Answer
$x\in(2,4)\cup(7,9)$
Work Step by Step
First of all, in order for the inequality to be defined
the logarithms must be defined (arguments must be positive):
$x-2>0\qquad and\qquad 9-x>0$
$x>2\qquad and\qquad x<9$
$x\in(2,9)\qquad (*)$
... on the LHS, apply rule: $\log_{a}(AB)=\log_{a}A+\log_{a}B$
... on the RHS, $1=\log 10$
$\log[(x-2)(9-x)]0$
Factor the trinomial.
Two factors of 28 whose sum is -11... $-7$ and $-4$
$(x-4)(x-7)>0$
The LHS would be graphed as a parabola turning upward, with 4 and 7 as zeros.
(see graph below)
The graph is above the x axis for $x<4$ and $x>7$.
Finally, because of (*) ,$x\in(2,9),$
$x\in(2,4)\cup(7,9)$
is the solution to the inequality.
