1. Solving Inequalities Using Multiplication and Division
Algebra 3-3

2. Multiplication Property of Inequality for c > 0 (multiplying by a positive #)
For every real number a, b, and for c > 0
If a > b then a ∙ c > b ∙ c ex. 4 > -1  4(5) > -1(5)
20 > -5
If a < b then a ∙ c < b ∙ c ex. -6 < 3  -6(5) < 3(5)
< 15
Property is also true for ≥ and ≤

3. Multiplication Property of Inequality for c < 0 (multiplying by a negative #)
For every real number a, b, and for c < 0
If a > b then a ∙ c < b ∙ c ex. 4 > -1  4(-2) < -1(-2)
< 2
If a < b then a ∙ c > b ∙ c ex. -6 < 3  -6(-2) > 3(-2)
12 > -6
Property is also true for ≥ and ≤
You have to flip the direction of the in equality if you multiply both sides by a negative number.

4. Division Property of Inequality for c > 0 (Dividing by a positive #)
For every real number a, b, and for c > 0
If a > b then 𝑎 𝑐 > 𝑏 𝑐 ex. 6 > 4  > 4 2
3 > 2
If a < b then 𝑎 𝑐 < 𝑏 𝑐 ex. 2 < 8  < 8 2
1 < 4
Property is also true for ≥ and ≤

5. Division Property of Inequality for c < 0 (Dividing by a negative #)
For every real number a, b, and for c < 0
If a > b then 𝑎 𝑐 < 𝑏 𝑐 ex. 6 > 4  6 −2 < 4 −2
-3 < -2
If a < b then 𝑎 𝑐 > 𝑏 𝑐 ex. 2 < 8  2 −2 > 8 −2
-1 > -4
Property is also true for ≥ and ≤
You have to flip the direction of the inequality when you divide both sides by a negative number.

6. Cliff’s Notes
Solving inequalities with multiplication and division is identical to solving equations, EXCEPT when you multiply or divide both sides by a negative number, you have to FLIP THE DIRECTION OF THE INEQUALITY.