Absolute Value

For Elementary Students

What is Absolute Value?

Absolute value tells you how FAR a number is from zero — not which direction, just the distance!

Think about it like this: If you walk 5 steps forward or 5 steps backward from your starting point, either way you're 5 steps away! Absolute value works the same way.

Starting point = 0

Walk 5 steps right → You're 5 steps away
Walk 5 steps left → You're ALSO 5 steps away!

The Symbol: | |

Vertical bars around a number mean "absolute value."

|5| means "the absolute value of 5"
|-5| means "the absolute value of negative 5"

The Big Rule

Absolute value is ALWAYS positive (or zero)!

Distance can't be negative — you can't be "-5 steps away" from something!

Examples with Positive Numbers

|8| = 8

8 is already positive, so it stays 8.

How far is 8 from zero? 8 units!

|25| = 25

25 is 25 units from zero.

Examples with Negative Numbers

|-7| = 7

Negative 7 becomes positive 7!

How far is -7 from zero? 7 units!

|-20| = 20

-20 is 20 units from zero.

The Special Case: Zero

|0| = 0

Zero is zero units from zero! (It's already at zero!)

Visualizing on a Number Line

<--5 units---|---5 units-->
    -5       0       5

Both -5 and 5 are 5 units away from 0!

So |-5| = 5 and |5| = 5 (same answer!)

More Examples

What's Inside Matters!

Sometimes you need to calculate what's INSIDE the absolute value bars first!

Example: |5 + 3|

Step 1: Add inside the bars

5 + 3 = 8

Step 2: Take absolute value

|8| = 8

Answer: 8

Example: |4 - 9|

Step 1: Subtract inside

4 - 9 = -5

Step 2: Take absolute value

|-5| = 5

Answer: 5

Comparing with Absolute Value

Question: Which is bigger: |-10| or |6|?

Step 1: Find absolute values

|-10| = 10
|6| = 6

Step 2: Compare

10 > 6

Answer: |-10| is bigger!

Even though -10 seems "smaller" (more negative), its DISTANCE from zero is larger!

Real-Life Absolute Value

Temperature: "It's 20 degrees below zero" means the temperature is -20°F, but the absolute value is 20 degrees. |-20| = 20

Owing Money: If you owe $50, your balance is -$50, but the absolute amount you owe is $50. |-50| = 50

Elevation: 300 feet below sea level is -300 feet, but the absolute elevation is 300 feet. |-300| = 300

Memory Trick

"Absolute value makes everything happy (positive)!"

Negative numbers turn positive! Positive numbers stay positive! Zero stays zero!

Quick Tips

Tip 1: Absolute value is ALWAYS positive or zero (never negative!)

Tip 2: |-x| = |x| (they're the same distance from zero!)

Tip 3: Calculate inside the bars FIRST, then take absolute value

Tip 4: Think "distance" — how far from zero?

Tip 5: The bars work like parentheses — do what's inside first!

For Junior High Students

Formal Definition

Absolute value of a number is its distance from zero on the number line, regardless of direction.

Mathematical definition:

|x| = {  x    if x ≥ 0
      { -x    if x < 0

Notation: |x| reads as "the absolute value of x"

Key property: |x| ≥ 0 for all real numbers x

Geometric Interpretation

On a number line, |x| represents the distance between x and the origin (0).

<----distance----|----distance---->
      -5    -3   0   3    5

|5| = 5 (5 units right of 0) |-5| = 5 (5 units left of 0) |-3| = 3 (3 units left of 0)

Key insight: Distance is always non-negative.

Basic Properties

Property 1: Non-negativity

|x| ≥ 0 for all x ∈ ℝ

Property 2: Zero

|0| = 0
|x| = 0 if and only if x = 0

Property 3: Symmetry

|x| = |-x|

Both x and -x have the same distance from zero.

Property 4: Identity for positive numbers

If x ≥ 0, then |x| = x
If x < 0, then |x| = -x

Evaluating Absolute Value Expressions

Example 1: Evaluate |8|

8 ≥ 0, so |8| = 8

Example 2: Evaluate |-12|

-12 < 0, so |-12| = -(-12) = 12

Example 3: Evaluate |0|

|0| = 0

Absolute Value with Operations

Order of operations applies: Evaluate inside absolute value bars first, then apply absolute value.

Example 1: |5 + 3|

= |8|
= 8

Example 2: |-4 + 2|

= |-2|
= 2

Example 3: |6 - 10|

= |-4|
= 4

Example 4: |-8 - 3|

= |-11|
= 11

Important Distinction

|a + b| ≠ |a| + |b| in general

Example:

Left side: |(-5) + 3| = |-2| = 2
Right side: |-5| + |3| = 5 + 3 = 8

2 ≠ 8

However, there's a special case (triangle inequality):

|a + b| ≤ |a| + |b|

This is always true!

Absolute Value Equations

Form: |x| = a

Solution depends on value of a:

If a > 0: Two solutions: x = a or x = -a

Example: |x| = 5

Solutions: x = 5 or x = -5

Check: |5| = 5 ✓ and |-5| = 5 ✓

If a = 0: One solution: x = 0

Example: |x| = 0

Solution: x = 0 only

If a < 0: No solution

Example: |x| = -3

No solution (absolute value can't be negative!)

More Complex Equations

Example: |x + 2| = 5

Solution:

x + 2 = 5  or  x + 2 = -5
x = 3      or  x = -7

Verification:

|3 + 2| = |5| = 5 ✓
|-7 + 2| = |-5| = 5 ✓

Comparing Absolute Values

Example: Compare |-15| and |8|

|-15| = 15
|8| = 8

Since 15 > 8, we have |-15| > |8|

Key principle: Comparing absolute values compares distances from zero, not the numbers themselves.

Operations with Absolute Values

Addition:

|-5| + |3| = 5 + 3 = 8

Subtraction:

|-10| - |4| = 10 - 4 = 6

Multiplication:

|a| · |b| = |a · b|

Example: |-3| · |4| = 3 · 4 = 12
Also: |-3 · 4| = |-12| = 12 ✓

Division:

|a| / |b| = |a / b|  (b ≠ 0)

Example: |-12| / |3| = 12 / 3 = 4
Also: |-12 / 3| = |-4| = 4 ✓

Distance Between Two Points

Distance formula on number line:

Distance between a and b = |a - b| = |b - a|

Example: Distance between -4 and 7

Method 1: |7 - (-4)| = |7 + 4| = |11| = 11
Method 2: |-4 - 7| = |-11| = 11

Both methods give the same result: 11 units apart.

Example: Distance between -8 and -2

|-8 - (-2)| = |-8 + 2| = |-6| = 6

Inequalities with Absolute Value

|x| < a (where a > 0) means: -a < x < a

Example: |x| < 5

Solution: -5 < x < 5
(All numbers between -5 and 5)

|x| > a (where a > 0) means: x < -a or x > a

Example: |x| > 3

Solution: x < -3 or x > 3
(Numbers farther than 3 from zero)

Applications

Temperature variation:

Find absolute difference: |T₁ - T₂|

Example: |25°C - (-10°C)| = |35°| = 35° difference

Error and tolerance:

Acceptable error: |actual - target| ≤ tolerance

Example: Manufacturing part: |length - 10cm| ≤ 0.1cm

Financial balance:

Debt magnitude: |-$500| = $500 owed

Physics:

Magnitude of displacement, force, velocity (absolute value of vector components)

Common Errors

Error 1: Assuming |x| = x always

❌ |-5| = -5
✓ |-5| = 5

Error 2: Distributing incorrectly

❌ |a + b| = |a| + |b| (not always true!)
✓ Use triangle inequality: |a + b| ≤ |a| + |b|

Error 3: Forgetting to consider both solutions

❌ |x| = 7 → x = 7 only
✓ |x| = 7 → x = 7 or x = -7

Error 4: Accepting negative absolute value

❌ |x| = -4 has solutions
✓ |x| = -4 has NO solutions

Tips for Success

Tip 1: Always evaluate inside absolute value bars first (order of operations)

Tip 2: Remember |x| ≥ 0 always (absolute value cannot be negative)

Tip 3: For |x| = a, check if a ≥ 0 before solving

Tip 4: Both x and -x give same absolute value: |x| = |-x|

Tip 5: Think geometrically: absolute value = distance from zero

Tip 6: For equations |x| = a (a > 0), remember TWO solutions: x = ±a

Tip 7: Verify solutions by substituting back into original equation

Summary

Definition: |x| = distance from x to 0 on number line

Key properties:

• |x| ≥ 0 for all x
• |x| = 0 if and only if x = 0
• |x| = |-x| (symmetry)
• |ab| = |a| · |b| (multiplicative)

Solving |x| = a:

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

Distance formula: Distance = |a - b|

Applications: Error measurement, temperature differences, financial magnitudes, physical quantities

Practice