Teacher gives me result
(-5/3; -1/2] [2/3; 5/3)
But how? Is a solution on screenshot incorrect?
I suspect that there are severe translation problems here. We cannot be sure of what the problem would say in English. Nor do we know exactly what your teacher said.
Let's try to clear up one source of confusion.
IMPLIED RESTRICTIONS.
If we are dealing with real numbers,
\(\displaystyle \sqrt{5 - 6x} \in \mathbb R \implies 0 \le 5 - 6x \implies 6x \le 5 \implies x \le \dfrac{5}{6}.\)
\(\displaystyle ln(4x^2 - a^2) \in \mathbb R \implies 0 < 4x^2 - a^2 \implies a^2 < 4x^2 \implies -\ 2|x| < a < 2|x|.\)
\(\displaystyle ln(2x + a) \in \mathbb R \implies 0 < 2x + a \implies -\ 2x < a.\)
Thus, whatever our answer may be, it must be subject to those restrictions.
Now \(\displaystyle x = \dfrac{5}{6} \implies \sqrt{5 - 6x} = 0 \implies 0 * ln(4x^2 - a^2) = 0 * ln(2x + a).\)
\(\displaystyle \therefore x = \dfrac{5}{6} \implies -\ 2 \left | \dfrac{5}{6} \right | < a < 2 \left | \dfrac{5}{6} \right | \implies -\ \dfrac{5}{3} < a < \dfrac{5}{3}.\)
This of course is an incomplete answer. It applies only to the special case of x = 5/6. If, however, the problem states or implies that a is to be a unique number, then x = 5/6 is excluded because a can take on many values. Otherwise, I do not see the rationale for the other restrictions on a. Nevertheless, without fully understanding the exact nuances of the problem in its original language, I have no basis to quibble with your teacher.
