INEQUALITIES INVOLVING ABSOLUTE VALUE

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. 

Solved Questions

Question 1 :

Solve for x : 

|3 - x|  <  7

Solution :

|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)

Question 2 :

Solve for x : 

|4x - 5|  >  -2

Solution :

|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. 

Question 3 :

Solve for x : 

|3 - (3x/4)|    1/4

Solution :

|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]

Question 4 :

Solve for x : 

|x| - 10  <  -3

Solution :

|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 5 :

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 6 :

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|    7/3

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 7 :

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

 ≥  -15/2

Combine the above two inequalities.

-15/2    x    11/2

So, the solution is

[-15/2,  11/2]

Question 8 :

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 9 :

Solve for x :

|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. 

Question 10 :

Solve for x : 

|x| < 0

Solution :

Here, there is no solution. 

Because, absolute value of any number will be positive and it can never be a negative value. 

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