Solve for $x$.
$\frac{3}{x-1}-\frac{1}{x^2-1}=\frac{5}{x-1}$
We are asked to solve the rational equation:
\[\frac{3}{x-1} - \frac{1}{x^2 - 1} = \frac{5}{x-1}\]
Before starting, we must identify the domain restrictions. The expression is undefined when denominators are zero. Specifically, \(x-1 = 0 \implies x = 1\), and \(x^2 - 1 = 0 \implies x = 1\) or \(x = -1\). Thus, any potential solution must satisfy \(x
eq 1\) and \(x
eq -1\). We observe that \(x^2 - 1 = (x-1)(x+1)\), which will be helpful for finding a common denominator.
To clear the denominators, we multiply every term in the equation by the common denominator, \((x-1)(x+1)\):
\[(x-1)(x+1) \left( \frac{3}{x-1} - \frac{1}{x^2 - 1} \right) = (x-1)(x+1) \left( \frac{5}{x-1} \right)\]
This simplifies the equation as follows:
\[3(x+1) - 1 = 5(x+1)\]
Expanding both sides, we get:
\[3x + 3 - 1 = 5x + 5\]
\[3x + 2 = 5x + 5\]
Now, we solve for \(x\) by isolating the variable:
\[2 = 5x - 3x + 5\]
\[2 = 2x + 5\]
\[2 - 5 = 2x\]
\[-3 = 2x\]
\[x = -\frac{3}{2}\]
Finally, we verify that our solution \(x = -\frac{3}{2}\) is not in our restricted set (\(x
eq 1, x
eq -1\)). Since it is valid, it is the correct solution. Among the given options, \(-\frac{3}{2}\) is indeed one of them.
Rational equations appear in many real-world scenarios such as calculating travel time. For instance, if a boat travels a certain distance upstream and downstream, the time taken involves expressions like \(\frac{d}{v-c}\) and \(\frac{d}{v+c}\), where \(v\) is the boat's speed and \(c\) is the current. Solving for these speeds often requires equating and solving rational expressions, similar to the method used here.