Absolute Value
Absolute value measures a number's distance from zero on the number line. Learn how to interpret |a|, compare numbers by magnitude, and see why distance is always positive.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on Absolute Value, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
You've seen negative numbers and learned to order them on a number line. Now it's time to think about numbers in a new way: by their distance from zero rather than their position. That distance has a special name—absolute value—and it changes how we compare numbers. In this lesson, you'll learn what absolute value means, how to interpret the notation |a|, and why a negative number can have the same absolute value as a positive number.
What Absolute Value Means
Absolute value measures how far a number is from zero on the number line. The symbol for absolute value is two vertical bars: |a| means "the absolute value of a." For example, |5| = 5 because 5 is 5 units away from zero. But here's the key insight: |−5| = 5 as well, because −5 is also 5 units away from zero, just in the opposite direction.
Distance is always non-negative. You can't have a negative distance. This means the absolute value of any number is always zero or positive. Even |0| = 0, since zero is zero units from itself. Think of absolute value as "how big the number is," ignoring whether it's positive or negative. It strips away the sign and tells you only the magnitude.
Absolute value appears everywhere in real life: the distance between two cities doesn't depend on which direction you measure; the size of an error is what matters, not whether you overestimated or underestimated; your distance from home is the same whether you walked north or south.
Distance is always non-negative. You can't have a negative distance. This means the absolute value of any number is always zero or positive. Even |0| = 0, since zero is zero units from itself. Think of absolute value as "how big the number is," ignoring whether it's positive or negative. It strips away the sign and tells you only the magnitude.
Absolute value appears everywhere in real life: the distance between two cities doesn't depend on which direction you measure; the size of an error is what matters, not whether you overestimated or underestimated; your distance from home is the same whether you walked north or south.
Reading and Writing Absolute Value Notation
When you see |−3|, you read it as "the absolute value of negative 3." To evaluate it, ask yourself: "How far is −3 from zero?" The answer is 3 units, so |−3| = 3.
Here are some examples:
Notice that |7| = |−7|. These are called opposite numbers or additive inverses—they're on opposite sides of zero but the same distance away. Every number except zero has an opposite that shares its absolute value.
Absolute value notation is also used inside larger expressions. For example, . Always evaluate the absolute value first, just like you'd evaluate what's inside parentheses.
Here are some examples:
| Expression | Value | Reasoning | ||
|---|---|---|---|---|
| 7 | 7 | 7 is 7 units from zero | ||
| −7 | 7 | −7 is 7 units from zero | ||
| 0 | 0 | 0 is 0 units from zero | ||
| −2.5 | 2.5 | −2.5 is 2.5 units from zero |
Absolute value notation is also used inside larger expressions. For example, . Always evaluate the absolute value first, just like you'd evaluate what's inside parentheses.
Absolute Value Compared to Order
Here's where students often get confused: absolute value and order are different ideas. Order compares which number is greater or smaller. Absolute value compares which number is farther from zero.
Look at −8 and −2. In order, −8 is smaller (farther left on the number line). But in absolute value, |−8| = 8 and |−2| = 2, so −8 is farther from zero. The number with larger absolute value is actually the one that's smaller in order.
Here's the key comparison:
Example: Compare −6 and 3.
By order: 3 > −6 (3 is to the right of −6 on the number line)
By absolute value: |−6| = 6 and |3| = 3, so |−6| > |3| (−6 is farther from zero)
Both statements are true at the same time. They answer different questions. When a problem asks "which is greater?" it's asking about order. When it asks "which has a greater absolute value?" or "which is farther from zero?" it's asking about magnitude.
Look at −8 and −2. In order, −8 is smaller (farther left on the number line). But in absolute value, |−8| = 8 and |−2| = 2, so −8 is farther from zero. The number with larger absolute value is actually the one that's smaller in order.
Here's the key comparison:
| Concept | Question Asked | Answer |
|---|---|---|
| Order | Which number is greater? | Compare positions left to right |
| Absolute Value | Which number is farther from zero? | Compare distances, ignoring sign |
By order: 3 > −6 (3 is to the right of −6 on the number line)
By absolute value: |−6| = 6 and |3| = 3, so |−6| > |3| (−6 is farther from zero)
Both statements are true at the same time. They answer different questions. When a problem asks "which is greater?" it's asking about order. When it asks "which has a greater absolute value?" or "which is farther from zero?" it's asking about magnitude.
Solving Absolute Value Equations
Sometimes you need to find the number (or numbers) that satisfy an absolute value statement. For example: "What number has an absolute value of 5?"
The answer is: both 5 and −5. Because |5| = 5 and |−5| = 5, this equation has two solutions. Except for |x| = 0, which has only one solution (x = 0), almost every absolute value equation will have two solutions—one positive and one negative.
Here's the general principle: if |x| = a (where a is positive), then x = a or x = −a.
Example: Solve |x| = 4.
Answer: x = 4 or x = −4. Both solutions are correct because both are exactly 4 units from zero.
But if the equation is |x| = −2, there is no solution. Absolute value is never negative, so no number's distance from zero can be −2. This is an important check: if you're solving |x| = a and a is negative, stop—there's no solution.
The answer is: both 5 and −5. Because |5| = 5 and |−5| = 5, this equation has two solutions. Except for |x| = 0, which has only one solution (x = 0), almost every absolute value equation will have two solutions—one positive and one negative.
Here's the general principle: if |x| = a (where a is positive), then x = a or x = −a.
Example: Solve |x| = 4.
Answer: x = 4 or x = −4. Both solutions are correct because both are exactly 4 units from zero.
But if the equation is |x| = −2, there is no solution. Absolute value is never negative, so no number's distance from zero can be −2. This is an important check: if you're solving |x| = a and a is negative, stop—there's no solution.
Key terms
- Absolute value.
- The distance of a number from zero on the number line, always written as |a|, and always non-negative.
- Magnitude.
- The size or extent of a number without regard to its sign; another word for absolute value.
- Distance from zero.
- How many units a number is away from 0 on the number line, measured as a non-negative value.
- Opposite numbers (additive inverses).
- Two numbers that are the same distance from zero but on opposite sides; for example, 6 and −6.
- Order.
- The arrangement of numbers from smallest to largest (or greatest to least) by their position on the number line.
- Evaluate.
- To find the value of a mathematical expression by performing the operations indicated.
Worked example
A submarine is at a depth of 120 meters below sea level (representing −120 on a vertical number line, where 0 is sea level). A helicopter is 95 meters above sea level (representing +95). Which is farther from sea level: the submarine or the helicopter? How far apart are they?
Let's break this into steps.
Step 1: Identify the positions.
Submarine position: −120 meters
Helicopter position: +95 meters
Step 2: Find the absolute value (distance from sea level) for each.
Submarine's distance from sea level: |−120| = 120 meters
Helicopter's distance from sea level: |95| = 95 meters
Step 3: Compare absolute values.
Since 120 > 95, the submarine is farther from sea level.
Step 4: Find the distance between them.
The total distance between them is the sum of their distances from sea level:
120 + 95 = 215 meters
Alternatively, you can think of it as the distance between −120 and 95 on the number line:
95 − (−120) = 95 + 120 = 215 meters
Answer: The submarine is farther from sea level (120 meters vs. 95 meters). They are 215 meters apart.
Step 1: Identify the positions.
Submarine position: −120 meters
Helicopter position: +95 meters
Step 2: Find the absolute value (distance from sea level) for each.
Submarine's distance from sea level: |−120| = 120 meters
Helicopter's distance from sea level: |95| = 95 meters
Step 3: Compare absolute values.
Since 120 > 95, the submarine is farther from sea level.
Step 4: Find the distance between them.
The total distance between them is the sum of their distances from sea level:
120 + 95 = 215 meters
Alternatively, you can think of it as the distance between −120 and 95 on the number line:
95 − (−120) = 95 + 120 = 215 meters
Answer: The submarine is farther from sea level (120 meters vs. 95 meters). They are 215 meters apart.
Practice questions
What is the value of |−9| + |4|?
Answer: 13
First, evaluate each absolute value. |−9| = 9 because −9 is 9 units from zero. |4| = 4 because 4 is 4 units from zero. Then add: 9 + 4 = 13. Remember, absolute value always gives a non-negative result.
Maria and Jordan are standing at point zero on a number line. Maria walks 8 steps to the left. Jordan walks 8 steps to the right. Without calculating, explain why they have the same absolute value but different positions.
Answer: They have the same absolute value because absolute value measures distance from zero, and both walked exactly 8 steps away. They end up at opposite positions (−8 and +8) because direction matters for location, but not for distance. Absolute value only cares about "how far," not "which way."
This question tests whether students understand the difference between position (order) and magnitude (absolute value). Maria is at −8 and Jordan at +8. Although −8 is less than +8 in order, both |−8| and |8| equal 8. The phrase "without calculating" encourages conceptual understanding over just finding the answer.
Which statement is true? (A) |−5| > |3| (B) −5 > 3 (C) |−5| < |3| (D) −5 = 3
- (A) |−5| > |3|
- (B) −5 > 3
- (C) |−5| < |3|
- (D) −5 = 3
Answer: (A) |−5| > |3|
Let's check each. (A): |−5| = 5 and |3| = 3, so 5 > 3 ✓ This is true. (B): −5 > 3 is false because −5 is to the left of 3 on the number line. (C): |−5| < |3| is false for the same reason as (A). (D): −5 = 3 is clearly false. The correct answer is (A). This question checks whether students can distinguish absolute-value comparison from order comparison.
FAQ
- Why is absolute value always positive or zero?
- Because absolute value measures distance, and distance cannot be negative. You can't walk −5 meters; you walk 5 meters in some direction. Similarly, a number cannot be negative units away from zero.
- Do I need to memorize absolute value symbols?
- You don't need to memorize them, but you should recognize that |a| means "the distance from a to zero." Once you understand that, the symbol makes sense: the bars act like they're measuring outward from the center in both directions.
- Can two different numbers have the same absolute value?
- Yes. Every number except zero has an opposite that shares the same absolute value. For example, |7| = |−7| = 7. In general, if a is any non-zero number, then |a| = |−a|.
- What's the difference between absolute value and order?
- Order tells you which number is greater or smaller by comparing their positions on the number line. Absolute value compares which number is farther from zero, ignoring sign. For example, −10 is less than −2 in order, but |−10| > |−2| in absolute value because −10 is farther from zero.
Learn this with a teacher, not a page
The Crimsora tutor teaches Absolute Value live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.