# Solve two step linear inequalities

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The opposite operation of addition is subtraction and vice versa.

Similarly, the opposite operation of multiplication is division and vice versa.

**Note :**whenever you multiply or divide both sides of an inequality by a negative number,flip the inequality sign.

## Example to solve two step linear inequalities

**Example 1 :**6x + 7 > 4x + 3

**Solution:**6x + 7 > 4x + 3

Transpose +7 to other side, we get,

6x > 4x + 3 -7

6x > 4x - 4

Now transpose 4x to left side so that like terms on one side and constants on other side

6x - 4x > -4

2x > -4 divide both side by 2

$\frac{2x}{2} > \frac{-4}{2}$

∴ x > -2

**Example 2 :**(x + 5) - 7(x -2) $\geq$ 4x + 9

**Solution :**(x + 5) - 7(x -2) $\geq$ 4x + 9

First, open the parenthesis

x + 5 - 7x + 14 $\geq$ 4x + 9

Add the like terms

-6x + 19 $\geq$ 4x + 9

Transpose +19 to other side

-6x $\geq$ 4x + 9 -19

Now transpose 4x to left side so that variable terms on one side and constants on other side

-6x - 4x $\geq$ -10

-10x $\geq$ -10

Divide both side by -10. Dividing by negative number

**flip the inequality sign**

$\frac{-10x}{-10} \leq \frac{-10}{-10}$

∴ x $\leq$ 1

**Example 3:**$\frac{3x -4}{2}\geq \frac{x + 1}{4} - 1$

**Solution :**$\frac{3x -4}{2}\geq \frac{x + 1}{4} - 1$

$\frac{3x -4}{2}\geq \frac{x + 1}{4} - \frac{1}{1}$ ( just 1 means one over one)

First we will simplify the right side. There are subtraction of two fractions so find the LCD of 4 and 1, it will be 4

$\frac{3x -4}{2}\geq \frac{x + 1}{4} - \frac{4}{4}$

$\frac{3x -4}{2}\geq \frac{x + 1 - 4}{4}$

$\frac{3x -4}{2}\geq \frac{x - 3 }{4} $

Since we have fractions on both side , we will cross multiply

4(3x - 4) $\geq$ 2(x -3 )

12x - 16 $\geq$ 2x - 6

12x - 2x $\geq$ - 6 + 16

10x $\geq$ 10

$\frac{10x}{10} \geq \frac{10}{10}$

∴ x $\geq $ 1

**Example 4 :**$\frac{x + 2}{4} - \frac{x}{5} \geq 3$

**Solution :**$\frac{x + 2}{4} - \frac{x}{5} \geq 3 $

On the left side we have subtraction of two fractions so find the LCD of 4 and 5, it will be 20

$\frac{5(x + 2) - 4x }{20} \geq 3$

$\frac{5x + 10 - 4x }{20} \geq 3 $

$\frac{x + 10 }{20} \geq 3 $

Now multiply both side by 20

(x + 10 ) $\geq$ 60

∴ x $\geq$ 50

From two step linear inequalities to Home

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