Question

Solve the rational equation: 9/x + 1/4 = 2/x

Answer 1

To solve the equation \( \frac{9}{x} + \frac{1}{4} = \frac{2}{x} \), we'll first eliminate the fractions by finding a common denominator. The common denominator for the terms involved (noting that \( x \) and \( 4 \) are the denominators) will be \( 4x \).

Now, we can rewrite each term with the common denominator:

1. \( \frac{9}{x} \) becomes \( \frac{9 \cdot 4}{x \cdot 4} = \frac{36}{4x} \)
2. \( \frac{1}{4} \) becomes \( \frac{1 \cdot x}{4 \cdot x} = \frac{x}{4x} \)
3. \( \frac{2}{x} \) becomes \( \frac{2 \cdot 4}{x \cdot 4} = \frac{8}{4x} \)

Rewriting the equation gives:

\[
\frac{36}{4x} + \frac{x}{4x} = \frac{8}{4x}
\]

Now, since all terms are over the common denominator \( 4x \), we can multiply the entire equation by \( 4x \) to eliminate the denominators:

\[
36 + x = 8
\]

Next, we will isolate \( x \):

\[
x = 8 - 36
\]
\[
x = -28
\]

Now we will check that this solution does not make any denominators zero. The original equation contained \( x \) in the denominators, and since \( x = -28 \neq 0 \), it is a valid solution.

Thus, the solution to the equation \( \frac{9}{x} + \frac{1}{4} = \frac{2}{x} \) is:

\[
\boxed{-28}
\]