CBSE Class 11 Mathematics Linear Inequalities Notes

Class XI

Chapter Notes

Linear Inequalities

Definitions

1. Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form an inequality.

2. Inequalities containing ‘<’, or ‘>’ are called strict inequalities.

3. Inequalities containing ‘≤’ or ‘≥’ are called slack inequalities.

4. An inequality containing any two of‘<’, ‘>’, ‘≤’ or ‘≥’is called double inequality.

5. Solution of an inequality in one variable is the value of the variable which makes it a true statement

6. A linear expression in one variable involving the inequality symbol is linear inequality in one variable. General forms:

ax + b < 0 (1)

ax + b > 0 (2)

ax + b ≤ 0 (3)

ax + b ≥ 0 (4)

7. A linear inequality involving two variables is known as a linear inequality in two variables. General forms

ax + by < c (5)

ax + by > c (6)

ax + by ≤ c (7)

ax + by ≥ c (8)

ax² + bx + c ≤ 0 (9)

ax² + bx + c ≥ 0 (10)

7. The region containing all the solutions of an inequality is called the solution region.

8. The solution region of the system of inequalities is the region which satisfies all the given inequalities in the system simultaneously.

9. Quadratic inequality is quadratic polynomial with inequality sign. Generic quadratic inequality is of the form ax²+bx+c > 0 Concepts

1. If two real numbers are related by the symbols ‘<’, ‘>’, ‘≤’ or ‘≥’ then the inequality is a numerical inequality and in case of algebraic expressions it is literal inequality. 2<3 is numerical inequality 5x+2≤7 is literal inequality

2. Rules for simplifying the inequalities

Rule 1: Equal numbers may be added to (or subtracted from) both sides of an equation.

If a < b then a + c < b + c.

Rule 2: Both sides of an equation may be multiplied (or divided) by the same non – zero number.

If a < b then ac < bc

Rule 3: Sign of inequality is reversed in case of multiplication (or division) by a negative number If a < b then ak > bk, where k is a negative number

Rule 4: Sign of inequality is reversed in case of taking the reciprocals  

5. In order to identify the half plane represented by an inequality, take any point (a, b) (not on line) and check whether it satisfies the inequality or not.
If it satisfies, then inequality represents the half plane and shade the region which contains the point, otherwise, the inequality represents the half plane which does not contain the point within it. For convenience, the point (0, 0) is preferred.

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