Linear Inequalities in One Variable: Solve and Graph Them

A linear inequality in one variable looks like ax + b > 0, where the unknown appears only to the first power and the relation is >, <, ≥, or ≤. Its solution is never a single number but a whole range — for example, every x greater than 2.25.

Inequalities matter whenever a quantity has a limit rather than an exact value: a budget you must stay under, a minimum score needed to pass, or a speed that must stay below a legal limit. They describe the possible, not just the exact.

Key Concepts

• Inequality symbols: < means "less than", > means "greater than", while ≤ and ≥ include equality.
• Solution set: the collection of all numbers satisfying the inequality, usually an interval.
• Interval notation: x > 9/4 is written (9/4, ∞); x ≤ 5 is (−∞, 5].
• Number-line picture: an open dot marks an excluded endpoint (> or <), a closed dot marks an included one (≥ or ≤).
• Equivalent inequalities: two inequalities with exactly the same solution set, reached by legal moves.

Formulas

You may add or subtract the same number from both sides freely. You may multiply or divide both sides by a positive number with no change. The critical rule: multiplying or dividing by a negative number flips the inequality sign. For example, −2x < 6 becomes x > −3 after dividing by −2 and flipping < to >.

• Add/subtract any number: direction unchanged
• Multiply/divide by a positive number: direction unchanged
• Multiply/divide by a negative number: direction flips (< becomes >, and so on)

Worked Example

Solve 3x − 5 > 4 − x. Add x to both sides: 4x − 5 > 4. Add 5 to both sides: 4x > 9. Divide by 4, which is positive, so the sign stays: x > 9/4, i.e., x > 2.25. Verify with a test value inside the solution set, x = 3: left side 3(3) − 5 = 4, right side 4 − 3 = 1, and 4 > 1 holds. Verify the boundary: at x = 2.25 both sides equal 1.75, confirming 2.25 is the exact threshold and is excluded by the strict inequality.

Common Misconceptions

• Dividing by a negative without flipping: −3x > 9 gives x < −3, not x > −3 — the most frequent inequality error.
• Answering with a single value: "x = 2.25" is wrong; the solution is every number above 2.25.
• Confusing > with ≥: x > 2.25 excludes 2.25 itself, drawn as an open dot; x ≥ 2.25 includes it, drawn closed.

Real-World Applications

• Budgeting: total spending must stay at or below income.
• Minimum qualifying marks and grade boundaries in exams.
• Speed and safety limits in traffic and machinery.
• Simple optimisation bounds, such as minimum material needed for a task.

Linear inequalities in one variable are solved like equations with one extra rule: flip the sign when multiplying or dividing by a negative number. The answer is always a range, pictured on a number line with open or closed dots at the boundary.