Training Signed Numbers Order of Operations with Signed Numbers
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Order of Operations with Signed Numbers

18 min Signed Numbers

Order of Operations with Signed Numbers

Order of operations — often remembered by the mnemonic PEMDAS — tells you the sequence in which to evaluate any mathematical expression. When signed numbers enter the picture, the rules stay exactly the same, but you must track positive and negative signs at every step.

A common mistake is to rush through an expression and lose a negative sign during an exponent or multiplication step. The safest approach is to work one operation at a time, rewriting the entire expression after each step so that every sign is visible. Parentheses and absolute-value bars add another layer, since they must be simplified from the inside out before you can proceed.

This lesson brings together all of the signed-number skills you have learned — addition, subtraction, multiplication, division, and exponents — and challenges you to apply them in multi-step expressions drawn from real-world contexts like temperature changes, account balances, and altitude differences.

Apply PEMDAS exactly as before, but be careful with signs!

Example 1

Evaluate: $-4 + 3 \times (-2)$

  1. Multiply: $3 \times (-2) = -6$
  2. Add:
  3. $-4 + (-6) = -10$
Example 2

Evaluate: $(-3)^2 - 4(-2)(5)$

  1. Exponent: $(-3)^2 = 9$
  2. Multiply:
  3. $4(-2)(5) = -40$
  4. Subtract: $9 - (-40) = 9 + 40 = 49$
Example 3

Evaluate: $\dfrac{-8 + 2}{-3}$

  1. Numerator: $-8 + 2 = -6$.
  2. Then $\dfrac{-6}{-3} = 2$.
Example 4

Evaluate: $-2x^2 + 3x - 5$ when $x = -4$

  1. $x^2 = (-4)^2 = 16$
  2. $-2(16) = -32$
  3. $3(-4) = -12$
  4. $-32 + (-12) - 5 = -49$
Example 5

Evaluate: $|{-5 + 2}| \times (-3) + 1$

  1. Inside absolute value:
  2. $-5 + 2 = -3$
  3. $|{-3}| = 3$
  4. $3 \times (-3) = -9$
  5. $-9 + 1 = -8$

Real-World Applications

Example 6

Temperature was $-8°F$ at 6 AM and rose $15°$ by noon. Noon temperature?

  1. $-8 + 15 = 7°F$
Example 7

Account balance: $\$250$. Withdraw $\$180$, deposit $\$75$. New balance?

  1. $250 + (-180) + 75 = \$145$
Interactive Explorer: Order of Operations with Signed Numbers
Step 1: E: −3² = −9
Step 2: M: 4 × (−2) = −8
Step 3: A: −9 + (−8)
Answer: −17

Practice Problems

1. $-5 + 2 \times (-3)$
2. $(-4)^2 - (-3)^2$
3. $\dfrac{-15 + 3}{-4}$
4. $3 - 2(-5 + 1)^2$
5. $(-1)^3 + (-1)^4 + (-1)^5$
6. $|{-8}| - |{-3}| \times 2$
7. $(-6)(2) - (-3)(4)$
8. Evaluate $x^2 - 2x + 1$ at $x = -3$
9. $\dfrac{(-4)^2 - 6}{-2 + 7}$
10. $-3|{-4 + 1}| + 7$
11. Temp is $5°C$, drops $12°$ overnight. New temp?
12. A submarine at $-200$ ft rises $75$ ft, then dives $120$ ft. Depth?
13. $(-2)^4 - 2^4$
14. $5(-3)^2 - 4(-2)^3$
15. $\dfrac{(-6)^2 - 36}{(-6) + 6}$
Show Answer Key

1. $-11$

2. $16 - 9 = 7$

3. $3$

4. $3 - 2(16) = -29$

5. $-1 + 1 + (-1) = -1$

6. $8 - 6 = 2$

7. $-12 + 12 = 0$

8. $9 + 6 + 1 = 16$

9. $\dfrac{10}{5} = 2$

10. $-9 + 7 = -2$

11. $-7°C$

12. $-200 + 75 - 120 = -245$ ft

13. $16 - 16 = 0$

14. $45 + 32 = 77$

15. Undefined ($0$ in denominator)

Multiplying and Dividing Signed Numbers Back to Module