# SOLVING SYSTEMS OF INEQUALITIES WITH ONE VARIABLE

Solving Systems of Inequalities with One Variable :

In this section, we will learn, how to solve system of inequalities with one variable.

## Solving Systems of Inequalities with One Variable - Steps

Step 1 :

Solve each given inequality and find the solution sets.Also represent the solution in the number line.

Step 2 :

Find the intersection of solution sets obtained in first step by taking the help of graphical representation of solution sets.

Step 3 :

The solution set obtained from step 2 is the required solution set of the given system of inequalities.

## Solving Systems of Inequalities with One Variable - Examples

Example 1 :

Solve the following system of linear inequalities

3x - 6 ≥ 0,  4x - 10 ≤ 6

Solution :

Solving the equations separately

 3x - 6 ≥ 0Add 6 on both sides3x ≥ 6Divide by 3 on both sidesx ≥ 6/3x ≥ 2 4x - 10 ≤ 6Add 10 on both sides4x ≤ 6 + 104x  ≤ 16Divide by 4 on both sidesx  ≤ 4

The solution set of first given inequality is [2,  ∞).

The solution set of second given inequality is (-∞, 4]

The intersection of these solution sets is the set [2, 4].

Example 2 :

Solve the following system of linear inequalities

(5x/4) + (3x/8)  >  39/8

(2x - 1)/12 - (x - 1)/3 < (3x + 1)/4

Solution :

Solving the first given inequality

(5x/4) + (3x/8)  >  39/8

(10x + 3x)/8 > 39/8

(13x/8)  > 39/8

Multiplying by 8 through out the equations

13x > 39

Divide by 13, we get

x > 39/13

x > 3

Solution set of the first given inequality is (3, ∞)

Solving the second given inequality :

(2x - 1)/12 - (x - 1)/3 < (3x + 1)/4

[(2x - 1) - 4(x - 1)]/12 < (3x + 1)/4

[(2x - 1 - 4x + 4)]/12 < (3x + 1)/4

(-2x + 3)/12 < (3x + 1)/4

Multiply 12 on both sides

(-2x + 3) < 3(3x + 1)

-2x + 3 < 9x + 3

Subtract 9x on both sides

-2x - 9x + 3 < 3

Subtract 3 on both sides

-11x  < 0

Divide by -1 on both sides

x > 0

The solution set of the given inequality is (0, ∞)

The intersection of the two solutions sets is (3, ∞).

Hence the solution of given inequalities is (3, ∞).

After having gone through the stuff given above, we hope that the students would have understood, how to solve systems of inequalities with one variable.

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