# Solving linear inequality

In this chapter we will learn method to solve linear inequality.

Let us first review the concept of linear inequality.

## What is Linear Inequality?

An expression is said to be linear inequality if;

(i) it contains inequality sign ” > “, “<“, etc.

(ii) the highest power of variables in given term is 1.

Some examples of linear inequality are;

4x + 3y < 7
2 + 2x > 3
5(x + 3) > 9

## How to solve linear inequality

In order to solve linear inequality, follow the below steps;

(01) Separate variables on one side and constant on other side.

(02) Movement of terms in inequality expression.

While moving any term from one side to another, you need to follow below rules;

Multiplication becomes division
Note: While moving negative number the inequality sign will reverse.

Division becomes multiplication
Note: While moving negative number the inequality sign will reverse.

The above rules are similar to rules applied for solving equations.

(03) Find the value of given variables.

### Important inequality rules for solving problems

(a) You can add same number on both sides of inequality.

For example
x + 7 > y

Adding 3 on both sides, we get;

x + 7 + 3 > y + 3

x + 10 > y + 3

(b) You can subtract same number on both sides of inequality.

For example;

y + 10 < 3

Subtracting number 2 on both sides;
y + 10 – 2 < 3 – 2

y + 8 < 1

Subtracting same number on both sides will not change the character of the given inequality.

(c) Multiplication & division of positive number

You can multiply or divide same positive number on both sides of inequality.

Multiplication example
Consider inequality 6x – 5 > 2

Multiply 3 on both sides.

3 (6x – 5) > 3( 2 )

18x – 15 > 6

Hence, multiplying positive number on both sides will not change the character of inequality.

Division Example
Consider inequality 2x – 18 < 20

Dividing 2 on both sides.

\mathtt{\frac{2x\ -\ 18}{2} \ < \ \frac{20}{2}}\\\ \\ \mathtt{x\ -\ 9\ < \ 10}

Hence, dividing positive number on both sides will not change character of inequality.

(d) Multiplication & division of Negative number

When we multiply or divide negative number to inequality, we need to reverse the sign of the given inequality.

For example;
Consider inequality 4x – 5 > 2y

Multiply -3 on both sides.
Since the number is negative, we will reverse ” > ” sign to ” < “

4x – 5 > 2y

-3 (4x – 5) < -3 (2y)

-12x + 15 < -6y

I hope you understood the above rules. Let us now solve some inequality questions for further understanding.

## Linear Inequality – Solved Problems

(01) Solve for value of x
6x – 15 < 2x – 3

Method 01
We will separate variable x on one side and constant on other side.

Moving 2x to the left

6x – 2x – 15 < -3

Move 15 to the right.

6x – 2x < -3 + 15

4x < 12

Move 4 to the right.
Multiplication becomes division.

x < 12 / 4

x < 3

Hence for all values of x < 3, the above inequality will satisfy.

Method 02
You can also solve the above inequality with following method.

6x – 15 < 2x – 3

Separate variable on left and constant on the right side.

Subtract 2x on both sides.
This step will eliminate variable from right side.

6x – 15 – 2x < 2x – 3 – 2x

4x – 15 < -3

This step will remove constant from left side.

4x – 15 + 15 < -3 + 15

4x < 12

Divide 4 on both sides

4x / 4 < 12 / 4

x < 3

Hence, the value of given inequality is x < 3.

Example 02
y – 15 > 6y – 3

Method 01
Separate variables on left and constant on right.

Move 6y towards left.

y – 15 – 6y > -3

Move 15 towards right.
y – 6y > -3 + 15

-5y > 12

Multiply -1 on both side to make the variable (-5y) positive.
Multiplication of negative number will reverse the sign of inequality.

– 5y > 12

-1 (-5y) < (-1) (12)

5y < -12

Move 5 towards right.
Multiplication becomes division.

y < -12 / 5

y < – 2.4

Hence, for given inequality the value of y < -2.4

Method 02

You can also solve the inequality by another method.

y – 15 > 6y – 3

We have to separate variable on left and constant on right.

Subtract 6y on both sides.
This step will eliminate variable form right side.

y – 15 – 6y > 6y – 3 – 6y

-5y – 15 > – 3

This step will remove constant -15 from the left.

-5y -15 + 15 > -3 + 15

-5y > 12

Divide by -5 on both sides.
Remember division with negative number will reverse inequality sign.

\mathtt{\frac{-5y}{-5} < \ \frac{12}{-5}}\\\ \\ \mathtt{y\ < \ -2.4}

Hence for inequality value of y < -2.4

Example 03
Solve the inequality
-6x > 180

Method 01
Move -6 towards right.
Multiplication becomes division.

Since number is negative, when you take number towards right, the inequality sign will reverse.

-6x > 180

x < 180 / -6

x < -30

Hence, for given inequality, value of x < -30.

Method 02
-6x > 180

Divide both sides by -6
Since number is negative, the sign of inequality will also change.

\mathtt{\frac{-6x}{-6} < \ \frac{180}{-6}}\\\ \\ \mathtt{x\ < \ -30}

Hence, x < -30 is the solution.

Example 04
Solve for value of x.
2 ( x – 6 ) < 4x – 7

Method 01
First open the bracket.
Opening up of bracket results in multiplication of numbers.

2.x – 2.6 < 4x – 7

2x – 12 < 4x – 7

Separate variables on left and constant on right side.

Move 4x towards left.

2x – 4x -12 < -7

-2x -12 < -7

Move 12 towards right.

-2x < -7 + 12

-2x < 5

Move -2 towards right.
The multiplication becomes division.
Since number is negative, the sign of equality will reverse.

x > -5/2

x > -2.5

Hence, value of x > -2.5

Method 02
2 ( x – 6 ) < 4x – 7

Open the bracket.
2x – 12 < 4x – 7

Separate variable on left and constant on right.

Subtract 4x on both sides.
This will eliminate variable from right side.

2x – 12 -4x < 4x – 7 – 4x

– 2x – 12 < – 7

This will eliminate constant term from left side.

-2x – 12 + 12 < – 7 + 12

-2x < 5

Divide -2 on both sides.
Division with negative number will reverse inequality sign.

-2x / -2 > 5 / -2

x > -2.5

Hence, value of x > -2.5

Example 05
Find the value of x.
– 3x + 7 > 0

Method 01
Separate variables on left and constant on right side.

Move 7 to the right.

-3x > -7

Move -3 to the right.
Multiplication will become division.
Since -3 is negative, the inequality will reverse.

x < -7 / -3

x < 7 / 3

x < 2.33

Hence, the value of x < 2.33

Method 02
– 3x + 7 > 0

Subtract 7 on both sides.
This will remove constant from left side.

– 3x + 7 – 7 > – 7

– 3x > – 7

Divide -3 on both sides.
Since the number is negative, the inequality will reverse.

-3x / -3 < -7 / -3

x < 7 / 3

x < 2.33

Hence, value of x < 2.33