Solving Absolute Value Equations
Solving an Equation of the Form  z  = a
We can use what we know about absolute value to solve equations that
contain absolute value.
For example, letâ€™s solve the equation x = 5.
The solutions are those numbers that are 5 units from 0 on the number line.
So, the equation x = 5 has two solutions: x = 5 and x = 5.
Now letâ€™s solve the equation x = 0.
The solution contains those numbers that are 0 units from 0.
So, the equation x = 0 has only one solution: x = 0.
Finally, letâ€™s consider the equation x = 5.
The solution contains those numbers that are 5 units from 0. Since
distance is never negative, there are no numbers that satisfy this equation.
Thus, the equation x = 5 has no solutions.
We can generalize these ideas to find the solutions of z = a.
PrincipleAbsolute Value Equations of the Form  z = a
When solving an equation of the form z = a:
If a > 0, there are two solutions, z = a and z = a.
If a = 0, there is one solution, z = 0.
If a < 0, there are no solutions.
Here, z represents an algebraic expression and a represents a real
number.
z
Example
Solve each equation:
a.  x  = 37
b.  y  = 0
c.  w  = .4
Solution
a. x = 37 has the form z = a where a = 37, a positive number.
So, the solutions of x = 37 are x = 37 and x = 37.
b. y = 0 has the form z = 0.
So, the solution of y = 0 is y = 0.
c. w = 4 has no solution because the absolute value of a number
cannot be negative.
Letâ€™s solve another equation
involving absolute value:

9y = 63 
First, letâ€™s rewrite the equation by substituting z for 9y.

z = 63 
This equation has two solutions. 
z = 63 and z = 63 
Now, to solve for y, we substitute back 9y for z. 
9y = 63 and 9y = 63 
Divide both sides by 9. 
y = 7 and y = 7 
So, the solutions of 9y = 63 are y = 7 and y = 7.
We can solve equations containing one absolute value expression using the
following procedure.
Procedure
To Solve an Equation Containing One Absolute Value Expression
Step 1 Isolate the absolute value expression to get an equation of
the form ax + b = c.
Step 2 Make the substitution z = ax + b.
Step 3 Use the Absolute Value Principle to solve for z.
Step 4 Replace z with ax = b.
Step 5 Solve for x.
Step 6 Check the answer.
