Absolute value equations and inequalities

The absolute value of integer is it numerical value with no regards to it signs. Symbol of absolute value is |  |. It is represented by two vertical lines.

Absolute value of | 3 |= 3
Absolute value of | -1 |= 1
Absolute  value of | 0 |=0

Since the integer 0 is neither positive nor negative , so absolute value of 0 is zero | 0 |= 0
So we can conclude

If a is any integer,
| a | = a , if a > 0
| a |= -a , if a < 0

Absolute value equations

To Understand absolute value equation  , we will take some equation

=> | x+ 3 |=4

To solve absolute value equation, we will consider two cases

=> (x+3) = 4   or  (x+3) = -4
=> x = 1 or x= -7

*** Important point to remember while solving equations, that Absolute equations are always positive, so if we equations have
| x+3 |=-4  , this will never be true . This equation will have no solutions.

Absolute value inequalities
Some important results in inequalities
Result 1) If a is a positive real number, then
(i) | x | <  a  =>  -a < x
(ii) | x | ≤  a => -a ≤ x ≤ a i.e x ∈  [-a, a]

Result 2) If a is a positive real number , then
(l) | x | > a => x< -a or x > a
(ii) | x |≥ a => x ≤ -a or x ≥ a

Result 3) Let r be a positive real number  and a be fixed real number , then
(i) | x – a | < r => a – r  < x <  a + r , i.e x ∈  (a-r , a+r)
(ii) | x – a | ≤ r => a – r  ≤ x ≤ a + r , i.e x ∈  [a-r , a+r]

Solving absolute value equations

Example 1) | 3x – 2 | ≤ 4

Here , in the given problem we will use the following result
| x – a | ≤ r => a – r  ≤ x ≤ a + r , i.e x ∈  [a-r , a+r]

So  according to result we have
=> 2-4 ≤ 3x ≤ 2+4
= > -2 ≤ 3x ≤ 6

Divide  whole equation by 3 , to isolate x
=> -2/3 ≤ x ≤ 6/3
=> -2/3 ≤ x ≤