Question

how do you use what you know about absolute value to add integers

Answer 1

Think of absolute value, |x|, as the distance of x from 0. That distance idea makes adding integers easy:

Rules
- If the integers have the same sign, add their absolute values and keep that sign.
- Reason: both move in the same direction on the number line.
- Formula: if sign(x) = sign(y) then x + y = sign(x)·(|x| + |y|).
- If the integers have opposite signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.
- Reason: the motions partially cancel; the leftover distance goes in the direction of the larger magnitude.
- Formula: if sign(x) ≠ sign(y) then x + y = sign of the larger·(| |x| − |y| |).
- Special case: if |x| = |y| and signs are opposite, x + y = 0.

Examples
- 6 + 3: same sign, |6| + |3| = 9 → answer 9.
- (−6) + (−3): same sign (negative), |−6| + |−3| = 9 → answer −9.
- 7 + (−4): opposite signs, subtract |7| − |4| = 3, sign of larger absolute (7) is positive → 3.
- (−7) + 4: subtract 7 − 4 = 3, sign of larger absolute (7) is negative → −3.
- 5 + (−5): |5| = |−5| so they cancel → 0.

Visual ways to think about it
- Number line: start at the first integer, move right for a positive addend and left for a negative addend.
- Counters: use red counters for negative and black for positive; pairs cancel.

Using absolute value gives a precise short rule and matches the intuitive number-line picture.