**Solving Inequalities with Variables on Both Sides :**

In this section, we will learn, how to solve inequalities with variables on both sides.

Use the binary operations addition, subtraction, multiplication and division to combine the like terms and isolate the variable (unknown).

**Note : **

We have to make the following changes, when we multiply or divide each side of the inequality by a negative value.

- If we have <, then change it as >
- If we have >, then change it as <
- If we have ≤, then change it as ≥
- If we have ≥, then change it as ≤

**Example 1 :**

Solve the following linear inequality

3x - 7 > x + 1

**Solution :**

**3x - 7 > x + 1**

**Subtract x on both sides**

**3x - x - 7 > x - x + 1**

**2x - 7 > 1**

**Add 7 on both sides**

**2x - 7 + 7 > 1 + 7**

**2x > 8**

**Divide by 2 on both sides**

**2x/2 > 8/2**

**x > 4**

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

**Example 2 :**

Solve the following linear inequality

x + 5 > 4x - 10

**Solution :**

**x + 5 > 4x - 10**

**Subtract 4x on both sides**

**x - 4x + 5 > 4x - 4x - 10**

**-3x + 5 > -10**

**Subtract 5 on both sides**

**-3x + 5 - 5 > -10 - 5**

**-3x > -15**

**Divide by -3 on both sides**

**x < 5**

**Hence the solution set of the given inequality is (-**∞, 5)

**Example 3 :**

Solve the following linear inequality

3x + 9 ≥ -x + 19

**Solution :**

3x + 9 ≥ -x + 19

**Add x on both sides**

**3x + x + 9 **≥ -x + x + 19

4x + 9 ** **≥ 19

Subtract 9 on both sudes

4x + 9 - 9 ** **≥ 19 - 9

4x ** **≥ 10

Divide by 4 on both sides

x ** **≥ 10/4

x ** **≥ 5/2

**Hence the solution set of the given inequality is [5/2, **∞)

Since we have grater than or equal sign, we have used closed parentheses to write 5/2.

**Example 4 :**

Solve the following linear inequality

3x + 17 ≤ 2(1 - x)

**Solution :**

3x + 17 ≤ 2(1 - x)

**By using distributive property on the right side, we get**

**3x + 17 ≤ 2 - 2x**

**Add 2x on both sides**

**3x + 2x + 17 ****≤ 2 - 2x + 2x**

**5x + 17 ****≤ 2**

**Subtract 17 on both sides**

**5x + 17 - 17 ****≤ 2 - 17**

**5x ****≤ -15**

**Divide by 5 on both sides**

**x ****≤ -15/5**

**x ****≤ -3**

**Hence the solution set of the given inequality is (-**∞, -3]

After having gone through the stuff given above, we hope that the students would have understood, how to solve inequalities with variables on both sides.

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