Questions: Subtracting Integers

Question 1

A student evaluates 3 − (−5) and writes the answer as −2. Which error most likely caused this?

• A. They added 3 and 5 to get 8, then incorrectly applied a negative sign
• B. They subtracted 5 from 3, ignoring the negative sign on −5
• C. They correctly applied the 'add the opposite' rule but made an arithmetic error
• D. They converted the problem to −3 + 5 instead of 3 + 5

Answer: (B)

The answer −2 results from computing 3 − 5 = −2 — the student dropped the negative sign on −5 and just subtracted 5. The correct application of 'add the opposite' converts 3 − (−5) to 3 + 5 = 8. Subtracting a negative number is the same as adding a positive, so the answer moves right on the number line (increases), not left.

Question 2

Which of the following is equivalent to −4 − (−9)?

• A. −4 − 9
• B. −4 + 9
• C. 4 − 9
• D. 4 + 9

Answer: (B)

Subtracting a number is the same as adding its opposite: a − b = a + (−b). So −4 − (−9) = −4 + 9 = 5. The opposite of −9 is +9. Options A and C fail to apply the conversion rule. Option D incorrectly flips the sign of −4 as well. Once you convert to −4 + 9, apply integer addition rules: different signs, so subtract absolute values (9 − 4 = 5) and keep the sign of the larger absolute value (positive), giving 5.

Question 3

The expression −7 − 3 and the expression −7 + (−3) are equivalent.

• T. True
• F. False

Answer: True

This is the 'add the opposite' rule directly: subtracting 3 is the same as adding −3. Both expressions give −7 + (−3) = −10. Same signs, so add the absolute values (7 + 3 = 10) and keep the negative sign. This example shows that the conversion rule always works, even when the subtracted number is positive.

Question 4

Subtracting a negative number generally produces a negative result.

• T. True
• F. False

Answer: False

Subtracting a negative is the same as adding a positive, which increases the value — the result depends on both numbers. 3 − (−5) = 3 + 5 = 8, which is positive. Even −2 − (−10) = −2 + 10 = 8, also positive. The result can be positive, negative, or zero depending on the starting value and the size of the subtracted negative. The 'two negatives' in a subtraction expression signal that you move right (add), not that the answer is negative.

Question 5

Explain in your own words why subtracting a negative number gives the same result as adding a positive number. Use either number line reasoning or algebraic reasoning.

Model answer: Algebraically: subtraction is defined as adding the opposite, so a − b = a + (−b). When b is already negative, say b = −5, the opposite of b is −(−5) = +5. So a − (−5) = a + 5. On a number line: subtracting a positive moves left; subtracting a negative reverses that direction, so you move right — exactly the same as adding a positive. Integer chips say the same: removing a negative chip (−) from a pile has the same net effect as adding a positive chip (+).

All three models — algebraic, geometric (number line), and physical (chips) — point to the same conclusion: the two negatives in 'subtract a negative' cancel out, producing a positive additive effect. The rule isn't arbitrary; it's a consequence of the definition of subtraction and the meaning of negative numbers.