A student wants to cancel out the term +8 in the equation x + 8 = 15. They ask: 'Which number do I add to +8 to make it disappear?' What is the correct answer, and why does it work?
A−8, because the additive inverse of 8 is −8, and 8 + (−8) = 0
B−8, because any number minus itself equals zero, which is a separate operation from addition
C1/8, because that is the inverse of 8
D0, because adding zero cancels any number
The additive inverse of 8 is −8, because 8 + (−8) = 0. Adding the opposite produces zero — the additive identity — which 'cancels' the term. Option B is a distractor: it gets the right answer but gives the wrong reason (this is addition, not subtraction). Option C confuses additive inverse with multiplicative inverse (reciprocal). Option D is wrong because 8 + 0 = 8, not 0.
Question 2 Multiple Choice
A student says 'the opposite of −12 must be −12 because it already has a negative sign.' What is the correct response?
AThe student is right — negative numbers are their own opposites
BThe opposite of −12 is +12, because opposites are equidistant from zero on opposite sides of the number line
CThe opposite of −12 is 0, because zero is the additive identity
DThe opposite of −12 is 1/12, because that is its multiplicative inverse
Opposites are mirror images across zero on the number line. −12 is 12 units to the left of zero; its opposite is 12 units to the right, which is +12. The confusion arises from thinking 'negative' already means 'opposite,' but the opposite of a negative number is always positive. Check: −12 + 12 = 0 ✓. The common misconception — that the opposite of a negative must also be negative — is exactly what option A represents.
Question 3 True / False
The double opposite of any number equals the original number — that is, −(−n) = n for all n.
TTrue
FFalse
Answer: True
Each application of 'take the opposite' reflects you across zero on the number line. Starting at n, one flip lands you at −n. A second flip from −n lands you back at n. This is not a trick — it is a direct consequence of what opposites mean geometrically. −(−7) = 7 because the opposite of '7 units left of zero' is '7 units right of zero,' which is just 7.
Question 4 True / False
The opposite of a number and the reciprocal of a number are the same thing.
TTrue
FFalse
Answer: False
These are inverses for completely different operations. The opposite (additive inverse) of 5 is −5, because 5 + (−5) = 0 — it undoes addition back to the additive identity (zero). The reciprocal (multiplicative inverse) of 5 is 1/5, because 5 × (1/5) = 1 — it undoes multiplication back to the multiplicative identity (one). Confusing the two is one of the most common errors in early algebra.
Question 5 Short Answer
Explain why subtraction can be understood as 'adding the opposite,' and give an example. Why is this reframing useful?
Think about your answer, then reveal below.
Model answer: Subtraction a − b equals a + (−b). For example, 7 − 3 = 7 + (−3) = 4. This reframing is useful because it reduces two operations (addition and subtraction) to a single operation (addition) plus the concept of additive inverse. In equation solving, every step that 'moves a term to the other side' is secretly using an additive inverse: to cancel +5, you add −5 to both sides. This unification makes the logic of algebra consistent and reduces the number of rules a student must memorize.
The reframing also clarifies why subtracting a negative number yields addition: 7 − (−3) = 7 + 3 = 10. Seen as 'add the opposite of −3,' the answer follows directly from the rule that the opposite of −3 is +3. Without the additive inverse concept, this double-negative rule appears arbitrary.