**Solving Systems of Inequalities with One Variable :**

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

**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.

**Example 1 :**

Solve the following system of linear inequalities

3x - 6 ≥ 0, 4x - 10 ≤ 6

**Solution :**

Solving the equations separately

3x - 6 ≥ 0 Add 6 on both sides 3x ≥ 6 Divide by 3 on both sides x ≥ 6/3 x ≥ 2 |
4x - 10 ≤ 6 Add 10 on both sides 4x ≤ 6 + 10 4x ≤ 16 Divide by 4 on both sides x ≤ 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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