Absolute Value Equations

An absolute value equation is an equation that contains an absolute value expression, such as $|ax + b| = c$. Because the absolute value of a number represents its distance from zero on a number line, both positive and negative values can have the same absolute value. For example, $|5| = 5$ and $|-5| = 5$.

The Golden Rule of Absolute Value

To solve an absolute value equation of the form $|X| = c$:

1. If $c > 0$: The equation splits into two separate cases: $X = c$ or $X = -c$.
2. If $c = 0$: There is only one case: $X = 0$.
3. If $c < 0$: There is no solution, because an absolute value can never result in a negative number.

Example 1: A Standard Equation

Let's solve the equation: $|2x - 3| = 7$

Since $7$ is positive, we split the equation into two cases:

Case 1: The inside is positive $2x - 3 = 7$ $2x = 10$ $x = 5$

Case 2: The inside is negative $2x - 3 = -7$ $2x = -4$ $x = -2$

The solutions are $x = 5$ and $x = -2$.

Example 2: Variables on Both Sides

When there are variables on both sides of the equation, you must always check your answers. This process can create "extraneous solutions"—answers that emerge from the algebra but don't actually work in the original equation.

Solve: $|x + 1| = 3x - 5$

Split into two cases:

Case 1: $x + 1 = 3x - 5$ $-2x = -6$ $x = 3$

Case 2: $x + 1 = -(3x - 5)$ $x + 1 = -3x + 5$ $4x = 4$ $x = 1$

Now, we must plug these back into the original equation $|x + 1| = 3x - 5$ to check them:

• Check $x = 3$: $|3 + 1| = 3(3) - 5$ $|4| = 9 - 5$ $4 = 4$ (This works!)

• Check $x = 1$: $|1 + 1| = 3(1) - 5$ $|2| = 3 - 5$ $2 = -2$ (This is false!)

Because $x = 1$ leads to a false statement, it is an extraneous solution. The only real solution is $x = 3$.