SAME signs → POSITIVE (+)
DIFFERENT signs → NEGATIVE (−)
Think: "Agree = Happy(+) | Disagree = Unhappy(−)"
This rule applies to multiplication and division only.
SAME sign: add the numbers, keep the sign
DIFFERENT sign: subtract, take the sign of the bigger number
(−5) + (−3) = −8 | (−7) + 4 = −3
Simplify: (−4) × (−3) ÷ (−2)
Step 1: (−4) × (−3) = +12 ← same sign rule
Step 2: (+12) ÷ (−2) = −6 ← different sign rule
📎 Your turn: (−5) × 2 × (−1) = ____
Same signs (both negative) → ADD the numbers, keep the sign.
8 + 5 = 13, both negative → (−8) + (−5) = −13
(−3)×(−4) = +12 (same sign).
Then (+12)×(−1) = −12 (different sign).
📎 Key: Odd number of negatives → result is NEGATIVE.
Subtracting a negative = ADDING a positive!
(−6) − (−2) = (−6) + 2 = −4
📎 Rule: "Minus minus = Plus" → – (–) = +
Left to right: (−24)÷(−6) = +4 (same sign).
Then (+4)÷(−2) = −2 (different sign).
Rational number = any number written as p/q (q ≠ 0)
Sign rules still apply! SAME → (+) · DIFFERENT → (−)
For adding fractions: COMMON DENOMINATOR first!
LCD = Least Common Denominator → smallest shared bottom number
−34 + 12 = ?
LCD of 4 and 2 is 4
= −34 + 24 = −14
LCD = 6. −2/3 = −4/6.
−4/6 + 1/6 = −3/6 = −1/2
Same signs (both −) → positive.
3/5 × 10/9 = 30/45 = 2/3.
Result: +2/3
Dividing fractions: FLIP the second and multiply.
4/5 × (−5/2) = −20/10 = −2
Different signs → negative.
Distributive Property:
a(b + c) = ab + ac
Think: "The outside number visits everyone inside the parentheses."
Watch signs! −a(b + c) = −ab − ac (negative distributes!)
3x + 5x = 8x ✔ | 3x + 5y = 3x + 5y ✗ (can't combine)
2x² + 3x ≠ 5x² ← different powers!
Expand: −3(2x − 5)
= (−3)(2x) + (−3)(−5)
= −6x + +15
⚠️ Negative × Negative = Positive (don't forget!)
4 × 3x = 12x AND 4 × 2 = 8
Result: 12x + 8
(−5)(2x) = −10x AND (−5)(−3) = +15
⚠️ Negative × Negative = Positive!
Result: −10x + 15
3(x+4) = 3x + 12
−2(x−1) = −2x + 2
Combine: (3x − 2x) + (12 + 2) = x + 14
½ × 6x = 3x AND ½ × (−4) = −2
Result: 3x − 2
5(x − 4) = 5x − 20 ✔
This is actually factoring in reverse! GCF = 5.
5x − 20 → factor out 5 → 5(x − 4)
FACTORING = opposite of distributing
Step 1: Find the GCF (Greatest Common Factor)
Step 2: Divide each term by GCF
Step 3: Write GCF × (remaining terms)
GCF = the biggest number/variable that divides into ALL terms evenly
Factor: 12x + 8
GCF of 12 and 8 = 4
12x ÷ 4 = 3x | 8 ÷ 4 = 2
= 4(3x + 2)
📎 Check: 4(3x+2) = 12x+8 ✔
6x² + 9x → GCF = 3x
= 3x(2x + 3)
Remember: x² ÷ x = x (subtract exponents)
GCF of 15 and 10 is 5.
15x ÷ 5 = 3x | 10 ÷ 5 = 2
Result: 5(3x + 2)
GCF = 4x (both numbers share 4, both have x).
8x² ÷ 4x = 2x | 12x ÷ 4x = 3
Result: 4x(2x − 3)
⚠️ Option D is NOT fully factored (4x is the true GCF, not 2x).
Both A and B are correct!
6(−x + 3) = −6x + 18 ✔
−6(x − 3) = −6x + 18 ✔
Both A and B are valid factored forms.
QUICK REFERENCE CHEAT SHEET
(+)(+) = + | (−)(−) = + | (+)(−) = − | (−)(+) = −
Subtract negative: – (–) = +
Distribute: a(b+c) = ab+ac
Factor: GCF first, then divide each term
Fractions: LCD to add/subtract · FLIP to divide
2(3x+1) = 6x + 2
3(x−4) = 3x − 12
Combine: 6x+3x = 9x | 2−12 = −10
Result: 9x − 10
(−1/3)(9x) = −3x
(−1/3)(−6) = +2 ← negative × negative = positive!
Result: −3x + 2
Numbers: GCF(14, 21) = 7
x: min power is x¹ → x
y: min power is y¹ → y
GCF = 7xy
Substitute a = −2:
3(−2 + 5) − (−2)
= 3(3) − (−2)
= 9 + 2 = 11
⚠️ − (−2) = +2 !
Factor: GCF = 3x → 3x(2x − 3)
Substitute x = −1:
3(−1)(2(−1) − 3)
= (−3)(−2 − 3)
= (−3)(−5) = +15
Same signs → positive!