Absolute Value Equations (One Variable)

Absolute Value Equations (One Variable)

How to solve absolute value equations problems: meaning, examples, and their solutions.

Meaning

The absolute value of a number means the distance between 0 and the number.

The absolute value of a number means
the distance between 0 and the number.
(regardless of its direction, sign)

So the absolute value sign
makes the number's sign to (+):
|(+)| = (+)
|(-)| = (+)

Example 1

Find the value of x. |x - 1| = 2

First, see if |x - 1| ≥ 0.

|x - 1| ≥ 0
because the result of the absolute value sign
cannot be (-).
(-) distance doesn't make sense.

So if it's (-), then the equation has no solution.

In this case, |x - 1| = 2 ≥ 0.

So move on to the next step.

|x - 1| = 2

So x - 1 can be 2.
(|2| = 2)

So x = 2 + 1 = 3.

|x - 1| = 2

So x - 1 can be -2.
(|-2| = 2)

So x = -2 + 1 = -1.

So x = -1, 3.

Example 2

Find the value of x. |2x + 8| + 1 = 0

First, see if |2x + 8| ≥ 0.

But |2x - 1| = -1 < 0.

So this equation has no solution.