**Solving and graphing inequalities :**

If the given inequality is in the form x <, >, ≤ (or) ≥ we can easily draw a graph.

In case we have the inequality like ax + b < bx + d, we have to solve for x and draw the graph.

For solving linear inequality, we follow the steps given below.

Let us see the steps involved in solving and graphing inequalities.

**Step 1 :**

Same number may be added or subtracted from both sides of an inequality without changing the sign of inequality.

**Step 2 :**

Both sides of an inequality can be multiplied or divided by same positive real number without changing the sign of inequality. However the sign of inequality is revered when both sides of an inequality are multiplied or divided by a negative number.

**Step 3 :**

Any term of an inequality may be taken to the other sides with its sign changed without affecting the sign of inequality.

In graphing inequalities on number line, we use the following sings.

- If we have one of the signs like < (less than) or > (greater than), we have to use the unfilled circle.
- If we have one of the signs like ≤ (less than or equal to) or ≥ (greater than or equal to), we have to use the filled circle.

Let us look into some examples to understand solving and graphing inequalities.

**Example 1 :**

Solve the following linear inequality and graph.

2x - 4 ≤ 0

**Solution :**

**2x - 4 ≤ 0**

**Add 4 on both sides**

**2x - 4 + 4 ****≤ 0 + 4**

**2x ****≤ 4**

**Divide by 2 on both sides**

**2x/2 ****≤ 4/2**

**x ≤ 2**

**Hence, any real number less than or equal to 2 is a solution of the given equation.**

The solution set of the given inequality is (-∞, 2].

**Example 2 :**

Solve the following linear inequality and graph.

-3x + 12 < 0

**Solution :**

-3x + 12 < 0

**Subtract 12 on both sides**

-3x + 12 - 12 < 0 - 12

-3x < -12

Divide by -4 on both sides

-3x/(-3) < -12/(-3)

x < 4

**Hence, any real number less 4 is a solution of the given equation.**

The solution set of the given inequality is (-∞, 2].

**Example 3 :**

Solve the following linear inequality and graph.

4x - 12 ≥ 0

**Solution :**

4x - 12 ≥ 0

**Add 12 on both sides**

4x - 12 + 12 ≥ 0 + 12

4x ≥ 12

Divide by 4 on both sides

4x/4 ≥ 12/4

x ≥ 3

**Hence, any real number greater than or equal to 3 is a solution of the given equation.**

The solution set of the given inequality is [3, ∞).

**Example 4 :**

Solve the following linear inequality and graph.

7x + 9 > 30

**Solution :**

7x + 9 > 30

**Subtract 9 on both sides**

7x + 9 - 9 > 30 - 9

7x > 21

Divide by 7 on both sides

7x/7 > 21/7

x > 3

**Hence, any real number greater than 3 is a solution of the given equation.**

The solution set of the given inequality is (3, ∞).

**Example 5 :**

Solve the following linear inequality and graph.

5x - 3 < 3x + 1

**Solution :**

5x - 3 < 3x + 1

**Subtract 3x on both sides**

**5x - 3 - 3x < 3x + 1 - 3x **

**2x - 3 < 1 **

**Add 3 on both sides**

**2x - 3 + 3 < 1 + 3**

**2x < 4**

**Divide by 2 on both sides**

**2x/2 < 4/2**

**x < 2**

**Hence, any real number lesser than 2 is a solution of the given equation.**

The solution set of the given inequality is (2, ∞).

**Example 6 :**

Solve the following linear inequality and graph.

3x + 17 ≤ 2(1 - x)

**Solution :**

3x + 17 ≤ 2(1 - x)

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

5x/5 ≤ -15/5

x ≤ -3

**Hence, any real number lesser than or equal to -3 is a solution of the given equation.**

The solution set of the given inequality is (-∞ , -3].

**Example 7 :**

Solve the following linear inequality and graph.

2(2x + 3) - 10 ≤ 6 (x - 2)

**Solution :**

2(2x + 3) - 10 ≤ 6 (x - 2)

4x + 6 - 10 ≤ 6 x - 12

4x - 4 ≤ 6 x - 12

Subtract 6x on both sides

4x - 4 - 6x ≤ 6 x - 12 - 6x

-2x - 4 ≤ - 12

Add 4 on both sides

-2x - 4 + 4 ≤ - 12 + 4

-2x ≤ - 8

Divide by -2 on both sides

-2x / (-2) ≤ - 8 / (-2)

x ≤ 4

**Hence, any real number lesser than or equal to 4 is a solution of the given equation.**

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

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