Absolute Value

Equations

Just like any operation, the absolute value function can be performed within an equation. Specifically, you can only apply it to individual expressions, not the whole equation. The point of absolute value is to change the output, and you can only change the output of a single expression, not a whole equation. What is odd though is that there is no algebraic way to reverse an absolute value. If you were to imagine normal operations like (PEMDAS) you can always reverse it by performing the opposite operation. Not only is there no opposite operation for absolute value, no operations that makes it all negative, if there were it still would not be an inverse because the absolute value takes both positive and negative values and inputs a positive, meaning it is fundamentally not invertible.

So how do you solve an equation with absolute value? The best way to do it is to use casework. Consider the equation \(|x|=1\). We know that the absolute value, or positive version, for some value \(x\) must be \(1\). There will be a negative and positive variant. Obviously, \(1\) is a positive variant, but \(-1\) is the negative variant as well. A simple absolute value equation will have two solutions, EXCEPT for if it is equal to a negative value, where it will never have a solution. For more complex equations, you will have more and more cases to cosnsider, such as in \(|x|+|y|=1\), you have to look at the positive and negative variants for \(y\) and then the positive and negative variants for \(x\) as well.

One important thing to note is that absolute value is not the only way to make an output strictly positive. A lot of these same ideas exist for even powers, which always produce a positive output. Although absolute value applies only to the direct equation and notation, there are a lot more cases in which it can be applied.

Graphing

When you graph an equation with absolute value, the output wil always be positive if \(|x|\) is isolated and outside of anything that could set it to be negative. Obviously there are offsets and phase shifts horizontally and negatavely (being adding and subtracting \(h\) outside of the absolute value, and \(-k\) being added or subtracted within the absolute value, respectively), the basic idea is that it reflects it around some axis.

One thing I do find important and interesting to know is what happens when you take \(|x|+|y|=1\). At first glance, this looks like it should just be \(x+y=1\), right? Not exactly, because this means that \(x\) can be negative with positive \(y\), or \(y\) can be negative with positve \(x\), and so on so on. Ths shape this creates is an enclosed diamond. A popular staple of the AMC 10 is to make these sorts of equations and solve for the area enclosed within an absolute value graph. You would do this by modeling the linear equations for all the different cases and seeing where they intersect. From there, you would just solve like a typical area problem, find the side lengths, etc.

Conclusion

Absolute Value functions can drastically change a lot about an expression or function. Although they are known just for making all values positive, it is important to remember that the key difference is that it is the distance from the number line. Absolute value equations should be computed with casework to consider both positive and negative variants. This is similar to how one would solve quadratics or even-power polynomials.