# Chapter 7 - Rational Functions - 7.1 Rational Functions and Variation - 7.1 Exercises: 36

$\text{all real numbers except }$ $x=\left\{ -4,9 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ The domain of the given rational function, $g(x)=\dfrac{2x-7}{(x+4)(x-9)} ,$ are the values of $x$ which will NOT make the denominator equal to $0.$ $\bf{\text{Solution Details:}}$ Solving for the values of $x$ that will make the denominator equal to $0$ results to \begin{array}{l}\require{cancel} (x+4)(x-9) .\end{array} Equating each factor to zero (Zero Product Property), then \begin{array}{l}\require{cancel} x+4=0 \\\\\text{OR}\\\\ x-9=0 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} x+4=0 \\\\ x=-4 \\\\\text{OR}\\\\ x-9=0 \\\\ x=9 .\end{array} Hence, the domain is the set of $\text{all real numbers except }$ $x=\left\{ -4,9 \right\} .$

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