Practice Maths

L33 — Adding & Subtracting Integers II

Key Terms

double sign
Two operation or sign symbols placed next to each other, e.g. −(−) or +(−). Simplify them before calculating.
same signs → positive
When two adjacent signs are the same — +(+) or −(−) — the combined effect is positive (addition).
different signs → negative
When two adjacent signs differ — +(−) or −(+) — the combined effect is negative (subtraction).

Double Sign Rules

  • +(+) = +  —  adding a positive: move right
  • +(−) = −  —  adding a negative: move left
  • −(+) = −  —  subtracting a positive: move left
  • −(−) = +  —  subtracting a negative: move right

Worked Example

Calculate 4 − (−3)

Step 1 — Identify the double sign: −(−) = +  (same signs → positive)

Step 2 — Rewrite: 4 − (−3) = 4 + 3

Step 3 — Calculate: 4 + 3 = 7

The Key Rule: Subtracting a Negative

The most important rule in this lesson: subtracting a negative number is the same as adding a positive number.

Written as a rule: a − (−b) = a + b

  • 5 − (−3) = 5 + 3 = 8
  • −4 − (−7) = −4 + 7 = 3
  • 2 − (−10) = 2 + 10 = 12

Think of it this way: “taking away a debt” increases your wealth. If you owe someone $3 and that debt is cancelled, you’re $3 better off.

Why Does This Work? The Number Line

On a number line, subtraction means moving left. Subtracting a negative is like “reversing a leftward move” — which sends you right (i.e. adding).

  • Start at 2. Subtract −5: instead of going left 5, go right 5. Land at 7.
  • Start at −3. Subtract −4: instead of going left 4, go right 4. Land at 1.

Mixed Operations with Multiple Integers

When you have a string of integers, simplify all double signs first, then work left to right:

  • −5 + 3 − (−2) + (−4): simplify → −5 + 3 + 2 − 4
  • Collect positives: 3 + 2 = 5. Collect negatives: −5 − 4 = −9.
  • Combine: 5 − 9 = −4

Simplifying Double Signs

  • + (+) = + (addition)
  • + (−) = − (subtraction)
  • − (+) = − (subtraction)
  • − (−) = + (addition) ← the key one!
When you see two signs next to each other: same signs → positive, different signs → negative. Write out the simplified expression before calculating to avoid confusion.

  1. Simplify: adding a negative

    1. 5 + (−3)
    2. 8 + (−2)
    3. 1 + (−6)
    4. −4 + (−3)
    5. 3 + (−9)
    6. −5 + (−8)
    7. 10 + (−4)
    8. −7 + (−2)

  2. Simplify: subtracting a negative

    1. 7 − (−2)
    2. 4 − (−5)
    3. −3 − (−1)
    4. −6 − (−8)
    5. 2 − (−8)
    6. −5 − (−9)
    7. 0 − (−3)
    8. −10 − (−4)

  3. Mixed double signs

    1. 3 + (−8)
    2. −2 + (−5)
    3. 10 − (−4)
    4. −1 − (−7)
    5. −4 + (−4)
    6. 5 − (−6)
    7. −9 − (−9)
    8. 0 + (−7)

  4. True or False: Double Sign Rules

    1. −(−5) = +5
    2. +(−3) = +3
    3. Adding a negative always gives a smaller result.
    4. Subtracting a negative is the same as adding a positive.

  5. Rewrite without brackets

    1. 6 + (−7)
    2. 9 − (−4)
    3. −5 + (−2)
    4. −3 − (−3)
    5. 0 − (−10)

  6. Debt Problem

    Mia owes $12 (represented as −12). She earns $8 and then borrows another $5 (adding −5). What is her balance?

  7. Temperature

    The temperature is −3°C. It “drops by negative 6 degrees” (i.e. −(−6)). What is the new temperature?

  8. Golf Scores

    In golf, under par is negative. Player A has a score of −4. Player B has a score of −7. What is the difference between their scores? (A − B)

  9. Find the Missing Integer

    1. ___ − (−3) = 5
    2. 7 + ___ = 2
    3. −4 − ___ = 1

  10. Exam Challenge

    1. Simplify: −8 − (−3) + (−2)
    2. Explain in words why −(−4) = +4.
    3. Is (−5) − (−5) = 0? Justify your answer.
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