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;

⟹ **Addition becomes subtraction**

⟹** Subtraction becomes addition**

⟹ **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

Addition becomes subtraction

6x – 2x – 15 < -3

Move 15 to the right.

Subtraction becomes addition

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

Add 15 on both sides.

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.

Addition becomes subtraction.

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

Add 15 on both sides.

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.

Addition becomes subtraction.

2x – 4x -12 < -7

-2x -12 < -7

Move 12 towards right.

Subtraction will become addition.

-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

Add 12 on both sides.

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.

Addition will become subtraction

-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**