Why wolfram cannot get the third solution? https://www.wolframalpha.com/input/?i=%5Csqrt%7B3x-4%7D%2B%5Csqrt%5B3%5D%7B5-3x%7D%3D1
This surely has the third solution as x= 13/3.
But:
https://www.wolframalpha.com/input/?i=x%3D%3D13%2F3%2C+%5Csqrt%7B3x-4%7D%2B%5Csqrt%5B3%5D%7B5-3x%7D-1
Why Wolfram fails to get it correct?
 A: Because it assumes the principal cube root. If you add the assumption that the root is the real root, it gives you all the solutions like here.
A: The Wolfram Language function Power[] (documentation) gives the principal root, meaning the one with the least complex angle.  For example, \sqrt[3]{-8}, which gives $1 + \mathrm{i}\sqrt{3}$, having angle $\pi/3$, instead of $-2$, having angle $\pi$.  We can see that the complex angle of $\sqrt[3]{5-3x}$ changes between 1.6 and 1.7.
WA: plot (5 - 3 x)^(1/3) and Arg((5 - 3 x)^(1/3)) for 0<=x<=5
in exactly the same way the result of Power[] does.
WA: plot Power[5 - 3 x,1/3] and Arg(Power[5 - 3 x,1/3]) for 0<=x<=5
Methods to avoid this


*

*Use a function that always returns the real root, like CubeRoot[] (doc).  \sqrt{3x-4}+CubeRoot[5-3x]=1.  Also cbrt() is a WA shortcut for this function.

*Use a form for which WA explicitly mentions that it is assuming a root, then tell it which assumption you actually want.  \sqrt{3x-4}+(5-3x)^(1/3)=1 , then click "the real-valued root instead".

*More generally than using CubeRoot[], there is surd() (docs), which also gives the real root, when there is one.  So here, we would use \sqrt{3x-4}+surd(5-3x,3)=1
