1st PUC Mathematics · Chapter 6

Linear Inequalities — The One Rule That Decides Whether You Get This Right

Linear Inequalities looks like a simplified version of linear equations — and for the most part, it is. Every technique from solving equations carries over directly, with one critical exception that costs more marks in this chapter than anything else: what happens to the inequality sign when you multiply or divide by a negative number.

The Rule That Changes Everything

When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. This is the single rule that separates inequalities from equations, and it has a genuine reason behind it, not just a memorised exception.

Consider 2 < 5 — clearly true. Multiply both sides by −1: you’d get −2 and −5. But −2 is actually greater than −5 on the number line (it’s further right), so the true statement is −2 > −5, not −2 < −5. The sign had to flip to keep the statement true. This is why the rule exists — it isn’t arbitrary, it’s what keeps the inequality mathematically honest after multiplying by something negative.

The Rules That Stay the Same

Everything else behaves exactly like an equation: you can add or subtract the same number from both sides without changing the inequality’s direction, and you can multiply or divide by a positive number without any change either.

Worked Example — Solving Algebraically

Solve: 3x − 5 < 2x + 1

Subtract 2x from both sides: x − 5 < 1. Add 5 to both sides: x < 6. No negative multiplication was needed here, so the sign never flips — solution is x < 6, or in interval notation, (−∞, 6).

Now a case where the flip matters: Solve −2x + 3 ≥ 7. Subtract 3 from both sides: −2x ≥ 4. Now divide both sides by −2 — since we’re dividing by a negative, the sign flips from ≥ to ≤: x ≤ −2, or (−∞, −2].

Graphing on the Number Line

A strict inequality (< or >) uses an open circle at the boundary point, since that exact value isn’t included in the solution. A non-strict inequality (≤ or ≥) uses a closed (filled) circle, since the boundary value is included. The direction of the shading or arrow matches the direction of the inequality — greater-than shades to the right, less-than shades to the left.

Worked Example — System of Two Inequalities

Solve: 2x + 1 > 5 and 3x − 2 < 10, and find the common solution.

First: 2x + 1 > 5 → 2x > 4 → x > 2. Second: 3x − 2 < 10 → 3x < 12 → x < 4. The values satisfying both conditions are 2 < x < 4, or the interval (2, 4) — this intersection is what a “system of inequalities” question is always asking for.

Where Students Actually Lose Marks

Forgetting to flip the sign when dividing by a negative is by far the most common error — and it’s easy to miss when the negative coefficient is buried mid-problem rather than obviously visible. The second common mistake is mixing up open and closed circles when graphing, especially under time pressure when ≤ and < start to look interchangeable.

Frequently Asked Questions

Does the flip rule apply to adding or subtracting a negative number too?

No — adding or subtracting never flips the sign, regardless of whether the number is positive or negative. The flip rule applies only to multiplication and division by a negative.

How is “solution set” different from a single answer?

An inequality’s solution is a whole range of values, not one number — which is why answers are written as intervals like (2, 4) or inequalities like x < 6, not as x = 6.

Where does this chapter come up again later?

Directly in 2nd PUC Linear Programming, where inequalities define the feasible region on a graph — the exact same open/closed circle and shading logic extends into two dimensions there.

Get the Rule Right the First Time

One flipped sign is the difference between full marks and a wrong answer here. Daniel Sir drills this exact distinction until it’s automatic. See the PUC Mathematics Programme →