Question

determine the value(s) for which the rational expression \frac{-11x + 1}{-30x^2 - 25x + 5} is undefined. if theres more than one value, list them separated by a comma, e.g. x = 2, 3.

provide your answer below:

x =

Explanation:

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<reasoning>

Identify the condition for an undefined rational expression

Using the Undefined Rational Expressions knowledge point
A rational expression is undefined when its denominator is equal to zero.
\[
-30x^2 - 25x + 5 = 0
\]

Factor out the greatest common factor

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} &-5(6x^2 + 5x - 1) = 0 \\ &6x^2 + 5x - 1 = 0 \end{aligned}$$

\]

Factor the quadratic trinomial

Using the Trinomial Factorization knowledge point
\[

$$\begin{aligned} &6x^2 + 6x - x - 1 = 0 \\ &6x(x + 1) - 1(x + 1) = 0 \\ &(6x - 1)(x + 1) = 0 \end{aligned}$$

\]

Solve for x using the zero product property

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &6x - 1 = 0 \quad \text{or} \quad x + 1 = 0 \\ &x = \frac{1}{6} \quad \text{or} \quad x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the value(s) for which the rational expression \(\frac{-11x + 1}{-30x^2 - 25x + 5}\) is undefined. If there's more than one value, list them separated by a comma, e.g. \(x = 2, 3\).

Provide your answer below:

\(x =\) <blank>\(-1, \frac{1}{6}\)</blank>
</answer>

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Answer:

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<reasoning>

Identify the condition for an undefined rational expression

Using the Undefined Rational Expressions knowledge point
A rational expression is undefined when its denominator is equal to zero.
\[
-30x^2 - 25x + 5 = 0
\]

Factor out the greatest common factor

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} &-5(6x^2 + 5x - 1) = 0 \\ &6x^2 + 5x - 1 = 0 \end{aligned}$$

\]

Factor the quadratic trinomial

Using the Trinomial Factorization knowledge point
\[

$$\begin{aligned} &6x^2 + 6x - x - 1 = 0 \\ &6x(x + 1) - 1(x + 1) = 0 \\ &(6x - 1)(x + 1) = 0 \end{aligned}$$

\]

Solve for x using the zero product property

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &6x - 1 = 0 \quad \text{or} \quad x + 1 = 0 \\ &x = \frac{1}{6} \quad \text{or} \quad x = -1 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the value(s) for which the rational expression \(\frac{-11x + 1}{-30x^2 - 25x + 5}\) is undefined. If there's more than one value, list them separated by a comma, e.g. \(x = 2, 3\).

Provide your answer below:

\(x =\) <blank>\(-1, \frac{1}{6}\)</blank>
</answer>

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