The statement is true and can be interpreted as "The distance between 3 and 5 is 2." This is why it is also true to write .

The idea can also be used to solve equations in which a distance is known to a point. For example, tells us that the distance between and 4 is 6. On a number line, it might look like this.

What values of make this statement true? Both and make the equation true, so they are solutions to the absolute value equation.

Another way to solve an equation involving an absolute value is to look at the piecewise meaning.

We can take the equation and split it into two cases.

• If , then the equation becomes . We can solve this using what we learned about linear equations. Adding 4 to each side, we can rewrite the equation as . We can check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is greater than or equal to 4.
• If , then the equation becomes . We can distribute the -1 on the left to get . From there, we can subtract 4 from each side and multiply each side of the result by -1 to get . We can also check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is less than 4.

Putting the results of the cases together, we again get that the solutions are -2 and 10.

Whether you prefer reasoning about distances on a number line or using the piecewise definition to write equations in different cases, the solutions should be the same.