Absolute Values in Algebra

Absolute values in algebra is a bit weird but aside from some specific rules they’re pretty fun to solve.

First off, you cannot change the inside signs of an absolute value. Here’s an example:

$$-3+|x-4|=2$$

When solving for this, the first step is to make sure that once you get just the absolute values on one side and the rest of the equation on the other, the rest of the equation must be positive, if it’s negative it’s not a number.

$$\begin{array}{c}-3+|x-4|=2\\ \downarrow \\ |x-4|=2+3\\ \downarrow \\ |x-4|=5\end{array}$$

If that works, then you must do two things rewrite the equation twice without the abs signs, one as normal and the other where the result is negative. Then solve the equation

$$\begin{array}{c}-3+|x-4|=2\\ \downarrow \\ |x-4|=2+3\\ \downarrow \\ |x-4|=5\\ \downarrow \\ x-4=5x-4=-5\\ x=9x=-1\\ x=-1,9\end{array}$$

Absolute Values in Equality Equations

Handling equality equations are pretty much the same, aside from a couple things.

First, rewrite the equality as a normal equation:

$$\begin{array}{c}-3+|x-4|\ge 2\\ \downarrow \\ -3+|x-4|=2\end{array}$$

Solve like normal, however instead of writing the solution like normal, you need to perform an additional check to see how many intervals it has.

An ABS value can result in either a single interval like $(x,y)$, or two intervals like $(-\infty ,x)u(y,\infty )$

In order to test, replace x in the equation with a number (or two, or three) and if it fails where it should be true (or vice versa) that will tell you all you need to know!

I personally test something between the two values, but any value can work.

$$\begin{array}{c}(-1,9)\\ -3+|(7)-4|\ge 2\\ 0\ge 2\end{array}$$

As you can see in the above, 7 results in 0, which fails the equality, therefore the answer is $(-\infty ,-1)u(9,\infty )$!