\frac{4}{9}^x + \frac{1}{5}^x = 87

Steps to Solve

1. Convert the bases
   To make it easier to work with, we can express the bases in a different form. Recall that $4 = 2^2$ and $9 = 3^2$. Thus, we have:
   $$ \left(\frac{4}{9}\right)^x = \frac{4^x}{9^x} = \frac{(2^2)^x}{(3^2)^x} = \frac{2^{2x}}{3^{2x}} $$
   So, our equation becomes:
   $$ \frac{2^{2x}}{3^{2x}} + \left(\frac{1}{5}\right)^x = 87 $$

2. Rewrite the equation properly
   Now, we can rewrite the $ \left(\frac{1}{5}\right)^x = 5^{-x} $ in the equation. This gives us:
   $$ \frac{2^{2x}}{3^{2x}} + 5^{-x} = 87 $$

3. Isolate one of the terms
   Before making further substitutions, let's isolate one term. To simplify the calculations, we will rearrange the equation as:
   $$ \frac{2^{2x}}{3^{2x}} = 87 - 5^{-x} $$

4. Substituting for easier calculation
   Let's substitute $y = 5^{-x}$, meaning $y$ can be replaced back later. The equation then becomes:
   $$ \frac{2^{2x}}{3^{2x}} = 87 - y $$
   We can also express $2^{2x}$ in terms of $y$:
   $$ 2^{2x} = (3^{2x})(87 - y) $$

5. Use numerical approximation/graphing if necessary
   Solving this equation analytically may be complex, so we can estimate or graph to find possible values of $x$. You could use numerical methods or graphing tools to find the intersection of the two parts on a graph.