**Solving inequalities with variables on both sides :**

To solve inequalities with variables on both sides, we have to follow the steps given below.

**Step 1 :**

By using basic operations like addition and subtraction, bring all "x" terms to the left side.

**Step 2 :**

Bring constants to the right side. If we have any fractions in the given inequality , we have to take L.C.M and simplify.

**Step 3 :**

Make the coefficient of x as positive 1.

For that, if we divide the given equation by any number through out the equation, we have to change the inequality sign to its opposite sign.

- If we have <, then convert it into >
- If we have >, then convert it into <
- If we have ≤, then convert it into ≥
- If we have ≥, then convert it into ≤

Let us look into some example problems to understand the above concept.

**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 "Solving inequalities with variables on both sides".

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