In this chapter we will understand the concept of solution set of inequality with solved examples.
What is solution set of inequality?
Solution set of inequality is the set of numbers which satisfies the given inequality expression.
For example;
Consider expression 2x < 6, where ” x ” is a whole number.
Solving the expression, we get;
2x < 6
x < 6 / 2
x < 3
Hence, x < 3 is the solution of given inequality.
Since x is a whole number, the solution set can be written as;
Solution Set ( S ) = { 0, 1, 2 }
When you put all the three numbers in the expression, the inequality will be satisfied.
Validating the solution.
Put x = 0 in 2x < 6
2. 0 < 6
0 < 6
Hence, the expression is satisfied.
Put x = 1 in 2x < 6
2. 1 < 6
2 < 6
Hence, expression is satisfied.
Put x = 2 in x < 6
2 . 2 < 6
4 < 6
Hence, expression is satisfied.
Conclusion
Solution set is the set of solution which satisfies the given inequality.
How to find solution set of Inequality ?
Follow the below steps;
(a) Solve the inequality equation to find the value of variable.
(b) Write down all the individual solution in the form of set.
(c) You can also use number line to graphically display the solution set.
Given below are examples of finding solution set for given given inequality.
Example 01
4x – 3 < 17, where ” x ” is natural number
Solution
Solving the given inequality.
4x – 3 < 17
4x < 17 + 3
4x < 20
x < 20 / 4
x < 5
Hence, x < 5 is the solution of given inequality.
Representing the solution in set form.
Solution set S = { 1, 2, 3, 4 }
Example 02
Find the solution set of given inequality.
8 < 2x + 6 < 30; where x is natural number.
Solution
The inequality consists of two parts;
8 < 2x + 6 and 2x + 6 < 30
Solving both the parts simultaneously.
8 – 6 < 2x and 2x < 30 – 6
2 < 2x and 2x < 24
1 < x and x < 12
Hence, we get the solution 1 < x < 12.
Writing all the solution in set form.
Solution set S = { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
Example 03
Find the solution set of given inequality
\mathtt{-2\ \leq x\ \leq 3} , where x is integer.
Solution
Since, \mathtt{-2\ \leq x\ \leq 3}
We can write solution set as;
S = { -2, -1, 0, 1, 2, 3}
Hence, all the six values in set will satisfy the expression.
Example 04
Find the value of x
9x – 3 > 33 ; where x is whole number.
Solution
Solving the given inequality expression.
9x > 33 + 3
9x > 36
x > 36 / 9
x > 4
Hence, x > 4 is the solution of given inequality.
Representing in solution set.
S = { 5, 6, 7, 8, 9, . . . . }
The solution can also be written in roster form.
S = { x > 4 ; x is whole number }
The solution is an infinite set since all the value above 4 is the solution.
Example 05
Find the solution set of given inequality.
11 < 6x – 7 < 41
Where x is a natural number.
Solution
The inequality consists of two parts.
11 < 6x – 7 and 6x – 7 < 41
Solving both the expression simultaneously.
11 + 7 < 6x and 6x < 41 + 7
18 < 6x and 6x < 48
18 / 6 < x and x < 48 / 6
3 < x and x < 6
Hence, we get the solution 3 < x < 6
Writing the solution in set form.
S = { 4, 5 }