Chapter 5: Linear Inequalities

In algebra so far you have solved equations: $2x + 3 = 7$. But many real-world conditions are inequalities: "the cost must be at most $\\₹500$", "the temperature must be at least 25^\\circ\\text{C}", "the score must lie between $60$ and $80$". Linear inequalities are the calculus of this kind of statement.

The rules for manipulating inequalities are almost the same as for equations , with one critical difference: multiplying or dividing by a negative number reverses the inequality. That single rule generates every common mistake.

In Class XI you extend inequalities from one variable to two. A linear inequality in two variables $ax + by \le c$ defines a half-plane in the Cartesian plane. A system of inequalities defines an intersection , a feasible region. This is the algebraic foundation for linear programming, which you will study in Class XII.

For competitive exams (JEE), linear inequalities also appear in absolute-value problems, in domain calculations, and as a step in deriving bounds for trigonometric, algebraic and probabilistic expressions.

What's inside

1. Inequalities: definitions and the rules of manipulation.
2. Algebraic solution of linear inequalities in one variable.
3. Graphical solution and number-line representation.
4. Inequalities involving absolute value.
5. Linear inequalities in two variables , half-planes.
6. Systems of linear inequalities and feasible regions.

Key results / Formula card

For $ax + by < c$: the open half-plane on one side of the line $ax + by = c$.

How to read this chapter

Solve every inequality on paper and sketch the result on a number line (or plane). The visual habit prevents sign errors. For absolute-value inequalities, learn the two case-splits cold.