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Equations with Moduli

  1. Dec 27, 2014 #1
    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. jcsd
  3. Dec 27, 2014 #2
    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.
     
  4. Dec 28, 2014 #3
    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)##

    ?
     
  5. Dec 28, 2014 #4
    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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