What is the interval of $A=\frac{2x-6}{x-4}$ provided that $5\lt x\lt 9$?

Question

$A=\frac{2x-6}{x-4}$

$5\lt x\lt9$

Find $?\lt A\lt ? $

I tried to solve this problem by trying to get $A$ into an expression that wouldn't have $x$ in the denominator.

I know that I can't just find the intervals of $2x-6$ and $x-4$ and divide these two intervals.

How can I solve this in the simplest way possible?

Answer 1

Hint: $A=2+\frac 2 {x-4}$. Can you use this to find the range?

Answer 2

Let $f(x)=\frac{2x-6}{x-4}$, then show that $f'(x)<0$ for $x \in [5,9]$. Since $f(5)=4$ , $f(9)=\frac{12}{5}$ and $f$ is continuous, we have $f([5,9])=[\frac{12}{5},4]$. Thus

$$f((5,9))=(\frac{12}{5},4).$$