**Definition of Linear Inqualities**

Two real numbers or two algebraic expressions related by the symbol

‘<’, ‘>’, ‘≤’ or ‘≥’ form an inequality

3 < 5; 7 > 5 are the examples of **numerical inequalities**

x < 5; y > 2; x ≥ 3, y ≤ 4 are some examples of **literal inequalities**

**Keys point of Inequalities**

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

**Example **Solve 5x – 3 < 3x +1 when

(i) x is an integer, (ii) x is a real number

**Solution **We have, 5x –3 < 3x + 1

or 5x –3 + 3 < 3x +1 +3

or 5x < 3x +4

or 5x – 3x < 3x + 4 – 3x

or 2x < 4

or x < 2

**Example **Solve 7x + 3 < 5x + 9. Show the graph of the solutions on number line

**Solution **We have 7x + 3 < 5x + 9 or 2x < 6 or x < 3

**Graphical Solution of Linear Inequalities in Two Variables**

- The region containing all the solutions of an inequality is called the solution region
- In order to identify the half plane represented by an inequality, it is just sufficient to take any point (a, b) (not on line) and check whether it satisfies the inequality or not. If it satisfies, then the inequality represents the half plane and shade the region which contains the point, otherwise, the inequality represents that half plane which does not contain the point within it. For convenience, the point (0, 0) is preferred
- If an inequality is of the type ax + by ≥ c or ax + by ≤ c, then the points on the line ax + by = c are also included in the solution region. So draw a dark line in the solution region.
- If an inequality is of the form ax + by > c or ax + by < c, then the points on the line ax + by = c are not to be included in the solution region. So draw a broken or dotted line in the solution region