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Subtract Using
Additive Inverse

Master the powerful idea that subtraction is really addition in disguise. Learn to rewrite \( p - q \) as \( p + (-q) \) and find distances on the number line.

7th Grade Math

What Is an Additive Inverse?

Every number has a partner that adds up to zero. That partner is called the additive inverse. For any number \( a \), its additive inverse is \( -a \), because:

\( 5 \)
Inverse: \( -5 \) because \( 5 + (-5) = 0 \)
\( -3 \)
Inverse: \( 3 \) because \( -3 + 3 = 0 \)
\( \tfrac{1}{2} \)
Inverse: \( -\tfrac{1}{2} \) because \( \tfrac{1}{2} + (-\tfrac{1}{2}) = 0 \)

Inverses on the Number Line

A number and its additive inverse are mirror images across zero. They sit at equal distances from zero, but on opposite sides.

−4 −3 −2 −1 0 1 2 3 4 −3 3 same distance from 0

Why Does This Matter?

Subtraction with negative numbers can feel confusing: what does \( 4 - (-3) \) even mean? The additive inverse rule gives you a single, reliable strategy: turn every subtraction into addition, then use the addition rules you already know.

Simplifies
One rule replaces many sign-combination cases.
Connects
Links subtraction and addition as inverse operations.
Extends
Works for integers, fractions, and decimals: all rational numbers.

The Core Rule

To subtract a number, add its opposite. This works for every pair of rational numbers: positive, negative, fractions, and decimals.

\( p - q \)
rewrite
\( p + (-q) \)

See It on the Number Line

Compute \( 2 - 5 \). Rewrite as \( 2 + (-5) \). Start at 2, then move 5 units left (adding a negative means moving left).

−5 −4 −3 −2 −1 0 1 2 3 4 5 start: 2 +(−5): move 5 left land: −3

Subtraction vs. Negative Sign

Students often confuse the subtraction operation with the negative sign. They look the same, but they play different roles. The additive inverse rule bridges both.

Subtraction (operation) Negative sign (attribute)
Symbol \( 8 \;\mathbf{-}\; 3 \) \( \mathbf{-}3 \)
Meaning "take away" or "find the difference" "the opposite of 3"
Connection \( 8 - 3 = 8 + (-3) \) : subtraction becomes adding a negative

Distance Equals Absolute Difference

The distance between two rational numbers on the number line is always the absolute value of their difference:

−4 0 3 −4 3 |3 − (−4)| = |7| = 7

Notice: \( 3 - (-4) = 3 + 4 = 7 \). The additive inverse rule is what lets us simplify \( 3 - (-4) \) into \( 3 + 4 \).

How to Apply the Rule

1
Identify the subtraction. Find the expression \( p - q \).
2
Find the additive inverse of \( q \). Flip its sign: \( q \to -q \).
3
Rewrite as addition. Replace \( p - q \) with \( p + (-q) \).
4
Use integer addition rules. Same signs: add magnitudes, keep sign. Different signs: subtract magnitudes, take the sign of the larger.

Example 1: \( 7 - 12 \)

1
Rewrite: \( 7 - 12 = 7 + (-12) \)
2
Different signs: \( |{-12}| - |7| = 12 - 7 = 5 \)
3
Larger magnitude is \( 12 \) (negative), so the answer is negative.
−6 −5 −4 −3 −2 −1 0 1 4 7 start: 7 +(−12): move 12 left −5

Example 2: \( -4 - (-9) \)

Subtracting a negative is where the additive inverse rule really shines.

1
Rewrite: \( -4 - (-9) = -4 + 9 \) (the inverse of \( -9 \) is \( 9 \))
2
Different signs: \( 9 - 4 = 5 \). Larger magnitude is \( 9 \) (positive).
−5 −4 −3 −2 −1 0 1 2 3 4 5 start: −4 +(9): move 9 right 5

Example 3: Fractions

The rule works for all rational numbers. Compute \( \frac{3}{4} - \frac{5}{4} \).

1
Rewrite: \( \frac{3}{4} - \frac{5}{4} = \frac{3}{4} + \left(-\frac{5}{4}\right) \)
2
Different signs, same denominator: \( \frac{3 + (-5)}{4} = \frac{-2}{4} \)
3
Simplify: \( \frac{-2}{4} = -\frac{1}{2} \)

Real-World Distance Problem

A scuba diver is at −15 meters (below sea level). A bird is flying at 8 meters above sea level. What is the distance between them?

0 m 8 m −15 m 23 m

Common Mistakes to Avoid

✕ Dropping a negative
Writing \( 5 - (-3) = 5 - 3 \). You must flip the sign: \( 5 - (-3) = 5 + 3 = 8 \).
✕ Sign of the result
Saying \( 3 - 7 = 4 \). Remember: \( 3 + (-7) = -4 \). The larger magnitude wins.
✕ Forgetting absolute value for distance
Getting \( -4 - 3 = -7 \) and saying the distance is \( -7 \). Distance is always positive: \( |-7| = 7 \).

Summary of Key Ideas

Concept Rule Example
Additive Inverse \( a + (-a) = 0 \) \( 6 + (-6) = 0 \)
Subtraction Rule \( p - q = p + (-q) \) \( 5 - 8 = 5 + (-8) = -3 \)
Subtracting a Negative \( p - (-q) = p + q \) \( -2 - (-6) = -2 + 6 = 4 \)
Distance \( |p - q| \) \( |-3 - 5| = |-8| = 8 \)

Question 1 of 5

Which expression is equivalent to \( 9 - 14 \)?

Question 2 of 5

What is \( -6 - (-11) \)?

Question 3 of 5

The temperature at dawn was \( -8^\circ\text{F} \). By noon it was \( 15^\circ\text{F} \). What is the distance (change) in temperature?

Question 4 of 5

Simplify: \( \frac{1}{3} - \frac{5}{3} \)

Question 5 of 5

Which statement correctly uses the additive inverse property?