Which of the following represents the solution of $\frac{3}{2}=\frac{3}{2 x}-\frac{6}{5 x}$ ?
The given equation is \(\frac{3}{2} = \frac{3}{2x} - \frac{6}{5x}\). Since the variable \(x\) appears in the denominator, we must ensure \(x eq 0\). This is our domain constraint.
To solve for \(x\), we first simplify the right-hand side of the equation. Both terms \(\frac{3}{2x}\) and \(\frac{6}{5x}\) have the variable \(x\) in the denominator. The least common denominator for \(2x\) and \(5x\) is \(10x\). Rewriting the terms:
\[\frac{3}{2x} = \frac{3 \times 5}{2x \times 5} = \frac{15}{10x}\]
\[\frac{6}{5x} = \frac{6 \times 2}{5x \times 2} = \frac{12}{10x}\]
Subtracting these, we get:
\[\frac{15}{10x} - \frac{12}{10x} = \frac{15 - 12}{10x} = \frac{3}{10x}\]
So the equation simplifies to:
\[\frac{3}{2} = \frac{3}{10x}\]
Now we have \(\frac{3}{2} = \frac{3}{10x}\). We can solve for \(x\) by cross-multiplying or by equating the denominators since the numerators are identical:
\[3 \times 10x = 3 \times 2\]
\[30x = 6\]
Dividing both sides by 30:
\[x = \frac{6}{30} = \frac{1}{5}\]
The value \(x = \frac{1}{5}\) satisfies the domain constraint \(x
eq 0\).
Comparing our result \(x = \frac{1}{5}\) with the given options, it matches the first choice. Thus, the solution to the equation is \(x = \frac{1}{5}\). Therefore, the correct option is \(\boxed{x = \frac{1}{5}}\).
This type of algebraic equation is commonly used in physics and engineering, such as calculating the equivalent resistance in parallel circuits or the time taken for two machines to complete a job when working together. For example, if one worker takes \(2x\) hours to complete a task and another takes \(5x\) hours, their combined rate of work is expressed using these types of fractional equations. Understanding how to solve such equations allows you to optimize schedules and determine resource allocation efficiently.