One-Step Inequalities

One-step linear inequalities are solved using addition, subtraction, multiplication, or division, similar to equations. However, there’s one critical rule:

When multiplying or dividing by a negative number, flip the inequality sign.

The inequality sign does not change when adding or subtracting.

The solutions to linear inequalities can be expressed in several ways: using inequalities, using a graph, or using interval notation.

The following diagrams give examples of solving a one-step linear inequalities. Scroll down the page for more examples and solutions.

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1. Addition and Subtraction
   Adding or subtracting the same number to both sides of an inequality does NOT change the direction of the inequality symbol.
   

Example 1: Using Addition
Solve: x − 7 < 10
Add 7 to both sides.
x − 7 + 7 < 10 + 7
Solution: x < 17

Example 2: Using Subtraction
Solve: y + 3 ≥ 5
Subtract 3 from both sides.
y + 3 − 3 ≥ 5 − 3
Solution: y ≥ 2

2. 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 3: 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 4: 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 5: 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 6: 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

One-Step Inequalities

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One-Step Inequalities 2

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One-step Inequality Examples

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Solving one-step inequalities

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