ABSOLUTE VALUE OF INTEGERS

The absolute value of an integer is the integer’s distance from 0 on a number line.

For example, the absolute value of -3 is 3.

To understand this, let us mark -3 on a number line. 

On the above number line, -3 is 3 units from 0.

Since -3 is 3 units from 0, we say that the absolute value of "-3" is 3.  

The absolute value of -3 is written |-3|.

And we have

|-3|  =  3

Because absolute value represents a distance and it is always positive. 

Solved Examples

Example 1 : 

Find the absolute value of the integer -9. 

Solution :

|-9|  =  9

Example 2 : 

Find the absolute value of the integer 9. 

Solution :

|9|  =  9

Example 3 : 

Find the absolute value of (-17 + 8). 

Solution :

|-17 + 8|  =  |-9|

|-17 + 8|  =  9

Example 4 : 

Find the absolute value of (28 - 13). 

Solution :

|28 - 13|  =  |15|

|28 - 13|  =  15

Example 5 : 

If |x| is an integer between 0 and 3, then, find all possible values of x.  

Solution :

Given : |x| is an integer between 0 and 3. 

Then, we have

|x|  =  1  and  |x|  =  2

Solve for x in |x|  =  1. 

Solve for x in |x|  =  2. 

The possible values of x are 

-2, -1, 1, 2

Example 6 :

If |2x - 1| is an integer between 3 and 6, then, find all possible values of x.  

Solution :

Given : |2x - 1| is an integer between 3 and 6. 

Then, we have

|2x - 1|  =  4  and  |2x - 1|  =  5

Solve for x in |2x - 1|  =  4. 

Solve for x in |2x - 1|  =  5. 

So, the possible values of x are 

-2, -3/2, 5/2, 3

Example 7 :

Which of the following expressions is equal to -1 for some values of x ?

a) |1 - x| + 6        b) |1 - x| + 4

c) |1 - x| + 2        d) |1 - x| - 2

Solution :

So, option d is correct.

Example 8 :

If |2x + 7| = 5, which of the following could be the value of x ?

a)  -6    b)  -4    c)  -2   d) 0

Solution :

|2x + 7| = 5

So, the value of x is -6, option a is correct.

Example 9 :

Evaluate the expression

|6² ÷ 4 + 21 | + 3 - |18 ÷ 6 + 4 ÷ 2|²

Solution :

= |6² ÷ 4 + 21 | + 3 - |18 ÷ 6 + 4 ÷ 2|²

= |36 ÷ 4 + 21 | + 3 - |18 ÷ 6 + 4 ÷ 2|²

Performing division first,

= |9 + 21 | + 3 - |3 + 4 ÷ 2|²

= |30| + 3 - |3 + 2|²

= 30 + 3 - |5|²

= 33 - 25

= 8

So, the answer is 8.

Example 10 :

If a, b and c are not equal to zero, what is the difference between the maximum and minimum values of S ?

S = 1 + |a|/a + 2|b|/b + 3 |ab|/ab - 4|c|/c

a) 12    b) 14   c) 22    d)  22   e) 18

Solution :

S = 1 + |a|/a + 2|b|/b + 3 |ab|/ab - 4|c|/c

To get maximum value, a, b and ab be positive and c be negative.

S = 1 + a/a + 2b/b + 3ab/ab - 4c/c

= 1 + 1 + 2 + 3 - 4(-1)

= 7 + 4

= 11

To get minimum value, a, b and ab be negative and c be positive.

S = 1 + (a/a) + 2(b/b) + 3(ab/ab) - 4c/c

= 1 + 1 + 2 + 3 - 4

= 7 - 4

= 3

The value of the expression will be minimum if we make as many terms negative as possible.

Higher the magnitude of the terms made negative, lower the value of the expression.

c has to be positive for S to be minimum. The last term will then be -4.

If ab is negative, then 3|ab|/ab = -3

If ab has to be negative, one of a or b has to be positive and the other has to be negative.

Possibility 1:

If a > 0 and b < 0, |a|/a = 1 and 2|b|/b = -2. The value of the expression is 1 + 1 - 2 - 3 - 4 = -7.

Possibility 2:

If a < 0 and b > 0, |a|/a = -1 and 2|b|b = 2. The value of the expression is 1 - 1 + 2 - 3 - 4 = -5.

Therefore, the minimum value is -7

Difference = 11 - (-7)

= 18

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