Absolute value equations are very useful when dealing with problems involving values that cannot be negative, like distance. For example, imagine that you go to visit your friend Tony who lives on the 4th floor of an apartment block that has 10 floors in total. When you arrive, you cannot remember what floor Tony lives on, so you make your way in, call him from the stairs and ask him what floor he is in. He says he is on the 4th floor, and you say that you are 2 floors away from him. Does that mean that you are on the 2nd floor or on the 6th floor? It could be either because both options are 2 floors away from Tony's apartment. No matter if you go up or down, the distance is the same.
In terms of absolute value inequalities, they are very handy when calculating margins of error or tolerance, which can be applied, for example, to measurements of weight, length, and temperature in a manufacturing process.
In this article, we will define what absolute value equations and inequalities are, and their rules, and we will also show you how to solve them using practical examples.
The absolute value of a number x is a number with the same magnitude, but positive. Absolute values are generically represented as ![]()
.
But what is the reasoning behind this? This happens because the absolute value represents the distance from zero to a number x on the number line.
The distance from zero to 2 is 2, and the distance from zero to -2 is also 2, therefore ![]()
, and![]()
.
Fig. 1: Absolute value example represented on the number line.
This is why ![]()
represents the value of a number x disregarding its sign.
If you have an expression inside the absolute value, calculate the value inside, then find the positive version of the result.
Absolute value notation
The absolute value for any real number x is denoted as follows:
![]()
From the expressions above, we can say that if the number inside the absolute value is already positive, you leave it like that, but if the number is negative, then the result will be the positive version of that number (as if you were multiplying the negative number by -1).
Properties of absolute values
The properties of absolute values are:
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Sum:
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Subtraction:
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Solving absolute value equations
Absolute value equations are equations that include absolute value expressions in them.
For any real numbers a and b, where b ≥ 0:
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As you can see in the expression above, when solving equations, absolute values involve an extra step. Keeping in mind that the value inside an absolute value could be positive or negative, you need to solve the equation considering both cases, therefore you will get two solutions.
The steps to solve absolute value equations are as follows:
- Find the solution for the case when a is positive
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- Find the solution for the case when a is negative
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- Check each solution by substituting their values into the original equation, to see if it remains true.
- Define the solution set.
- Graph the solutions on the number line, if required.
For the equation ![]()
, we can obtain 2 possible solutions as follows:
1. Solution 1 ![]()
:
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2. Solution 2 ![]()
:
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3. Check the solutions:
a) Check solution 1 ![]()
![]()
![]()
✔
b) Check solution 2 ![]()
![]()
![]()
✔
4.
Solution set: Both solutions have been proved to make the original equation true. Therefore, we can say that the
solution set is
![]()
5. Graph on the number line:
If we represent the solution on the number line, we can see that they are both 4 units away from 5.
Fig. 2: Solutions of an absolute value equation on the number line.
An equation like ![]()
will never be true, because the absolute value of a number x will always be a positive number. Therefore, this type of equation has no possible solution. In this case, we can say that the solution set is the empty set, which can be denoted as { } or ∅.
solving absolute value inequalities
Absolute value inequalities are inequalities that involve absolute value expressions.
You can solve absolute value inequalities by rewriting them as compound inequalities.
Compound inequalities are two inequalities joined together by the words and or or.
For all real numbers a and b, where b ≥ 0:
![]()
The symbols > (greater than) and < (less than) exclude the specific value as part of the solution. The symbols ≥ (greater than or equal) and ≤ (less than or equal) include the specific value as part of the solution, instead of excluding it.
The solution of an inequality can be represented on the number line, using an empty circle to represent that the value of x is not part of the solution, and a closed circle if the value of x is part of the solution.
Example 1: Solve ![]()
This is the second case: ![]()
Therefore, we can say the following:
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Now we need to find both solutions:
1. Solution 1:
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2. Solution 2:
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3. Solution set:
The solution set is ![]()
4. Graph on the number line:
Fig. 3: Solution set of an absolute value inequality on the number line - Example 1.
Example 2: Solve ![]()
This is the first case: ![]()
Therefore, ![]()
In this case, we can write the inequality as a compound inequality and join them together with the word and, then solve each one separately.
1. Solutions 1 and 2:
![]()
2. Solution set:
The solution set is ![]()
3. Graph on the number line:
Fig. 4: Solution set of an absolute value inequality on the number line - Example 2.
Absolute Value Equations and Inequalities - Key takeaways
- The absolute value of a number x will be a number with the same magnitude, but positive.
- The absolute value of a number x represents the distance from zero to that number x on the number line.
- Absolute value equations are equations that include absolute value expressions in them.
- When solving equations, absolute values involve an extra step. Keeping in mind that the value inside an absolute value could be positive or negative, solve the equation considering both cases.
- You can solve absolute value inequalities by rewriting them as compound inequalities.
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