**Inequalities Involving Absolute Value :**

In this section, we will learn, how to solve absolute value inequalities.

**Case 1 : **

Inequality in the form |x - a| < r.

We can write the given absolute value inequality into two branches as shown below.

x - a < r x < a + r |
x - a > - r x > a - r |

Combine the above two inequalities.

(a - r) < x < (a + r)

So, the solution to |x - a| < r is

(a-r, a+r)

**Case 2 : **

Inequality in the form |x - a| > r.

We can write the given absolute value inequality into two branches as shown below.

x - a > r x > a + r |
x - a < - r x < a - r |

We can cannot combine the above two inequalities.

So, the solution to |x - a| > r is

(-∞, a - r) U (a + r, ∞)

**Case 3 : **

Inequality in the form |x - a| ≤ r.

We can write the given absolute value inequality into two branches as shown below.

x - a ≤ r x ≤ a + r |
x - a ≥ - r x ≥ a - r |

Combine the above two inequalities.

(a - r) ≤ x ≤ (a + r)

So, the solution to |x - a| ≤ r is

[a-r, a+r]

**Case 4 : **

Inequality in the form |x - a| ≥ r.

We can write the given absolute value inequality into two branches as shown below.

x - a ≥ r x ≥ a + r |
x - a ≤ - r x ≤ a - r |

We can cannot combine the above two inequalities.

So, the solution to |x - a| ≥ r is

(-∞, a - r] U [a + r, ∞)

**Case 5 : **

Inequality in the form |x - a| > - r.

Here, the solution is all real numbers.

Because, the absolute value of any number will be positive and also it is greater than a negative value.

**Case 6 : **

Inequality in the form :

|x - a| < - r

or

|x - a| ≤ - r

Here, there is no solution.

Because, the absolute value of any number will be positive and it can never be less than or equal to a negative value.

**Question 1 :**

(i) |3 - x| < 7

(ii) |4x - 5| ≥ -2

(iii) |3 - (3x/4)| ≤ 1/4

(iv) |x| - 10 < -3

**Solution (i) : **

|3 - x| < 7

We can write the above absolute value inequality into two branches as shown below.

3 - x < 7 - x < 4 x > -4 |
3 - x > -7 -x > -10 x < 10 |

Combine the above two inequalities.

-4 < x < 10

So, the solution is

(-4, 10)

**Solution (ii) :**

|4x - 5| > -2

Here, the solution is all real numbers.

Because absolute value of any number will be positive and also it is greater than a negative value.

**Solution (iii) : **

|3 - (3x/4)| ≤ 1/4

We can write the above absolute value inequality into two branches as shown below.

3 - (3x/4) ≤ 1/4 -3x/4 ≤ -11/4 3x/4 ≥ 11/4 3x ≥ 11 x ≥ 11/3 |
3 - (3x/4) ≥ -1/4 -3x/4 ≥ -13/4 3x/4 ≤ 13/4 3x ≤ 13 x ≤ 13/3 |

Combine the above two inequalities.

11/3 ≤ x ≤ 13/3

So, the solution is

[11/3, 13/3]

**Solution (iv) :**

|x| - 10 < -3

Add 10 to each side.

|x| < 7

We can write the above absolute value inequality into two branches as shown below.

x > 7 |
x < -7 |

Combine the above two inequalities.

-7 < x < 7

So, the solution is

(-7, 7)

**Question 2 :**

Solve (1/|2x - 1|) < 6 and express the solution using interval notation.

**Solution : **

(1/|2x - 1|) < 6

Multiply each side by |2x - 1|.

1 < 6|2x - 1|

Divide each side by 6.

1/6 < |2x - 1|

|2x - 1| > 1/6

We can write the above absolute value inequality into two branches as shown below.

2x - 1 > 1/6 12x - 6 > 1 12x > 7 x > 7/12 |
2x - 1 < -1/6 12x - 6 < -1 12x < 5 x < 5/12 |

We can not combine the above two inequalities.

So, the solution is

(-∞, 5/12) U (7/12, ∞)

**Question 3 :**

Solve −3|x| + 5 ≤ −2 and graph the solution set in a number line

**Solution :**

**-3|x| + 5 ≤ -2**

Subtract 5 from each side.

**-3|x| ≤ -7**

**Divide each side by (-3).**

** |x| **≥

**We can write the above absolute value inequality into two branches as shown below. **

x ≥ 7/3 |
x ≤ -7/3 |

We can not combine the above two inequalities.

So, the solution is

(-∞, -7/3] U [7/3, ∞)

**Question 4 :**

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

**Solution :**

2|x + 1| - 6 ≤ 7

Add 6 to each side.

2|x + 1| ≤ 13

Divide each side by 2.

|x + 1| ≤ 13/2

x + 1 ≤ 13/2 x ≤ 11/2 |
x + 1 ≥ -13/2 x ≥ -15/2 |

Combine the above two inequalities.

-15/2 ≤ x ≤ 11/2

So, the solution is

[-15/2, 11/2]

**Question 5 :**

Solve (1/5) |10x − 2| < 1.

**Solution :**

** (1/5) |10x − 2| < 1**

**Multiply each side by 5. **

** |10x - 2| < 5**

10x - 2 < 5 10x < 7 x < 7/10 |
10x - 2 > -5 10x > -3 x > -3/10 |

Combine the above two inequalities.

-3/10 < x < 7/10

So, the solution is

(-3/10, 7/10)

**Question 6 :**

Solve |5x − 12| < −2

**Solution :**

Here, there is no solution.

Because, absolute value of any number will be positive and it can never be less than or equal to a negative value.

After having gone through the stuff given above, we hope that the students would have understood, how to solve absolute value inequalities.

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