Linear Inequalities • Topic 1 of 4

Solving Inequalities

Solving a linear inequality uses the same balance moves as an equation, with one crucial rule: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. So −2x < 6 becomes x > −3 after dividing by −2. Adding or subtracting never flips the sign. The solution is a range of values, often written with an inequality symbol. Checking a test value from your solution range in the original inequality confirms the direction. This single flip rule is the most common SAT slip in inequality questions.

✅ Solved examples

1. Solve x + 5 > 9.
Subtract 5: x > 4.
2. Solve 3x ≤ 15.
Divide by 3 (positive, no flip): x ≤ 5.
3. Solve −2x < 6.
Divide by −2 and flip the sign: x > −3.
4. Solve 2x − 3 ≥ 7.
Add 3: 2x ≥ 10; divide by 2: x ≥ 5.

✏️ Practice — try these, take hints as needed

1. Solve x − 4 < 2.
Add 4 to both sides.
No flip (adding).
2 + 4.
x < 6.
2. Solve 5x ≥ 20.
Divide by 5 (positive).
No flip.
20 ÷ 5.
x ≥ 4.
3. Solve −3x > 12.
Divide by −3.
Dividing by a negative flips the sign.
12 ÷ (−3).
x < −4.
4. Solve 4x + 1 ≤ 9.
Subtract 1: 4x ≤ 8.
Divide by 4.
No flip.
x ≤ 2.
5. Solve 7 − x > 3.
Subtract 7: −x > −4.
Multiply by −1 and flip.
—
x < 4.

📝 Topic test — 8 questions

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