# What is an Inequality?

In classes IX and X we have studied equations in one variable and two variables and also learnt how to solve them.

Example: 1

Example: 2

We have also solved some statement problems by translating them in the form of equations.

Example: 3
The sum of the ages of A and B is 50 years. The difference of their ages is 10 years. Find their ages.
Solution:

These equations can be solved algebraically or graphically.
Now let us examine some statements which we shall try to translate in the form of an equation. (Is it possible?)
In this chapter, we will study linear inequalities in one and two variables. This study is very useful in solving problems in the field of Science, Mathematics, Statistics, Optimisation Problems in Economics, Business, Psychology etc,

Examples:
1. The tallest student in the class has height 165cm
2. The least mark obtained by students in XI B in Mathematics is 50
3. Think of all the natural numbers less than 10
4. If each pen costs Rs. 6/-, the maximum number of pens that can be bought with Rs. 100/- is what?
Let us take example 1 to 4 and try to form equations out of them
1. Statement means that all the students in the class have height less than or equal to 165cm
So we can write . Here stands for the height of any student in the class
Read the statement as "is less than or equal to 165".'' means less than or
equal to. This is an inequality.
Similarly,
2. , where stands for the mark obtained by a student in Mathematics, '' stands for "greater than or equal to"
3. ('<' strictly less than)
4. , here since cannot be equal to 100 (stands for the number of pens). Therefore we can also write it as (strictly less than)
In general the symbols used in inequalities are

Definition
Two real numbers or two algebraic expressions related by the symbol '<', '>', '' or '' form an inequality
Examples:
1. 10>5, -3<3, 0<2 are numerical inequalities. There is no solution. It is only a fact that has been illustrated in the inequality.
2. are literal (Algebraic) inequalities. Solutions are possible and must be specified
3.  are examples of double inequalities
If integers, then has solution 2, 3, 4 and -1<y<5 has
olution 0, 1, 2, 3, 4.
4. , then can take all values
, then can take
all the real values below 3 (not including 3)
Now we shall see how to solve an inequality.