Absolute Value — Definition, Examples, and How to Solve

Grade: 6–7 | Topic: Arithmetic

What You Will Learn

By the end of this page, you will understand what absolute value means as a measure of distance, be able to evaluate absolute value expressions, and solve straightforward absolute value equations. You will also see how absolute value is used in real-life situations involving temperature, elevation, and money.

Theory

What Is Absolute Value?

The absolute value of a number is its distance from zero on the number line. Distance is always positive (or zero), so absolute value is always zero or positive.

Absolute value is written with vertical bars: $|x|$

$|-7| = 7 \qquad |7| = 7 \qquad |0| = 0$

Both $-7$ and $7$ are 7 units away from zero, so they have the same absolute value.

Formal definition: $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

In plain English: if the number is positive or zero, keep it. If the number is negative, remove the negative sign.

Absolute Value in Expressions

When you see absolute value bars in an expression, evaluate what is inside first, then take the absolute value.

$|3 - 8| = |-5| = 5 \qquad -|4| = -(4) = -4$

Note: $-|4|$ is negative 4, not the absolute value of $-4$. The negative sign is outside the bars.

Solving Absolute Value Equations

An equation like $|x| = 5$ means "what number is 5 units from zero?" There are two answers: $x = 5$ or $x = -5$.

For $|x + 2| = 7$:

Case 1: The expression inside is positive: $x + 2 = 7 \implies x = 5$

Case 2: The expression inside is negative: $x + 2 = -7 \implies x = -9$

Both are valid solutions: $x = 5$ or $x = -9$.

Worked Examples

Example 1: Evaluating Absolute Value Expressions

Problem: Evaluate $|{-13}| + |6| - |{-2}|$.

Step 1: Evaluate each absolute value. $|{-13}| = 13 \qquad |6| = 6 \qquad |{-2}| = 2$

Step 2: Substitute and calculate. $13 + 6 - 2 = 17$

Answer: 17

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Example 2: Absolute Value with Expressions Inside

Problem: Evaluate $|2 \times (-6) + 5|$.

Step 1: Work out the expression inside the bars. $2 \times (-6) + 5 = -12 + 5 = -7$

Step 2: Take the absolute value. $|-7| = 7$

Answer: 7

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Example 3: Solving an Absolute Value Equation

Problem: Solve $|3x - 6| = 12$.

Step 1: Write two cases.

Case 1: $3x - 6 = 12$ $3x = 18 \implies x = 6$

Case 2: $3x - 6 = -12$ $3x = -6 \implies x = -2$

Step 2: Verify both solutions. $|3(6) - 6| = |12| = 12 \checkmark$ $|3(-2) - 6| = |-12| = 12 \checkmark$

Answer: $x = 6$ or $x = -2$

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Example 4: Real-World Context

Problem: The temperature in Oslo is $-8°\text{C}$ and in Rome is $+14°\text{C}$. Which city is farther from freezing point (0°C), and by how much?

Step 1: Find the absolute values (distance from 0°C). $|-8| = 8 \qquad |14| = 14$

Step 2: Compare. $14 > 8$

Answer: Rome is farther from freezing by $14 - 8 = \mathbf{6°\text{C}}$.

Common Mistakes

Mistake 1: Thinking $-|x|$ equals $|{-x}|$

❌ $-|5| = |{-5}| = 5$

✅ $-|5| = -(5) = -5$, but $|{-5}| = 5$. These are different! The negative sign outside the bars is applied after taking the absolute value.

Mistake 2: Only Writing One Solution for Absolute Value Equations

❌ $|x| = 9 \implies x = 9$ (only one answer)

✅ $|x| = 9 \implies x = 9$ or $x = -9$ (two answers)

Mistake 3: Trying to Solve When the Result is Negative

❌ Trying to solve $|x| = -4$ by writing two cases.

✅ No solution exists. Absolute value can never equal a negative number.

Practice Problems

Try these on your own before checking the answers:

1. Evaluate $|{-20}| - |8|$.
2. Evaluate $|5 - 11| + |{-3}|$.
3. Solve $|x| = 15$.
4. Solve $|2x + 1| = 9$.
5. A submarine is at $-340$ metres and a hot air balloon is at $+270$ metres. Which is farther from sea level (0 metres)?

Click to see answers

1. $20 - 8 = \mathbf{12}$
2. $|-6| + 3 = 6 + 3 = \mathbf{9}$
3. $x = 15$ or $x = -15$
4. Case 1: $2x + 1 = 9 \implies x = 4$. Case 2: $2x + 1 = -9 \implies x = -5$. Answer: $\mathbf{x = 4}$ or $\mathbf{x = -5}$
5. $|-340| = 340$, $|270| = 270$. The submarine is farther from sea level.

Summary

• Absolute value $|x|$ measures the distance from zero — always zero or positive.
• To evaluate: work out the expression inside first, then remove any negative sign.
• To solve $|expression| = k$: write two cases — expression $= k$ and expression $= -k$.
• If $k < 0$, there is no solution (absolute value can never be negative).

Related Topics

• Integers — Operations, Number Line, and Word Problems
• Comparing and Ordering Integers
• Adding and Subtracting Integers

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