Absolute Value Function Solving Absolute Value Equations

 

Solving Absolute Value Inequalities

 

In this lesson, we are going to learn how to solve absolute value inequalities using the standard approach usually taught in an algebra class. That is, learn the rules and apply it correctly. There are four cases involved when solving absolute value inequalities.

warning image to show that "a" must be positive In all cases, the assumption is that the value of "a" is positive, that is, a>0.

 

Four (4) Cases in Absolute Value Inequality

Case 1 :   |x|<a  or |x| ≤ a
If |x|<a  implies -a<x<a, or if |x|≤a  implies -a ≤x≤a
 
Case 2:    |x|>a  or |x|≥a
If |x|>a then x<-a or x>a, in addition,  if |x|≥a then x≤-a or x≥a
 
Case 3:   |x|<-a or |x|≤-a
  • The absolute value of any number is either zero (0) or positive which can never be less than or equal to a negative number.
  • The answer for this case is always no solution.
Case 4:   |x|>-a or |x|≥-a
  • The absolute value of any number is either zero (0) or positive. It makes sense that it must always be greater than any negative number.
  • The answer for this case is always all real numbers.

 


 

Example 1: Solve the absolute value inequality |x+3|<-2.

If you're not familiar yet with the different cases, I suggest that you keep a copy of the list of cases above as reference. This will definitely help you solve the problems easily.

The problem suggests that there exists a value of "x" that can make the statement true. Well, the absolute value of something is always zero or positive which is never less than a negative number. This statement must be false, therefore, there is no solution. This is an example of case 3.

Pick some test values to verify:

 

If x is positive, say, x = 5; If x is zero; If x is negative, say, x = −5;
8<-2 is a false inequality statement 3 < -2 is a false inequality statement 2<-2 is a false inequality statement

 


 

Example 2: Solve the absolute value inequality |x-2|>-1.

If you think about it, any values of "x" can make the statement true. Test some numbers including zero, and any negative or positive number. What do you get?

Remember, the absolute value expression will yield a zero or positive answer which is always greater than a negative number. Therefore, the answer is all real numbers. This is case 4.

 


 

Example 3: Solve the absolute value inequality |x+6|<2.

This is a "less than" absolute value inequality which is an example of case 1. Get rid of the absolute value symbol by applying the rule. Then solve the linear inequality that arises.

use the inequality formula that if |x|<b then -b<x<b

The goal is to isolate the variable "x" in the middle. To do that, we subtract the left, middle and right parts of the inequality by 6.

|x+6|<2 then the solution is -8<x<-4

The answer in the form of the inequality symbol states that the solution are all values of x between -8 and -4 but not including -8 and -4 themselves.

We can also write the answer in interval notation using a parenthesis to denote that -8 and -4 are not part of the solutions.

(-8,-4)

Or, write the answer in a number line where we use open circles to exclude -8 and -4 from the solution.

showing a number line where points -8 and -4 are not shaded but the line segment between then is

 


 

Example 4: Solve the absolute value inequality |3x-6|≤15.

This is a "less than or equal to" absolute value inequality which still falls under case 1. Clear out the absolute value symbol using the rule and solve the linear inequality.

image showing the procedure how to solve a "less than or equal to" situation in absolute value inequality

Isolate the variable "x" in the middle by adding all sides by 6 and then dividing by 3 (coefficient of x).

|3x-6|≤15, where the solution is  -3 ≤ x ≤ 7

The inequality symbol suggests that the solution are all values of x between -3 and 7 , and also including the endpoints -3 and 7. We include the endpoints because we are using the symbol less than or equal to.

To write the answer in interval notation, we will utilize the square brackets instead of the regular parenthesis to denote that -3 and 7 are part of the solution.

[ -3 , 7 ]

And finally, we will use closed or shaded circles to show that -3 and 7 are included.

 

a number line showing that points -3 and 7 are shaded , and the line segment between them

 


 

Example 5: Solve the absolute value inequality |x-4|>7.

This is an example of a "greater than" absolute value inequality which is an example of case 2. Let's eliminate the absolute value expression using the rule below.

if |y|>c then y<-c or y>+c

 

As you can see, we are solving two separate linear inequalities.

x<-3 or x>11

 

In interval notation, the word "or" is replaced by the symbol "U" to mean "union". The union of sets means that we are putting together the non-overlapping elements of two or more sets of solutions.

(-inf,-3) union with (11,+inf)

The answer in interval notation makes more sense if you see how it looks in the number line. In case 2, the arrows will always be in opposite directions. The open circles imply that -3 and 7 are not included in the solutions which is the consequence of the symbol ">".

number line showing where the line to the left of -3 is shaded, and the line to the right of 11 is shaded as  well

 


 

Example 6: Solve the absolute value inequality absolute value of 4x-2 is greater than or equal to 10.

Break this up into two linear inequalities, and then solve each separately. Here's the rule for case 2.

formulas to solve absolute value inequalties

Here's the solution.

x is less than or equal to -2 or x is greater than or equal to 3

For the interval notation, we use the square brackets to include -2 and 3 in the solution.

(-inf, -2] union with [3 , +inf)

The shaded or closed circles signifies that -2 and 3 are part of the solution. In case 2, the arrows will always point to opposite directions.

number line with x≤ -2 and x≥3

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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