Solving a Rational Equation without Factoring

Solve the following rational equation:

\frac{2}{x} - \frac{3}{2} = \frac{7}{2 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
We have three denominators: $x,$ $2,$ and $2 x .$ No factoring is required. The product of the first two denominators is equal to the third denominator, so, the LCD is $2 x .$ Only one value is excluded from a solution set, 0. Next, multiply the whole equation (both sides of the equal sign) by $2 x .$
$\begin{matrix} 2 x (\frac{2}{x} - \frac{3}{2}) & = & (\frac{7}{2 x}) 2 x \\ 2 x (\frac{2}{x}) - 2 x (\frac{3}{2}) & = & (\frac{7}{2 x}) 2 x & Distribute 2 x . \\ 2 (2) - 3 x & = & 7 & Denominators cancel out . \\ 4 - 3 x & = & 7 \\ −3 x & = & 3 \\ x & = & −1 \\ or {−1} \end{matrix}$
The proposed solution is −1, which is not an excluded value, so the solution set contains one number, $−1,$ or ${−1}$ written in set notation.

Your turn

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

Solve the rational equation: $\frac{2}{3 x} = \frac{1}{4} - \frac{1}{6 x} .$

How did it go?