Lesson 22

Identify all values of \(x\) that make the equation true.

1. \(\frac{2x+1}{x}=\frac{1}{x-2}\)
2. \(\frac{1}{x+2}=\frac{2}{x-1}\)
3. \(\frac{x+3}{1-x} = \frac{x+1}{x+2}\)
4. \(\frac{x+2}{x+8}= \frac{1}{x+2}\)

Kiran is solving \(\frac{2x-3}{x-1} = \frac{2}{x(x-1)}\) for \(x\), and he uses these steps:

\(\begin{align} \frac{2x-3}{x-1} &= \frac{2}{x(x-1)}\\ (x-1)\left(\frac{2x-3}{x-1} \right) &= x(x-1) \left( \frac{2}{x(x-1)} \right)\\ 2x-3 &= 2\\ 2x &= 5 \\ x &= 2.5 \\ \end{align} \)

He checks his answer and finds that it isn't a solution to the original equation, so he writes “no solutions.” Unfortunately, Kiran made a mistake while solving. Find his error and calculate the actual solution(s).

Identify all values of \(x\) that make the equation true.

1. \(x=\frac{25}{x}\)
2. \(x+2= \frac{6x-3}{x}\)
3. \(\frac{x}{x^2} = \frac{3}{x}\)
4. \(\frac{6x^2+18x}{2x^3} = \frac{5}{x}\)

Rewrite the rational function \(g(x) = \frac{x-9}{x}\) in the form \(g(x) = c + \frac{r}{x}\), where \(c\) and \(r\) are constants.

Elena has a boat that would go 9 miles per hour in still water. She travels downstream for a certain distance and then back upstream to where she started. Elena notices that it takes her 4 hours to travel upstream and 2 hours to travel downstream. The river’s speed is \(r\) miles per hour. Write an expression that will help her solve for \(r\).