Absolute Value
Absolute value is a number’s distance from zero on the number line. Distance is never negative, so an absolute value is always positive or zero. It is written with two straight bars. For example, |-6| = 6 and |6| = 6, because both numbers are 6 units from 0.
Before this lesson: You should know how to find negative numbers on a number line and what opposites are.
Extra practice: The Pre-Algebra Unit 1 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.
What Does Absolute Value Mean?
Imagine you start at 0 on a number line and walk to a number. The absolute value is how many steps you took. It doesn’t matter which way you walked.
Look at −5 and 5. One is to the left of 0 and one is to the right. But each one is 5 units away, so they have the same absolute value.
So absolute value tells you how far a number is from 0. The sign of the number tells you which way.
You’ll use absolute value in the next lesson, adding and subtracting integers. It also helps you find distances on the coordinate plane.
The Absolute Value Symbol and Rule
The symbol |x| is read “the absolute value of x.” The bars go on both sides of the number.
- If a number is positive or 0, its absolute value is the number itself.
- If a number is negative, its absolute value is its opposite (the same number without the minus sign).
|8| = 8 \qquad |-8| = 8 \qquad |0| = 0
Two more things to know:
- Find each absolute value first. Then do any adding or subtracting outside the bars.
- A minus sign outside the bars stays. So -|8| = -8.
Worked Examples
Example 1: Find absolute values
Find |-14|, |14| and |0|.
Step 1. −14 is 14 units from 0, so its absolute value is 14.
Step 2. 14 is also 14 units from 0.
Step 3. 0 is 0 units from itself.
Answer: |-14| = 14, |14| = 14 and |0| = 0.
Example 2: Add absolute values
Find |-9| + |4|.
Step 1. Find each absolute value: |-9| = 9 and |4| = 4.
|-9| + |4| = 9 + 4 = 13
Step 2. Add the results.
Answer: 13.
Example 3: A minus sign outside the bars
Find -|-6|.
|-6| = 6
Step 1. Work inside the bars first.
-|-6| = -(6) = -6
Step 2. The minus sign outside the bars means “the opposite of.” The opposite of 6 is −6.
Answer: −6. The bars only change what is inside them.
Example 4: Compare absolute values
Compare |-11| and |7|. Then compare −11 and 7.
|-11| = 11
Step 1. Find the distance of −11 from 0.
|7| = 7
Step 2. Find the distance of 7 from 0.
|-11| \gt |7|
Step 3. Compare. 11 is greater than 7.
-11 \lt 7
Step 4. Compare without the bars. A negative number is less than a positive number.
Answer: −11 is the smaller number, but it is farther from 0, so it has the larger absolute value.
Example 5: Word problem
A diver is at −18 feet, which means 18 feet below the surface of the water. A seagull is flying 25 feet above the water. Which one is farther from the surface? By how much?
|-18| = 18
Step 1. Find the diver’s distance from the surface, in feet.
|25| = 25
Step 2. Find the seagull’s distance from the surface, in feet.
Step 3. Compare: 25 is greater than 18, so the seagull is farther away.
25 - 18 = 7
Step 4. Subtract to find the difference in feet.
Answer: The seagull is farther from the surface, by 7 feet.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Letting the bars change a sign outside them | -|5| = 5 | The minus outside stays: -|5| = -5. |
| Giving a negative answer | |-3| = -3 | Distance is never negative: |-3| = 3. |
| Mixing up absolute value and opposite | |4| = -4 | The opposite of 4 is −4, but |4| = 4. |
| Comparing the numbers, not the distances | |-9| \lt |2| | Find each absolute value first: 9 \gt 2, so |-9| \gt |2|. |
| Thinking the absolute value of 0 is 1 | |0| = 1 | 0 is 0 units from itself: |0| = 0. |
Absolute Value vs. Opposite
Absolute value and opposite are easy to mix up. The opposite switches a number to the other side of 0. The absolute value just gives the distance, which is never negative.
| Number | Opposite | Absolute value |
|---|---|---|
| 7 | −7 | 7 |
| −7 | 7 | 7 |
| −12 | 12 | 12 |
| 3 | −3 | 3 |
| 0 | 0 | 0 |
For a negative number, the opposite and the absolute value are the same. Absolute value equations, such as |x| = 5, are taught in Absolute Value Equations in Algebra 1.
Practice Problems
Find each value.
- |-20|
- |15|
- |0|
- |-7| + |-3|
- |-12| - |5|
- -|-4|
- Write \lt, \gt or = between |-13| and |9|.
- Write \lt, \gt or = between −13 and 9.
- Order these numbers by their absolute value, from least to greatest: −8, 3, −1, 6.
- Maya’s bank balance is −$35. Leo’s bank balance is −$50. Who owes more money, and how much more?
Show answers
1.
|-20| = 20
Find the distance of −20 from 0.
2.
|15| = 15
Find the distance of 15 from 0.
3.
|0| = 0
0 is 0 units from 0.
4.
|-7| + |-3| = 7 + 3
Find each absolute value.
7 + 3 = 10
Add.
5.
|-12| - |5| = 12 - 5
Find each absolute value.
12 - 5 = 7
Subtract.
6.
|-4| = 4
Find the absolute value inside the bars.
-|-4| = -4
The minus sign outside stays.
7.
|-13| = 13
Find the first absolute value.
|9| = 9
Find the second absolute value.
|-13| \gt |9|
Compare, since 13 is greater than 9.
8.
-13 \lt 9
A negative number is always less than a positive one.
9.
The absolute values are 8, 3, 1 and 6. From least to greatest: −1, 3, 6, −8.
10.
|-50| = 50
Find Leo’s debt.
|-35| = 35
Find Maya’s debt.
50 - 35 = 15
Leo owes more. Subtract to find how much more.
Answer: Leo owes $15 more than Maya.
Want more practice? The Pre-Algebra Unit 1 Practice Pack has 36 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.
Absolute Value FAQ
Can absolute value be negative?
No. An absolute value is a distance, and distance is never negative. But a minus sign outside the bars, as in -|4|, makes the final answer −4.
What is the absolute value of 0?
The absolute value of 0 is 0, because 0 is no distance from itself: |0| = 0.
What is the absolute value of −5?
The absolute value of −5 is 5, because −5 is 5 units from 0: |-5| = 5.
Is absolute value the same as the opposite?
Only for negative numbers. The absolute value of −3 is 3, which is also its opposite. But the absolute value of 3 is 3, while its opposite is −3.
Summary
- Absolute value is a number’s distance from 0 on the number line.
- It is written with bars: |x|.
- An absolute value is always positive or 0.
- A number and its opposite have the same absolute value: |-5| = |5| = 5.
- Find the absolute value first, and keep any minus sign that is outside the bars.
Previous lesson: Negative Numbers
Next lesson: Adding and Subtracting Integers
Back to the Pre-Algebra Index