Q1. Solve 3x + 5 < 14.
3x < 14 − 5 = 9
x < 9/3
x < 3
Linear Inequalities — Class 11
Inequalities behave like equations — except for one crucial rule. Solve any linear inequality and see the boundary value instantly.
Linear inequalityA statement like ax + b > c (or <, ≤, ≥) instead of an equals sign.
Rules of inequalitiesAdding/subtracting the same number keeps the direction; multiplying/dividing by a NEGATIVE number REVERSES the inequality sign.
Solution setUsually an interval of numbers (e.g. x > 3), not a single value like in an equation.
Boundary-value calculator Interactive
For ax + b > c, find the critical value x = (c − b)/a. Remember: if a is negative, flip the inequality direction.
Boundary value, x = (c−b)/a—
If a is negative, the inequality sign must be reversed when you divide — a > b becomes < after dividing by a negative.
Solved examples
Q2. Solve −2x + 3 ≤ 11.
−2x ≤ 11 − 3 = 8
Divide by −2 (a negative number) — REVERSE the inequality:
x ≥ −4
Formula sheet
| Quantity | Formula |
|---|---|
| ax + b > c (a > 0) | x > (c − b)/a |
| ax + b > c (a < 0) | x < (c − b)/a (sign flips!) |
Common mistakes & exam wins
- Multiplying or dividing both sides by a NEGATIVE number reverses the inequality sign — the most common mistake in this chapter.
- A solution is usually a range of values, shown as an open circle (strict <, >) or filled circle (≤, ≥) on the number line.
- For a system of two inequalities, the solution is where both number-line regions overlap.
Frequently asked questions
What happens when you multiply an inequality by a negative number?
The direction of the inequality sign reverses (e.g. > becomes <).
What is the solution of a linear inequality?
Usually a range (interval) of values that satisfy it, not a single number.
How do you represent the solution of an inequality on a number line?
Use an open circle for strict inequalities (<, >) and a filled/closed circle for ≤ or ≥, shading the valid region.