## Intermediate Algebra: Connecting Concepts through Application

$\text{all real numbers except }$ $a=-\dfrac{7}{2},3 .$
$\bf{\text{Solution Outline:}}$ The domain of the given rational function, $h(a)=\dfrac{3a-1}{(2a+7)(a-3)} ,$ are the values of $a$ which will NOT make the denominator equal to $0.$ $\bf{\text{Solution Details:}}$ Solving for the values of $a$ that will make the denominator equal to $0$ results to \begin{array}{l}\require{cancel} (2a+7)(a-3)=0 .\end{array} Equating each factor to zero (Zero Product Property), then \begin{array}{l}\require{cancel} 2a+7=0 \\\\\text{OR}\\\\ a-3=0 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 2a=-7 \\\\ a=-\dfrac{7}{2} \\\\\text{OR}\\\\ a-3=0 \\\\ a=3 .\end{array} Hence, the domain is the set of $\text{all real numbers except }$ $a=-\dfrac{7}{2},3 .$