Absolute Value

How far is a number from zero? Absolute value measures distance on the number line, always giving a positive result!

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What Is Absolute Value?

Absolute value is the distance a number is from zero on the number line.

Key concept: Distance is ALWAYS positive or zero!

Symbol: | | (vertical bars around the number)

Read: |5| as "the absolute value of 5"

Think: "How far from zero?"

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Absolute Value on the Number Line

Number line helps visualize:

Example: |5| and |-5|

On number line:

• 5 is 5 units to the RIGHT of 0
• -5 is 5 units to the LEFT of 0
• Both are 5 units from 0

Therefore:

• |5| = 5
• |-5| = 5

Same distance, different directions!

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Definition

Formal definition:

|x| = x if x ≥ 0 (positive or zero stays same) |x| = -x if x < 0 (negative becomes positive)

In simple terms:

• If number is positive or zero → stays same
• If number is negative → drop the negative sign

Absolute value "removes" the negative!

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Basic Examples

Example 1: |7| = 7 (7 is 7 units from zero)

Example 2: |-7| = 7 (-7 is 7 units from zero)

Example 3: |0| = 0 (0 is 0 units from zero)

Example 4: |-15| = 15 (-15 is 15 units from zero)

Example 5: |100| = 100 (positive stays positive)

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Absolute Value Is Always Non-Negative

Key property: |x| ≥ 0 for all x

Absolute value is NEVER negative!

Examples:

• |8| = 8 (positive)
• |-8| = 8 (positive)
• |0| = 0 (zero, not negative)

Even if input is negative, output is positive or zero!

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Opposite Numbers Have Same Absolute Value

Opposites are same distance from zero!

Examples:

• |6| = 6 and |-6| = 6 (same!)
• |25| = 25 and |-25| = 25 (same!)
• |100| = 100 and |-100| = 100 (same!)

Different numbers, same absolute value!

Think: Mirror images across zero

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Comparing with Absolute Value

Compare |3| and |-5|:

|3| = 3 |-5| = 5

So: |-5| > |3|

Even though -5 < 3, the absolute value of -5 is greater!

Absolute value ignores which side of zero!

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Absolute Value of Zero

|0| = 0

Zero is the ONLY number whose absolute value equals itself AND its opposite!

Why? Zero is exactly 0 units from zero!

Special case to remember!

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Solving Absolute Value Equations

Simple equation: |x| = 5

Meaning: "What number is 5 units from zero?"

Answer: x = 5 or x = -5 (both!)

Both 5 and -5 are 5 units from zero.

Absolute value equations often have TWO solutions!

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Example: Solving |x| = 8

Question: What values of x make |x| = 8?

Think: What numbers are 8 units from zero?

Answer: x = 8 (8 units right of zero) x = -8 (8 units left of zero)

Check: |8| = 8 ✓ |-8| = 8 ✓

Both work!

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No Solution Cases

Equation: |x| = -3

Think: Can a distance be negative?

NO! Distance is never negative.

Therefore: No solution!

|x| = negative number has NO solution!

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Absolute Value in Expressions

Evaluate: |6 - 10|

Step 1: Calculate inside first 6 - 10 = -4

Step 2: Take absolute value |-4| = 4

Answer: 4

Always do operations inside | | first!

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More Expression Examples

Example 1: |-3| + |5| = 3 + 5 = 8

Example 2: |8| - |-2| = 8 - 2 = 6

Example 3: |7 - 9| + |2 + 1| = |-2| + |3| = 2 + 3 = 5

Evaluate each absolute value separately!

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Absolute Value and Multiplication

Property: |a × b| = |a| × |b|

Example: |-3 × 4| = |-12| = 12

Or: |-3| × |4| = 3 × 4 = 12

Same answer!

Example 2: |5 × (-2)| = |-10| = 10 Or: |5| × |-2| = 5 × 2 = 10

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Absolute Value and Addition

Note: |a + b| does NOT always equal |a| + |b|

Example: |-3 + 5| = |2| = 2

But: |-3| + |5| = 3 + 5 = 8

Different answers!

Must evaluate inside absolute value FIRST!

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Distance Between Two Numbers

Distance between a and b:

Distance = |a - b| or |b - a|

Same result either way!

Example: Distance between 8 and 3 |8 - 3| = |5| = 5 Or: |3 - 8| = |-5| = 5

Both give 5 units apart!

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Real-World: Temperature

Temperature change uses absolute value:

Started: -5°F Ended: 10°F

Change: |10 - (-5)| = |10 + 5| = |15| = 15°F

Temperature changed by 15 degrees!

Absolute value shows magnitude of change!

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Real-World: Elevation

Elevation differences:

Death Valley: -282 feet (below sea level) Mt. Whitney: 14,505 feet (above sea level)

Difference: |14,505 - (-282)| = |14,505 + 282| = |14,787| = 14,787 feet

Absolute value gives total distance!

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Real-World: Error/Deviation

Estimated: 100 Actual: 95

Error: |100 - 95| = |5| = 5

Don't care if over or under estimate! Just care HOW FAR off!

Absolute value measures error magnitude!

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Comparing Absolute Values

Which is greater?

Compare: |-20| and |15|

|-20| = 20 |15| = 15

20 > 15

So |-20| > |15|

Even though -20 < 15, its absolute value is greater!

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Nested Absolute Values

Example: | |-6| |

Step 1: Inside first |-6| = 6

Step 2: Outside |6| = 6

Answer: 6

Work from inside out!

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Absolute Value with Fractions

Example 1: |-3/4| = 3/4

Example 2: |1/2 - 3/4| = |-1/4| = 1/4

Example 3: |-2.5| = 2.5

Works with any number type!

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Order of Operations with Absolute Value

Absolute value acts like parentheses:

Do inside first, then take absolute value!

Example: 2 + |3 - 8|

Step 1: Inside absolute value 3 - 8 = -5

Step 2: Absolute value |-5| = 5

Step 3: Add 2 + 5 = 7

Answer: 7

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Common Mistakes to Avoid

❌ Mistake 1: Thinking |-5| = -5

• Wrong! |-5| = 5
• Absolute value makes it positive!

❌ Mistake 2: |a + b| = |a| + |b|

• Not always true!
• Must evaluate inside first

❌ Mistake 3: Forgetting two solutions

• |x| = 7 has TWO solutions: 7 and -7
• Don't forget the negative!

❌ Mistake 4: Thinking |x| can be negative

• Absolute value is NEVER negative
• |x| ≥ 0 always!

❌ Mistake 5: Not doing inside operations first

• Always evaluate inside | | before taking absolute value

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Properties of Absolute Value

Always non-negative: |x| ≥ 0 for all x

Zero only for zero: |x| = 0 only if x = 0

Same for opposites: |x| = |-x| for all x

Triangle inequality: |a + b| ≤ |a| + |b|

Multiplication: |a × b| = |a| × |b|

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Solving Strategy

For |x| = a:

• If a > 0: Two solutions (x = a or x = -a)
• If a = 0: One solution (x = 0)
• If a < 0: No solution

For expressions:

1. Calculate inside | | first
2. Take absolute value of result
3. Continue with other operations

For comparisons:

1. Find each absolute value
2. Compare the results

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Quick Reference

Definition:

• Distance from zero
• Always ≥ 0
• Symbol: | |

Basic:

• |positive| = positive
• |negative| = positive (drop sign)
• |0| = 0

Solving |x| = a:

• a > 0: x = a or x = -a
• a = 0: x = 0
• a < 0: no solution

Properties:

• |-x| = |x|
• |x × y| = |x| × |y|
• Do inside first!

Applications:

• Distance
• Error/deviation
• Temperature change
• Elevation difference

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Practice Tips

Tip 1: Visualize on number line

• Helps understand distance concept
• Makes it concrete

Tip 2: Remember two solutions

• For equations, check both positive and negative
• Both might work!

Tip 3: Work inside out

• Always evaluate inside | | first
• Then take absolute value

Tip 4: Check reasonableness

• Answer should be ≥ 0
• If negative, you made a mistake!

Tip 5: Practice with real situations

• Temperature changes
• Distances
• Errors in measurements

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Summary

Absolute value measures distance from zero:

Definition:

• |x| = distance of x from 0 on number line
• Always positive or zero
• Symbol: | | (vertical bars)

Key properties:

• |positive| = positive (stays same)
• |negative| = positive (becomes positive)
• |0| = 0
• |x| = |-x| (opposites have same absolute value)
• |x| ≥ 0 (never negative)

Solving equations:

• |x| = a (where a > 0) has two solutions: x = a and x = -a
• |x| = 0 has one solution: x = 0
• |x| = negative has no solution

In expressions:

• Evaluate inside absolute value first
• Then take absolute value
• Then continue with operations

Applications:

• Finding distance between numbers
• Measuring change (temperature, elevation)
• Calculating error or deviation
• Any situation where magnitude matters, not direction

Important skills:

• Understanding distance concept
• Working with negative numbers
• Solving equations with two solutions
• Order of operations

Mastering absolute value is essential for understanding distance, magnitude, and working with positive and negative numbers!