**Linear inequalities in one variable with fractions :**

**Let us see some example problems on linear inequalities in one variable with fractions.**

**Example 1 :**

**Solve the following linear inequality **

**Solution :**

(6x - 9 - 16x)/12 ≥ -6

(-10 x - 9)/12 ≥ -6

Multiply by 12 on both sides

[(-10 x - 9)/12] **x** 12 ≥ -6 **x** 12

-10x - 9 ≥ -72

Add 9 on both sides

-10 x - 9 + 9 ≥ -72 + 9

-10x ≥ -63

Divide by -10 on both sides

-10x/(-10) ≤ -63/(-10)

x ≤ 63/10

x ∈ (-∞, 63/10]

Hence the solution set of the given inequality is (-∞, 63/10].

**Example 2 :**

**Solve the following linear inequality **

**Solution :**

(25x - 10 - 21x + 9)/15 > x/4

(4x - 1)/15 > x/4

Multiply by 15 on both sides

4x - 1 > (x/4) **x** 15

4x - 1 > 15x/4

Multiply by 4 on both sides

4(4x - 1) > 15x

16x - 4 > 15x

Subtract 15x on both sides

16x - 15x - 4 > 15x - 15x

x - 4 > 0

Add 4 on both sides

x - 4 + 4 > 0 + 4

x > 4

x ∈ (4, ∞)

Hence the solution set of the given inequality is (4, ∞)

**Example 3 :**

**Solve the following linear inequality **

**Solution :**

(3x + 20)/10 ≥ (x - 6)/3

Multiply by 10 on both sides

3x + 20 ≥ 10(x - 6)/3

Multiply by 3 on both sides

3(3x + 20) ≥ 10(x - 6)

9x + 60 ≥ 10x - 60

Subtract 10 on both sides

9x - 10x + 60 ≥ 10x - 10x - 60

-x + 60 ≥ - 60

Subtract 60 on both sides

-x + 60 - 60 ≥ - 60 - 60

-x ≥ - 120

Divide by (-1) on both sides

x ≤ 120

x ∈ (-∞, 120]

Hence the solution set of the given inequality is (-∞, 120]

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