## Intermediate Algebra: Connecting Concepts through Application

$\text{all real numbers except }$ $x=\left\{ 1,8 \right\}$
$\bf{\text{Solution Outline:}}$ 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 $\text{all real numbers except }$ $x=\left\{ 1,8 \right\} .$