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\( - \)
\( + \)
\( | \; | \)

Absolute Value
vs. Order

Learn the difference between how far a number is from zero and where it sits on the number line.

6th Grade Math

What is Order on a Number Line?

Order tells us which number is greater or lesser. On a number line, numbers increase as you move to the right and decrease as you move to the left. So \( -3 < 2 \) because \( -3 \) is to the left of \( 2 \).

\( -4 \)
\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( -3 \)
\( 2 \)
\( -3 < 2 \) (left is less)

What is Absolute Value?

Absolute value tells us how far a number is from zero, ignoring direction. We write it with bars: \( |{-3}| = 3 \) and \( |3| = 3 \). Both are 3 units from zero.

\( -4 \)
\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
 3 units
\( 3 \)
 3 units

Why Does This Distinction Matter?

In everyday life, we use both ideas without thinking about it. Imagine two people standing on opposite sides of a building:

Order (Position)
"Person A is further left than Person B." This is about where they stand relative to each other.
Absolute Value (Distance)
"Both are the same distance from the door." This is about how far, ignoring direction.

The Core Rule

\( -5 \)
\( -4 \)
\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( -5 \)
\( 2 \)
Order: \( -5 < 2 \)
But \( |-5| = 5 > |2| = 2 \)
 5 units 2 units

Smaller Number, Bigger Distance?

A negative number can be less than a positive number but farther from zero. The key insight:

\( -7 < 4 \)
\( -7 \) is to the left of \( 4 \). It is the lesser number.
\( |{-7}| > |4| \)
\( -7 \) is 7 units from zero, while \( 4 \) is only 4 units.

Two Different Questions

Order (Position) Absolute Value (Distance)
Question Which is greater or lesser? Which is farther from zero?
Symbol \( < \), \( > \), \( = \) \( | \; | \)
Direction Matters (left vs right) Ignored (only distance)
Result Can be negative Always \( \geq 0 \)
Example \( -6 \) \( < \) \( 1 \) \( |{-6}| \) \( = 6 > \) \( |1| \) \( = 1 \)

How to Apply This Skill

1
Read the statement carefully. Is it about position (\( < \), \( > \)) or distance (\( | \; | \))?
2
If it uses \( < \) or \( > \): think about where numbers sit on the number line (left/right).
3
If it uses \( | \; | \): ignore the sign and think about distance from zero.
4
Watch for trick statements that mix both ideas. Translate each part separately.

Example 1: Interpreting a Statement

\( -5 \)
\( -4 \)
\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( 3 \)
\( 4 \)
 4 units 3 units

Example 2: Comparing Debts

Maya owes $8 (account balance: \( -8 \)). Leo owes $3 (account balance: \( -3 \)). Who has the lower balance? Who owes more?

Order (Lower Balance)
\( -8 < -3 \), so Maya has the lower balance. She is further left on the number line.
Absolute Value (More Debt)
\( |{-8}| = 8 > |{-3}| = 3 \), so Maya owes more. Her balance is farther from zero.

Example 3: Temperature Comparison

City A: \( -12^\circ\text{F} \). City B: \( 5^\circ\text{F} \). Which city is colder? Which temperature is farther from \( 0^\circ\text{F} \)?

\( -15 \)
\( -12 \)
\( -5 \)
\( 0 \)
\( 5 \)
\( 10 \)
\( -12^\circ\text{F} \)
\( 5^\circ\text{F} \)
 12 units 5 units
1
Colder (order): \( -12 < 5 \), so City A is colder. This is about position.
2
Farther from \( 0^\circ\text{F} \) (absolute value): \( |{-12}| = 12 > |5| = 5 \). City A is farther from \( 0^\circ\text{F} \). This is about distance.

Example 4: Two Negatives

Compare \( -2 \) and \( -9 \). Which is greater? Which has the greater absolute value?

\( -10 \)
\( -9 \)
\( -7 \)
\( -5 \)
\( -3 \)
\( -2 \)
\( 0 \)
\( -9 \)
\( -2 \)
 9 units 2 units
Order
\( -9 < -2 \). So \( -2 \) is greater (further right).
Absolute Value
\( |{-9}| = 9 > |{-2}| = 2 \). So \( -9 \) has the greater absolute value (farther from zero).

Reading Carefully

In word problems and test questions, certain phrases signal whether you need order or absolute value:

Order Phrases
"greater than," "less than," "warmer," "colder," "higher," "lower," "to the left/right of"
Absolute Value Phrases
"farther from zero," "closer to zero," "greater distance," "owes more," "magnitude"

Common Mistakes

✕ Mixing Up the Two Ideas
Saying "\( -6 < 2 \), so \( -6 \) is closer to zero." Wrong: \( < \) tells you position, not distance. \( |{-6}| = 6 > |2| = 2 \).
✕ "Less Than" Means "Smaller Absolute Value"
Thinking a lesser number always has a smaller absolute value. \( -10 < 1 \), but \( |{-10}| = 10 \gg |1| = 1 \).
✕ Forgetting Negatives With Two Negatives
For \( -3 \) and \( -7 \): \( -3 > -7 \) (order), but \( |{-3}| < |{-7}| \) (absolute value). These go in opposite directions!

Summary Table

Numbers Order Comparison Absolute Value Comparison
\( -5 \) and \( 3 \) \( -5 < 3 \) \( |{-5}| = 5 > |3| = 3 \)
\( -2 \) and \( 8 \) \( -2 < 8 \) \( |{-2}| = 2 < |8| = 8 \)
\( -7 \) and \( -1 \) \( -7 < -1 \) \( |{-7}| = 7 > |{-1}| = 1 \)
\( 4 \) and \( -4 \) \( -4 < 4 \) \( |{-4}| = 4 = |4| = 4 \)
\( -3 \) and \( -9 \) \( -9 < -3 \) \( |{-9}| = 9 > |{-3}| = 3 \)

Question 1 of 5

Which statement correctly describes the relationship between \( -6 \) and \( 2 \)?

Question 2 of 5

Sam says: "Since \( |{-3}| < |{-8}| \), we know that \( -3 < -8 \)."
Is Sam correct?

Question 3 of 5

The temperature on Monday was \( -15^\circ\text{F} \) and on Tuesday it was \( 10^\circ\text{F} \). Which day had the temperature farther from \( 0^\circ\text{F} \)?

Question 4 of 5

Jenna owes \( \$12 \) (balance: \( -12 \)). Kyle has \( \$5 \) (balance: \( +5 \)). Which statement about absolute value is correct?

Question 5 of 5

For which pair of numbers is the greater number also farther from zero?