

Absolute value
equations 
Solving absolute value
equations 





Absolute value
equations 
Solving absolute value
equations 
Recall
that the absolute value of a real number a,
denoted 
a ,
is the number without its
sign and represents the distance between 0 (the origin) and that
number on the real number line. 
Thus, regardless of the value of a number
a
its absolute value is
always either positive or zero, never negative that is,  a

> 0.

To
solve an absolute value equation, isolate the absolute value on
one side of the equation, and use the definition of absolute
value. 
If the number on the other side of the equal sign is positive, we will need to set up two
equations to get rid of the absolute value, 
 the first equation that set the expression inside the absolute value symbol equal to the other side of the equation, 
 and the second equation that set the expression inside the absolute value equal to the
opposite of the number on the other side of the equation. 
Solve
the two equations and verify solutions by plugging the solutions
into the original equation. 
If
the number on the other side of the absolute value equation is negative then
the equation has no solution. 

Example:

1
 2x
 = 17

Solution:
1

2x =
17
or
1 
2
x = 17 
2
x =  16
2x =
18 
x = 
8
x = 9 

The
solutions to the given equation are
x =  8
and
x = 9. 

Example:

3
 x
 = 5

Solution:
3

x = 5
or
3

x = 5 
x = 
8
x = 2 

The
solutions to the given equation are
x =  8
and
x = 2. 

Example:

x + 1  =
2x 
3

Solution:
x + 1 = 2x

3
or
x + 1 = (2
x 
3) 
x  2x
=  4
x +
2x = 3 
1 
x = 4
3x = 2
=> x = 2/3 

Check
solutions: 
x = 4
=>

x + 1  =
2x 
3,
x = 2/3
=>

x + 1  =
2x 
3 

4
+ 1  =
2 · 4 
3

2/3 + 1  =
2 · 2/3 
3

5 = 5
5/3 is not equal
5/3 
The
check shows that x =
2/3 is not a solution,
because the right side of the equation becomes negative. There is a single solution to this equation: x = 4. 

Example:

x + 2  =
 2x 
5 

Solution: As both
sides of the equation contain absolute values the
only way the two sides are equal is, the two quantities
inside the absolute value bars are equal or equal but with
opposite signs.

x + 2 = 2x

5
or
x + 2 = (2
x 
5) 
x 
2x
= 
5 
2
x + 2 = 2
x +
5 
x =
7
3x = 3 
x = 7
x = 1 

Check
solutions: 
x =
7 =>

x + 2  =
 2x 
5 , x = 1
=>

x + 2  =
 2x 
5  

7 + 2  =
 2 · 7 
5


1 + 2  =  2 ·
1 
5 

9 = 9
3 =  3  

Therefore,
the
solutions to the given equation are x =
7 and x =
1. 










Precalculus
contents A 



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