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Introduction

Linear Inequalities

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

Examples : 20x < 300, 50x + 20y ≤ 110 and 40x + 20y < 120 etc.

2 < 7; 8 > 3 are the examples of numerical inequalities.

x < 3; y > 1; x ≥ 2, y ≤ 5 are some examples of literal inequalities.

2 < 6 < 8, 1 < x < 3 and 1 < y < 3 are the examples of double inequalities.

The values of x, which make an inequality a true statement, are called solutions of the inequality. Thus, any solution of an inequality in one variable is a value of the variable which makes it a true statement. And the set {0,1,2,3,4,5} is called its solution set.

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

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

**Rule - 1: ** Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality.

**Rule - 2: ** Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed.

The region containing all the solutions of an inequality is called the solution region. The solution region of a system of inequalities is the region which satisfies all the given inequalities in the system simultaneously.

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