Solving a Rational Equation by Factoring the Denominator

Solve the following rational equation: $\frac{1}{x} = \frac{1}{10} - \frac{3}{4 x} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First find the common denominator. The three denominators in factored form are $x, 10 = 2 \cdot 5,$ and $4 x = 2 \cdot 2 \cdot x .$ The smallest expression that is divisible by each one of the denominators is $20 x .$ Only $x = 0$ is an excluded value. Multiply the whole equation by $20 x .$

$\begin{matrix} 20 x (\frac{1}{x}) & = & (\frac{1}{10} - \frac{3}{4 x}) 20 x \\ 20 & = & 2 x - 15 \\ 35 & = & 2 x \\ \frac{35}{2} & = & x \end{matrix}$
The solution is $\frac{35}{2} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve the rational equation: $- \frac{5}{2 x} + \frac{3}{4 x} = - \frac{7}{4} .$

How did it go?