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Absolute Value

Understanding and using absolute value

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Absolute Value

Definition

The absolute value of a number is its distance from zero on the number line.

Symbol: ∣a∣|a|

Key point: Distance is always positive (or zero)!

Examples

  • ∣5∣=5|5| = 5 (5 is 5 units from zero)
  • ∣−5∣=5|-5| = 5 (-5 is also 5 units from zero)
  • ∣0∣=0|0| = 0 (0 is 0 units from zero)

Formal Definition

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Opposite Numbers

Numbers like 5 and -5 are opposites (same absolute value, different signs).

If ∣a∣=∣b∣|a| = |b| and a≠ba \neq b, then a=−ba = -b

Absolute Value with Operations

Evaluate inside first, then take absolute value:

  • ∣3+(−7)∣=∣−4∣=4|3 + (-7)| = |-4| = 4
  • ∣−6∣+∣2∣=6+2=8|-6| + |2| = 6 + 2 = 8
  • ∣5−8∣=∣−3∣=3|5 - 8| = |-3| = 3

Comparing Absolute Values

To compare ∣−8∣|-8| and ∣5∣|5|: ∣−8∣=8,∣5∣=5|-8| = 8, \quad |5| = 5 8>58 > 5

📚 Practice Problems

1Problem 1easy

❓ Question:

Find: ∣−12∣|-12|

💡 Show Solution

The absolute value is the distance from zero.

−12-12 is 12 units from zero.

∣−12∣=12|-12| = 12

Answer: 1212

2Problem 2easy

❓ Question:

Find: ∣−12∣|-12|

💡 Show Solution

The absolute value is the distance from zero.

−12-12 is 12 units from zero.

∣−12∣=12|-12| = 12

Answer: 1212

3Problem 3medium

❓ Question:

Evaluate: ∣7−10∣|7 - 10|

💡 Show Solution

Step 1: Evaluate inside the absolute value bars 7−10=−37 - 10 = -3

Step 2: Take absolute value ∣−3∣=3|-3| = 3

Answer: 33

4Problem 4medium

❓ Question:

Evaluate: ∣7−10∣|7 - 10|

💡 Show Solution

Step 1: Evaluate inside the absolute value bars 7−10=−37 - 10 = -3

Step 2: Take absolute value ∣−3∣=3|-3| = 3

Answer: 33

5Problem 5hard

❓ Question:

Evaluate: ∣−8∣−∣3−7∣|-8| - |3 - 7|

💡 Show Solution

Step 1: Evaluate each absolute value

∣−8∣=8|-8| = 8

For ∣3−7∣|3 - 7|: 3−7=−43 - 7 = -4 ∣−4∣=4|-4| = 4

Step 2: Subtract 8−4=48 - 4 = 4

Answer: 44

6Problem 6hard

❓ Question:

Evaluate: ∣−8∣−∣3−7∣|-8| - |3 - 7|

💡 Show Solution

Step 1: Evaluate each absolute value

∣−8∣=8|-8| = 8

For ∣3−7∣|3 - 7|: 3−7=−43 - 7 = -4 ∣−4∣=4|-4| = 4

Step 2: Subtract 8−4=48 - 4 = 4

Answer: 44

7Problem 7medium

❓ Question:

Evaluate: |7 - 10|

💡 Show Solution

Step 1: Evaluate inside the absolute value first. 7 - 10 = -3

Step 2: Find the absolute value. |-3| = 3

Remember: Always do operations inside the bars first, then take absolute value.

Answer: 3

8Problem 8hard

❓ Question:

Solve for x: |x - 4| = 9

💡 Show Solution

Step 1: Understand what this means. The distance between x and 4 is 9 units.

Step 2: Set up two equations. Case 1: x - 4 = 9 x = 13

Case 2: x - 4 = -9 x = -5

Step 3: Check both solutions. |13 - 4| = |9| = 9 ✓ |-5 - 4| = |-9| = 9 ✓

Answer: x = 13 or x = -5

Explain using:

⚠️ Common Mistakes: Absolute Value

Avoid these 3 frequent errors

🌍 Real-World Applications: Absolute Value

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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📌 Related Topics in Integers and Rational Numbers

Operations with IntegersAbsolute ValueComparing and Ordering Rational Numbers

❓ Frequently Asked Questions

What is Absolute Value?▾
Understanding and using absolute value
How can I study Absolute Value effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 8 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Absolute Value study guide free?▾
Yes — all study notes, flashcards, and practice problems for Absolute Value on Study Mondo are free to access. No account is needed.
What course covers Absolute Value?▾
Absolute Value is part of the Pre-Algebra course on Study Mondo, specifically in the Integers and Rational Numbers section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Absolute Value?▾
Yes, this page includes 8 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.