In this chapter we will learn basic concepts of linear inequality with its properties and solved examples.
Let us first understand the meaning of inequality first.
What is inequality ?
When numbers or variables are not equal to each other, they are represented by inequality sign.
Signs of Inequality
There are mainly 5 types of inequality sign which is commonly used in mathematics.
” > ” ⟹ ” greater than ” sign
” < ” ⟹ ” less than ” sign
“ \mathtt{\geq } ” ⟹ “greater than equal to” sign
“ \mathtt{\leq } ” ⟹ “less than equal to” sign
” \mathtt{\neq } ” ⟹ ” not equal to ” sign
These 5 symbols form the basics of inequality.
Let us now move to understand the concept of linear inequality.
What is Linear Inequality ?
The following points explains the concept of linear inequality.
(a) Linear equality is same as linear equation, except that instead of “equal to” sign, the expression has “inequality sign”.
Consider linear equation 4x + 3 = 3y.
The “=” sign states that left part 4x + 3 is equal to right part 3y.
Now consider linear inequality 4x + 7 > 8y.
Here ” > ” sign states that the left side 4x + 7 is greater than right side 8y.
(b) In linear inequality the highest power of variable in a given term is always equal to or less than 1.
The above two points are the characteristics of linear inequality. Let us see some examples for conceptual understanding.
Examples of Linear Inequality
Given below are some examples of linear inequality
(i) 9x + 2y < 7
It’s a linear inequality since;
(a) it contains inequality sign ” < “
(b) the highest power of variable is 1
9x ⟹ power is 1
2y ⟹ power is 1
7 ⟹ power is 0
(ii) \mathtt{11\ +\ 4x\ \geq \ 3x}
It’s a linear inequality since;
(a) it contains inequality sign “ \mathtt{\geq } ”
(b) highest power of terms is 1
11 ⟹ Power is 0
4x ⟹ Power is 1
3x ⟹ Power is 1
(iii) 10x + 3xy < 4
The given expression is not a linear inequality since the highest power of one of its terms is 2.
10x ⟹ Power is 1
3xy ⟹ Power is 2
4 ⟹ power is 0
Types of Linear Inequality
Linear inequality are classified on the basis of types of variable present in the expression.
Some of types of linear inequality commonly asked in school curriculum are;
(a) Linear Inequality with one variable
If the Inequality expression contains only one variable then it is called Linear Inequality with one variables.
Some examples are;
⟹ 4x + 3 > 2
The expression contains only one variable x.
⟹ 9y + 6 < 3y
The expression contains only one variable y.
⟹ \mathtt{13\ +\ 6x\ \geq \ 4x}
The expression contains only one variable x.
Linear Inequality with two variable
If the Linear inequality expression contains two variables then the expression is known as Linear Inequality with two variables.
Some examples are;
⟹ 6x + 3y > 7
The inequation contains two variable types x & y.
⟹ 16x + 9 < 13z
The inequation contains two variable types x and z.
Linear Inequality – Solved Problems
(01) Identify if the below expressions are linear inequality expressions or not.
(i) 18x + 10z > 3
(ii) 6xy + 20 < 3x
(iii) \mathtt{9\ +\ 15z\ \geq \ 3}
(iv) \mathtt{x^{2} \ +\ 7x\ +\ 5\ \leq \ 15}
(v) x ( y – 4 ) < 2
Solution
(i) 18x + 10z > 3
Its a linear inequality since;
(a) It contains inequality sign ” > “
(b) The highest power of all the terms is 1
18x ⟹ power is 1
10z ⟹ power is 1
3 ⟹ power is 0
(ii) 6xy + 20 < 3x
It’s not an inequality expression since the highest power of variable is 2.
6xy ⟹ power of variable is 2
20 ⟹ power is 0
3x ⟹ power is 0
(iii) \mathtt{9\ +\ 15z\ \geq \ 3}
It’s a linear inequality since the highest power of variable is 1.
9 ⟹ power is 0
15z ⟹ power is 1
3 ⟹ power is 0
(iv) \mathtt{x^{2} \ +\ 7x\ +\ 5\ \leq \ 15}
It’s not a linear inequality since the highest power of variable is 2.
\mathtt{x^{2}} ⟹ variable power is 2
7x ⟹ power is 1
5 ⟹ power is 0
15 ⟹ power is 0
(v) x ( y – 4 ) < 2
Let’s simplify the expression and remove the bracket.
x ( y – 4 ) < 2
x. y – 4x < 2
This expression is not a linear inequality since the highest power of variable is 2.
x. y ⟹ variable power is 2
4x ⟹ variable power is 1
2 ⟹ variable power is 0