Now you can add and subtract and multiply and divide signed numbers. Real problems mix all four, so two old ideas come back: the order of operations, and choosing the right sign rule for each step.
Key idea
Work in order: parentheses, exponents, multiply and divide (left to right), add and subtract (left to right). At each step, use the sign rule for that operation.
Why it works
Each operation keeps its own meaning. In $-2 - 3$, the 3 is subtracted: a move of 3 to the left, ending at $-5$. In $(-2)(-3)$, two negatives are multiplied, and the product is $6$. Mixing them up changes the problem. Doing one step at a time keeps each meaning clear.
Worked examples
Worked example: Multiply first
Find $-2 - 3 \times (-4)$.
- Multiply first: $3 \times (-4) = -12$.
- Now subtract: $-2 - (-12) = -2 + 12 = 10$.
Answer: $10$
Worked example: Parentheses first
Find $\frac{-6 + 2}{-2}$.
- The fraction bar groups the top: $-6 + 2 = -4$.
- Divide: $-4 \div (-2) = 2$.
Answer: $2$
Worked example: A diver
A diver is at $-12$ m. She swims down 3 m per minute for 4 minutes, then up 5 m. Where does she end?
- Write one expression: $-12 + 4 \times (-3) + 5$.
- Multiply: $4 \times (-3) = -12$.
- Add left to right: $-12 + (-12) = -24$, then $-24 + 5 = -19$. Estimate: she went down more than up, so she should end deeper than $-12$ ✓.
Answer: $-19$ m
Common mistake
Using a multiplying rule in an adding step, such as reading $-2 - 3$ as "two negatives make a positive." Name each operation before you use a rule: adding and subtracting are moves; multiplying and dividing use the sign rules for products.