Inequalities • Topic 1 of 4

Linear Inequalities

A linear inequality is solved exactly like a linear equation, with one rule that changes everything: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. So −2x > 6 becomes x < −3, not x > −3. Adding or subtracting any number, or multiplying by a positive number, leaves the direction unchanged. CAT often phrases these as systems: "find all integers x satisfying 3 < 2x − 1 ≤ 9", where you solve the compound inequality to get 2 < x ≤ 5 and then count integer solutions {3, 4, 5}. The fast habit is to isolate x while watching every multiplication for a hidden negative. With two variables, treat each constraint as a boundary line and reason about the feasible region — the same logic that underlies linear programming.

✅ Solved examples

1. Solve −3x + 5 > 14.
−3x > 9. Divide by −3 and flip: x < −3.
2. Find all integers x with 3 < 2x − 1 ≤ 9.
Add 1: 4 < 2x ≤ 10. Divide by 2: 2 < x ≤ 5. Integers: 3, 4, 5 (three values).
3. Solve (x − 2)/(−4) ≥ 3.
Multiply by −4 and flip: x − 2 ≤ −12 ⇒ x ≤ −10.
4. If 5 − 2x ≥ 1 and x + 3 > 2, find the range of x.
First: −2x ≥ −4 ⇒ x ≤ 2. Second: x > −1. Combined: −1 < x ≤ 2.

✏️ Practice — try these, take hints as needed

1. Solve −4x ≤ 20.
Divide by −4.
Flip the sign.
20/(−4) = −5.
x ≥ −5
2. Count integers x with −2 ≤ 3x + 1 < 13.
Subtract 1 throughout.
−3 ≤ 3x < 12.
Divide by 3: −1 ≤ x < 4.
5 integers (−1, 0, 1, 2, 3)
3. Solve 7 − x/2 < 4.
Subtract 7: −x/2 < −3.
Multiply by −2, flip.
x > 6.
x > 6
4. If 2x − 3 ≤ 5 and 4 − x ≤ 6, find the range.
First gives x ≤ 4.
Second: −x ≤ 2 ⇒ x ≥ −2.
Intersect.
−2 ≤ x ≤ 4
5. Solve (3 − 2x)/5 > 1.
Multiply by 5: 3 − 2x > 5.
−2x > 2.
Divide by −2, flip.
x < −1

📝 Topic test — 8 questions

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