Linear Inequalities

The mathematical statement which says that one quantity is not equal to another quantity is called Inequalities.

For example, if and are two quantities such that then any one of the following four conditions will be true:

i.e., either i) ii) iii) iv)

Each of the four conditions, given above, is an inequality. In the same way each of the following also represents an inequality:

etc.

Linear inequality in one variable:

1. 
2. 
3. 
4. 

Solving Linear Inequality

To solve a given linear inequality means to find the value or values of the variable used in it.

The following working rules must be adopted for solving a given linear inequality:

Rule 1: On transferring a positive term from one side of an inequality to its other side, the sign of the term becomes negative.

e.g.

Rule 2: On transferring a negative term from one side of an inequality to it’s other side, the sign of the term becomes positive.

e.g.

Rule 3: If each term of an inequality be multiplied or divided by the same positive number, the sign of the inequality remains the same.

e.g

Rule 4: If each term of an inequality be multiplied or divided by the same negative number, the sign of inequality changes.

e.g

Rule 5: If signs of each term on both the sides of an inequality are changed, the sign of inequality reverses.

e.g

Rule 6: If either sides of an inequality are positive or both are negative, then on taking their reciprocals, the sign of inequality reverses.

e.g

Replacement set & Solution set

The set, from which the absolute value of the variable is to be chosen, is called replacement set and its subset, whose elements satisfy the given inequality ,is called the solution set.

Example: Let the given inequality be :

1. If the replacement set , the set of natural numbers,the solution set = {1,2}
2. If the replacement set , the set of whole numbers, the solution set = {0,1,2}
3. If the replacement set or ,the set of integers, the solution set ={ 2, 1, 0, -1, -2…}

Example 1: If the replacement set is the set of natural numbers (N) find the solution of:

Solution:

the replacement set (natural numbers)

solution set= {1, 2, 3}

Example 2: If the replacement set is the set of whole numbers ( ), find the solution set of :

Solution:

the replacement set is the set of whole numbers

the solution set = {0, 1, 2, 3, 4}

Example 3: If the replacement set is the set of integers between -6 and 8, find the solution set of

Solution:

the replacement set is the set of integers between -6 and 8

the solution set = {-5,-4,-3,-2,-1, 0, 1, 2, 3, 4}

Example 4: If the replacement set is the set of real numbers (R), find the solution set of:

Solution:

the replacement set is the set of real numbers

solution set = { }

Example 5: Solve the following inequality: if

1. (integers)
2. (real numbers)

Solution:

and

or, and

or, and

or,

When.

solution set = {-2, -1 ,0 ,1 ,2, 3}

When

solution set = { and }

Exercise

1. State true or false:
   1. 
   2. 
2. State whether the following statements are true or false given are real numbers and .
   1. If , then
   2. If , then
   3. If then
   4. If then
   5. If then
3. If , find the solution set of the following inequalitys:
   1. 
   2. 
4. If the replacement set is the set of whole numbers, solve:
   1. 
   2. 
5. Solve the inequality: given that .
6. If , find:
   1. The smallest value of , when is a real number.
   2. The smallest value of , when is an integer.
7. If the replacement set is the set of real numbers, solve:
   1. 
   2. 
8. Find the smallest value of for which , where is an integer.
9. Find the largest value of for which and .
10. Solve the inequality: and .
11. Given (integers), find the solution set of:
12. Given (whole numbers), find the solution set of : .