# Equations with Moduli

1. Dec 27, 2014

### Astudious

How does one go about solving equations in one variable which contain moduli? For instance, those of the form f(x,|x|)=0 or f(x,|g(x)|) more generally.

Obviously I don't expect a completely "one-size fits all" solution, but a general approach to dealing with the moduli is what I'm looking for. (i.e. let's assume that, once the moduli are gone, I will be able to deal satisfactorily with the remaining equation.)

2. Dec 27, 2014

### MohammedRady

Get the modulus on one side, and everything else on the other side.

$|f(x)| = g(x)$

Using the definition of the absolute value, $|a| = a$ if $a$ is positive (or 0), and $|a| = -a$ if $a$ is negative, we have two equations:

$g(x) = f(x)$
$g(x) = -f(x)$

You might get extraneous solutions though, so always plug in the values you found into $g(x)$ and make sure $g(x)$ is positive (or zero). If you find that $g(x)$ is negative for a particular value, ignore this solution, since the absolute value of any real number is greater than or equal to zero by definition.

Hope this helps.

By the way, the functions $f$ and $g$ I used in my explanation are in no way related to those in your post, so don't get confused.

3. Dec 28, 2014

### Astudious

Thanks. So we just rearrange into the form above, and then solve the two equations

$g(x) = f(x)$
$g(x) = -f(x)$

and use the superset of the solutions, removing any solutions which lead to g(x)<0 since no modulus of f(x) can equal them.

What if it were something like

$|f(x)| + |g(x)| = h(x)$

?

4. Dec 28, 2014

### MohammedRady

Square both sides.

$[|f(x)| + |g(x)|]^2 = h(x)^2$
$|f(x)|^2 + 2|f(x)||g(x)| + |g(x)|^2 = h(x)^2$

Recall that $a^2 = |a|^2$ and $|a||b| = |ab|$

$f(x)^2 + 2|f(x)g(x)| + g(x)^2 = h(x)^2$
$|f(x)g(x)| = \frac{1}{2} [h(x)^2 - f(x)^2 - g(x)^2]$

You now have the equation in the form you mentioned in your first post.

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