1Cademy - Finding Undefined Values of $$\frac{9y}{x}$$, $$\frac{4b-3}{2b+5}$$, and $$\frac{x+4}{x^2+5x+6}$$

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Rational Expression

Example

Finding Undefined Values of $\frac{9y}{x}$ , $\frac{4b-3}{2b+5}$ , and $\frac{x+4}{x^2+5x+6}$

Determine the values for which each rational expression is undefined by setting the denominator equal to zero and solving.

ⓐ $\frac{9y}{x}$ : The denominator is $x$ . Setting it equal to zero gives $x = 0$ . Therefore $\frac{9y}{x}$ is undefined for $x = 0$ .

ⓑ $\frac{4b - 3}{2b + 5}$ : The denominator is $2b + 5$ . Set it equal to zero and solve:

$2b + 5 = 0$ $2b = -5$ $b = -\frac{5}{2}$

Therefore $\frac{4b - 3}{2b + 5}$ is undefined for $b = -\frac{5}{2}$ .

ⓒ $\frac{x + 4}{x^2 + 5x + 6}$ : The denominator is the trinomial $x^2 + 5x + 6$ . Set it equal to zero and factor:

$x^2 + 5x + 6 = 0$ $(x + 2)(x + 3) = 0$ $x + 2 = 0 \text{ or } x + 3 = 0$ $x = -2 \text{ or } x = -3$

Therefore $\frac{x + 4}{x^2 + 5x + 6}$ is undefined for $x = -2$ or $x = -3$ .

These three parts illustrate increasing complexity: a monomial denominator yields a single obvious restricted value, a binomial denominator requires solving a one-step linear equation, and a trinomial denominator requires factoring and applying the Zero Product Property to find two restricted values.

Updated 2026-04-21

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OpenStax Elementary Algebra

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