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Opposites &
Absolute Value

Learn how every number has a mirror image on the number line, and how absolute value measures distance from zero: no matter which direction.

Pre-Algebra

What Are Opposites?

Two numbers are opposites if they are the same distance from zero on the number line but on opposite sides. For any number \( a \), its opposite is \( -a \).

-4 -2 0 +2 +4 2 units 2 units

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 never negative.

Symbol
Written with vertical bars: \( |a| \). Read as "the absolute value of \( a \)."
Always \( \geq 0 \)
No matter the input, the output is zero or positive: \( |a| \geq 0 \).

Seeing \( |a| \) on the Number Line

-5 -4 -3 -2 -1 0 1 2 3 4 5 -3 +3 3 units 3 units

The Formal Rule

The absolute value of a number \( a \) is defined piecewise:

If \( a \geq 0 \)
The number is already non-negative, so \( |a| = a \). Example: \( |7| = 7 \).
If \( a < 0 \)
Negate the negative to make it positive: \( |a| = -a \). Example: \( |-7| = -(-7) = 7 \).

The Opposites Rule

-5 0 +5 + (-5) + (+5)

Opposites cancel each other out: they pull equally in opposite directions, landing right back at zero.

Opposite vs. Absolute Value

Opposite \( -a \) Absolute Value \( |a| \)
What it does Flips the sign Strips the sign
Result sign Can be positive, negative, or zero Always \( \geq 0 \)
Example: \( a = 4 \) \( -4 \) \( 4 \)
Example: \( a = -4 \) \( 4 \) \( 4 \)
Example: \( a = 0 \) \( 0 \) \( 0 \)

Real-World Meaning: Distance

Absolute value appears whenever we talk about distance, because distance is always non-negative.

Temperature Change
If the temperature drops \( 12^\circ \), the size of the change is \( |-12| = 12 \) degrees.
Elevation
A submarine at \( -200 \) ft is \( |-200| = 200 \) feet from sea level.
Debt
A balance of \( -\$50 \) means you owe \( |-50| = \$50 \).

Interpreting Elevation

Sea Level (0) +850 ft Mountain Peak 850 ft Sub -200 ft 200 ft

How to Find Opposites & Absolute Value

1
Identify the number. Is it positive, negative, or zero?
2
For the opposite: flip the sign. Positive becomes negative, negative becomes positive. The opposite of \( 0 \) is \( 0 \).
3
For absolute value: ask "How far from zero?" Drop any negative sign. If already positive or zero, keep it.
4
Check your answer. Absolute value must be \( \geq 0 \). Opposite must have the opposite sign (unless the number is \( 0 \)).

Example 1: Simple Numbers

Find the opposite and absolute value of each:

\( a = 6 \)
Opposite: \( -6 \)
Absolute value: \( |6| = 6 \)
\( a = -9 \)
Opposite: \( 9 \)
Absolute value: \( |-9| = 9 \)
\( a = 0 \)
Opposite: \( 0 \)
Absolute value: \( |0| = 0 \)

Example 2: Temperature Context

At midnight, the temperature was \( -8^\circ \)F. By noon it rose to \( +15^\circ \)F. What is each temperature's distance from \( 0^\circ \)?

-15 -10 -5 0 5 10 15
\( -8^\circ \)
\( +15^\circ \)
 8 units 15 units

Example 3: Nested Expressions

Evaluate: \( -|{-12}| \) and \( |{-(-5)}| \).

A
\( -|{-12}| \): First, find \( |-12| = 12 \). Then apply the negative sign outside: \( -12 \).
B
\( |{-(-5)}| \): First, simplify inside: \( -(-5) = 5 \). Then take absolute value: \( |5| = 5 \).

Distance Between Two Points

The distance between two numbers \( a \) and \( b \) on a number line is: $$ |a - b| $$

-6 -4 -2 0 2 4 6 -3 +5 8 units

Common Mistakes

✕ Absolute value can be negative
Wrong: \( |-7| = -7 \). Absolute value is always \( \geq 0 \). Correct: \( |-7| = 7 \).
✕ Confusing \( -|a| \) with \( |-a| \)
\( -|5| = -5 \) (negative of the absolute value), but \( |-5| = 5 \). The negative sign outside the bars is not removed.
✕ "Opposite" means "negative"
The opposite of \( -3 \) is \( +3 \), not \( -3 \). "Opposite" means flip the sign, not "make it negative."

Summary Cheat Sheet

Number \( a \) Opposite \( -a \) \( |a| \) \( -|a| \)
\( 10 \) \( -10 \) \( 10 \) \( -10 \)
\( -7 \) \( 7 \) \( 7 \) \( -7 \)
\( 0 \) \( 0 \) \( 0 \) \( 0 \)
\( -\frac{1}{2} \) \( \frac{1}{2} \) \( \frac{1}{2} \) \( -\frac{1}{2} \)

Question 1 of 5

What is the value of \( |-13| \)?

Question 2 of 5

Which number is the opposite of \( -\frac{3}{4} \)?

Question 3 of 5

A diver is at an elevation of \( -45 \) feet. How far is the diver from sea level?

Question 4 of 5

What is the value of \( -|{-8}| \)?

Question 5 of 5

What is the distance between \( -6 \) and \( 4 \) on the number line?