Absolute Value and Absolute Value Equations

Absolute value represents the distance between two points on a number line, such as the distance from home to a grocery store. This distance is always a positive value.
If point $A$ is at $1$ and point $B$ is at $4$ , the distance is $|1 - 4| = |-3| = 3$ . Conversely, $|4 - 1| = |3| = 3$ .
For a single number, the absolute value indicates its distance from zero. For example, $|4| = 4$ and $|-10| = 10$ .

When solving for a variable inside an absolute value, such as $|x| = 5$ , there are two potential solutions ( $x = 5$ and $x = -5$ ) because both are five units away from zero.
To solve a complex equation like $3|x + 2| - 1 = 8$ :
- Isolate the absolute value expression first. Adding $1$ and dividing by $3$ yields $|x + 2| = 3$ .
- Do not distribute coefficients into absolute value bars.
- Split the isolated absolute value into two separate equations: $x + 2 = 3$ and $x + 2 = -3$ .
- Solving these gives $x = 1$ and $x = -5$ .

In some cases, solving the two equations resulting from an absolute value split produces an extraneous solution—a result that does not satisfy the original equation.
Example: $|3x + 2| = 4x + 5$ .
- Equation 1: $3x + 2 = 4x + 5$ leads to $x = -3$ .
- Equation 2: $3x + 2 = -(4x + 5)$ leads to $x = -1$ .
- Testing $x = -3$ : The left side becomes $7$ and the right side becomes $-7$ . Since $7 \neq -7$ , $x = -3$ is extraneous.
- Testing $x = -1$ : Both sides equal $1$ , so $x = -1$ is the only valid solution.

An absolute value cannot result in a negative number algebraically. If an isolated absolute value is equal to a negative constant, such as $|3x + 2| = -15$ , there is no solution.
It is critical to isolate the absolute value before making this determination:
- For $3|2x| = -12$ , isolating the bars results in $|2x| = -4$ , which has no solution.
- For $-3|2x| = -12$ , isolating the bars results in $|2x| = 4$ , which can be solved for $x = 2$ and $x = -2$ .