Desmos error? Incorrect evaluation of expression?

Question

This is the graph plotted by Desmos for the inequality $~\log_{0.2}\left(x^{2}-x-2\right)>\log_{0.2}\left(-x^{2}+2x+3\right)$

Here, you can see that plotted interval is $~x \in [2, 2.5)~$but log is not defined at x=2 (It becomes $~\log_{0.2}{0}$ which is undefined

I believe the answer should have been $x \in (2, 2.5)$

Answer 1

The answer is on $(2, 2.5)$. There aren't dash marks on $x=2$ because the function is undefined there and desmos doesn't know what to do with it. If it were really $[2, 2.5)$, you would see a bolded line on $x=2$ as shown below.

Answer 2

$\log_{1/5}\left(x^{2}-x-2\right)>\log_{1/5}\left(-x^{2}+2x+3\right)$ $\implies x^2-x-2>0 \& -x^2+2x+3>0 \implies -1> x ~\text{or}~ x>2 \& -1<x<3 \implies x \in (2,3)$

When $\log_{1/5}$ removed sign of in-equation changes, we get $x^2-x-2<-x^2+2x+3 \implies 2x^2-3x-5<0 \implies 2(x-5/2)(x+1)<0 \implies -1 <x<5/2 \implies x\in (-1,5/2)$

Finally the overlap of $(2,3) \& (-1,5/2)$ gives the interval $(2,5/2)$ as the solution of the in-equation.