Answer
$\text{all real numbers except }$ $
x=\left\{ 1,8 \right\}
$
Work Step by Step
$\bf{\text{Solution Outline:}}$
The domain of the given rational function, $
f(x)=\dfrac{2x+7}{x^2-9x+8}
,$ are the values of $
x
$ which will NOT make the denominator equal to $0.$
$\bf{\text{Solution Details:}}$
The values of $
x
$ that will make the denominator equal to $0$ are
\begin{array}{l}\require{cancel}
x^2-9x+8=0
.\end{array}
Using the FOIL Method, which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(x-8)(x-1)=0
.\end{array}
Equating each factor to zero (Zero Product Property), then
\begin{array}{l}\require{cancel}
x-8=0
\\\\\text{OR}\\\\
x-1=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-8=0
\\\\
x=8
\\\\\text{OR}\\\\
x-1=0
\\\\
x=1
.\end{array}
Hence, the domain is the set of $
\text{all real numbers except }$ $
x=\left\{ 1,8 \right\}
.$
