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.
Absolute 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
Solve each equation:
a. | x | = 37
b. | y | = 0
c. | w | = .4
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
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.