**How to Solve Absolute Value Inequalities on a Number Line :**

In this section, we will learn, how to solve an absolute value inequality on a number line.

**Example 1 : **

Solve 2|x + 1| − 6 ≤ 7 and graph the solution set in a number line

**Solution : **

2|x + 1| − 6 ≤ 7

Add 6 on both sides

2|x + 1| − 6 + 6 ≤ 7 + 6

2 |x + 1| ≤ 13

Divide by 2 on both sides

|x + 1| < 13/2

Now we may split this into two branches as shown below.

(x + 1) > - 13/2 Subtract 1 throughout the equation x > (-13/2) - 1 x > -15/2 |
(x + 1) < 13/2 Subtract 1 throughout the equation x < (13/2) - 1 x < 11/2 |

Hence, the solution is

-15/2 < x < 11/2

By graphing the above solution set, we get

By representing the above graph in solution set, we get

(-∞, -15/2) U (11/2, ∞)

**Example 2 :**

Solve : (1/5) |10x - 2| < 1

**Solution : **

(1/5) |10x - 2| < 1

Multiply by 5 throughout the equation

|10x - 2| < 5

Now we may split the above inequality into two parts

10x - 2 > -5 Add 2 on both sides 10x - 2 + 2 > -5 + 2 10x > -3 Divide by 10 on both sides x > -3/10 |
10x - 2 < 5 Add 2 on both sides 10x - 2 + 2 < 5 + 2 10x < 7 Divide by 10 on both sides x < 7/10 |

Hence, the solution is

-3/10 < x < 7/10.

**Example 3 :**

Solve : |5x - 12| < -2

**Solution : **

|5x - 12| < -2

In the right side, we have numerical value -2 and in the left side, we have a absolute value function.

By applying any values to x, we get only positive values on the right side. So the given statement will never become true for any value of x.

Hence, there is no solution for the given inequality.

After having gone through the stuff given above, we hope that the students would have understood, how to solve an absolute value inequality on a number line.

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