Solve Linear Inequalities by Multiplication or Division

When solving linear inequalities with multiplication or division, there’s one crucial rule that differs from equations:

When multiplying or dividing both sides by a negative number, you must reverse (flip) the inequality sign.

The following diagrams give examples of solving a linear inequalities using multiplication and division. Scroll down the page for more examples and solutions.

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Multiplication and Division by a Positive Number
Multiplying or dividing both sides of an inequality by a positive number does NOT change the direction of the inequality symbol.

Example 1: Multiplying by a Positive Number
Solve: \(\frac{w}{2} \le 4 \)
Multiply both sides by 2.
\(\frac{w}{2} × 2 ≤ 4 × 2 \)
Solution: w ≤ 8

Example 2: Dividing by a Positive Number
Solve: 5z > 20
Divide both sides by 5.
\(\frac{5z}{5} > \frac{20}{5} \)
Solution: z > 4

Multiplying/Dividing by a Negative Number
Multiplying or dividing both sides of an inequality by a negative number MUST reverse the direction of the inequality symbol.

Example 3: Multiplying by a Negative Number
Solve: \(\frac{a}{-3} < 6 \)
Multiply both sides by -3 (which is negative).
\(\frac{a}{-3} × (−3) > 6 × (−3) \) <– Remember to flip the symbol
Solution: a > −18

Example 4: Dividing by a Negative Number
Solve: −4b ≤ 12
Divide both sides by -4 (which is negative).
\(\frac{-4b}{-4} × (−4) ≥ 12 ×(−4) \) <– Remember to flip the symbol
Solution: b ≥ −3

How to Solve Single-Step Inequalities by Division?
Example:
-4x ≤ 48

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Solving Single-Step Inequalities by Multiplication
Example:
-x/12 ≥ 7

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Solving and Graphing Inequalities using Multiplication or Division
How to solve and graph one variable inequalities using multiplication or division?
Example:
-0.5x ≤ 7.5
75x ≥ 125
x/1-3 > -10/9

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Solving one-step inequalities by multiplying or dividing
There are three examples shown.
Examples:

1. 5m ≤ 40
   
2. -3x < 12
   
3. y/5 ≥ -6.5

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