EDGE Math 8

Example

Determine the domain of each rational expression.

1.

\frac{3 x}{2 x - 5}

2.

\frac{x - 1}{x^{2} - 5 x + 6}

Solution

1. The denominator is $2 x - 5.$ Equate it to 0, and then solve.

$2 x - 5 = 0$
$2 x - 5$	Add 5 to both sides of the equation.
$x = \frac{5}{2}$	Divide both sides of the equation by 2.

Thus, the denominator is equal to 0 when $x = \frac{5}{2} .$ Consequently, the domain of $\frac{3 x}{2 x - 5}$ is the set of real numbers except $\frac{5}{2} .$

2. Let the denominator $x^{2} - 5 x + 6$ be equal to 0; that is, $x^{2} - 5 x + 6 = 0 .$ Factor the left-hand side of the equation.

(x - 2) (x - 3) = 0

Apply the Zero Product Principle. Let each factor be equal to zero, and then solve.

$x - 2 = 0$	or	$x - 3 = 0$
$x = 2$		$x = 3$

The denominator is equal to 0 when $x = 2$ or $x = 3$ Thus, the domain of $\frac{x - 1}{x^{2} - 5 x + 6}$ is the set of real numbers except 2 and 3.

Domains of Rational Expressions