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# 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.

Principle

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 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.