Given a real
number a, we define its absolute value,
, as 
This means
the absolute value of a real number is either positive or zero.
To solve an equation involving only one absolute
value, proceed as follows:
a)
Isolate the absolute value to one side of the
equation, getting
|expression|
= Right Side
b)
Two situations could arise
i)
if Right Side is negative
number, the equation has no solution
ii) Otherwise, remove
the absolute value sign rewritten the equation as two linear
equations
expr=(Right
Side) or expr=- (Right Side)
c)
Solve these two equations separately
d)
Check the answers into the original equation.
Example
1:
Solve
|2x - 5| = 3
Removing the
absolute value 2x
- 5 = 3
Removing the
absolute value
2x-5 = 3
or 2x-5
= -3
Adding
5
2x-5+5 =
3+5 or
2x-5+5
= -3+5
Collecting
2x
= 8
or
2x
= 2
Simplifying
x
= 4 or
x
= 1
Check for x =4Check
for x = 1
|
|2x
- 5|
|
=
|
3 |
|
|2x
- 5|
|
=
|
3 |
|
|
|2(4)-5|
|
?
|
3 |
|
|2(1)-5|
|
?
|
3 |
|
|
|8
- 5|
|
?
|
3 |
|
|2
- 5|
|
?
|
3 |
|
|
|3|
|
=
|
3 |
True! |
|-
3|
|
=
|
3 |
True! |
So,
the solution is x = 1 or x = 4.
Remark:
If
when isolating the absolute value you get
an expression involving x instead of a Number,
solutions can be find only for values of x making it non negative
(see example 3), this means you have to reject any solution not
satisfying this restriction. Examples
4 & 5. shows this situation.
Example
4:
Solve 
Since for
12 - x < 0 there is no solution, we can assume that 12 - x
>= 0 , so
Removing absolute
value .............. 5x - 8 = + (12 - x) or 5x - 8 = - (12 - x)
Removing the
parentehsis .............. 5x - 8 =12 - x or 5x - 8 = - 12 + x
Grouping the
x in one side ............ 5x + x =12 + 8 or 5x - x =-12+ 8
Collecting
like terms ..................... 6x = 20 or 4x = - 4
Isolating
the x .............................. x = 10/3 or x = -1
Checking the
solutions we'd see that both solutions satisfy the original equation!
So solutions
are x = 10/3 or x = -1
Example
5:
Solve 
