Inequality : Definition, Property and Examples

What is inequality in Math?

The word “inequality” means that the expression is not equal to another expression.
In Math the inequality is used to form a comparison between two numbers

Tools for Inequality are
” > ” {Greater than}
” < ” {Less than}
” \leqq ” {less than equal to}
” \leqq ” {greater than equal to}

All these tools are used for comparison of different math expressions, for example
a > b, means “a” is greater than “b”
a < b, means “a” is less than “b”
a \leqq b, , means “a” is less than equal to “b”

Types of Inequality

Numerical Inequality

The comparison of two numbers come under numerical inequality
Example:
2 < 5 [ 2 is less than 5]
6 > 3 [6 is greater than 3]

Literal Inequality

The comparison in which one element is variable come under literal inequality

Example 01
x > 6 [ x is greater than 6]
Here x is variable and its value can be { 7, 8, 9, 10, . . . . . infinity}

Example 02
y < 4 [y is less than 4]
Here y is variable and its value can be { 0, 1, 2, 3, -1, -2, etc..)

Double Inequality

When there is more than two relation between the variable and number, we can term it as double inequality

Example
4 < x < 8 [x is greater than 4 and less than 8]

-3 < y < 0 [ y is greater than -3 and less than 0}
Hence, you will find two kinds of inequality sign in this form.

Slack Inequalities

Inequalities with greater than equal to or less than equal to sign

ax + by ≥ c
ax + by ≤ c

The presence of “≥” and “≤” makes it slack inequality

Strict Inequalities

Inequality with strict less than or greater than sign is known as strict inequality
ax + b < 0
ax + b > 0
ax + by > c

Linear Inequality with one variable

ax + b > 0
ax + b ≤ 0
ax + b ≥ 0
Here the variable is x

Linear inequality with two variables

ax + by < c
ax + by > c
ax + by ≤ c
In all the equations there are two variables x & y

Quadratic Inequalities with one variable

Both the above equations are quadratic equation with one variable x

Rules for solving inequality

For Grade 11 we have to restrict ourselves to Linear Inequality with one variable, so we will discuss this topic in detail

Rules for solving Linear Inequality

Suppose you have been provided with the following inequality function
30x < 200

For solving the above expression remember the following rules of inequality
1. Equal Number can be added (or subtracted) to both side of the inequality equation
For Example
30x + 20 < 200+20

2. Both side can be multiplied (or divided) by the same positive number without change in the sign of inequality
For Example
(30x * 2) < (200 *2)

3. Both side can be multiplied (or divided) by same negative number, but the sign would change
For Example
(30x * -2) > (200 * -2)
Notice how the sign is changed from ” < ” to ” >” due to multiplication with -2

Questions

(1) Solve 5x – 3 < 3x + 1, when
(a) x is integer
(b) x is natural number

Solution
Let’s solve the equation first
5x – 3 < 3x +1
5x – 3x < 1+3
2x < 4
x < 2

(a) when x is integer the value of x can be anywhere between minus infinity to 1

(b) x is a natural number
Natural Number (N) = 1, 2, 3, 4, 5, 6, 7 ……
Since x < 2
The required value of x = 1 is the answer

(02) Solve 7x + 3 < 5x + 9

Solution
Solving the inequality equation
7x + 3 < 5x + 9
7x – 5x < 9 – 3
2x < 6
x < 3

Hence the value of x can be anywhere from minus infinity to less than three ( as shown in below number line)

(03) Solve the following expression
{\frac{3x-4}{2} \ \geqq \frac{x+1}{4} -1}

Solution

\mathbf{\frac{3x-4}{2} \ \geqq \frac{x+1}{4} -1}\\\ \\
\mathbf{\frac{3x-4}{2} \ \geqq \frac{x+1-4}{4}}\\\ \\

\mathbf{\frac{3x-4}{2} \ \geqq \frac{x-3}{4}}\\\ \\
\mathbf{3x-4\ \geqq \frac{x-3}{2}}\\\ \\
\mathbf{6x-8\ \geqq x-3}\\\ \\
\mathbf{6x-x\ \geqq -3\ +8}\\\ \\
\mathbf{5x\ \geqq \ 5}\\\ \\
\mathbf{x\ \geqq \ 1}\\

For the above inequality the value of x can be from 1 to infinity.