Since opposites make zero, you can add any two signed numbers. Think of each number as a move on the number line: positive moves right, negative moves left.
Key idea
Same signs: add the absolute values and keep the sign. Different signs: subtract the smaller absolute value from the larger, and give the answer the sign of the number farther from $0$.
Why it works
With different signs the moves go opposite ways, so part of one cancels part of the other. In $-7 + 3$, three of the seven steps left are undone by the three steps right (three zero pairs). The 4 steps left that remain give $-4$. That is why you subtract: the cancelled part disappears.
Worked examples
Worked example: Different signs
Find $-7 + 3$.
- The signs are different. Subtract the absolute values: $7 - 3 = 4$.
- $-7$ is farther from $0$, so the answer is negative.
Answer: $-4$
Worked example: Both negative
Find $-5 + (-8)$.
- Both moves go left. Add the absolute values: $5 + 8 = 13$.
- Keep the negative sign.
Answer: $-13$
Worked example: The positive one is smaller
Find $9 + (-14)$.
- Different signs: $14 - 9 = 5$.
- $-14$ is farther from $0$, so the sum is negative.
Answer: $-5$
Common mistake
Adding the absolute values when the signs are different, so that $-7 + 3$ comes out as $-10$. Picture the move: from $-7$, adding $3$ moves right, toward $0$, so the answer must be bigger than $-7$. Check your answer's size against a quick sketch.