Example 7 from Algebra and Trigonometry, 2.2 Linear Equations in One Variable

Solving a Rational Equation with Factored Denominators and Stating Excluded Values

Solve the rational equation after factoring the denominators: $\frac{2}{x + 1} - \frac{1}{x - 1} = \frac{2 x}{x^{2} - 1} .$ State the excluded values.

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 must factor the denominator $x^{2} −1.$ We recognize this as the difference of squares, and factor it as $(x - 1) (x + 1) .$ Thus, the LCD that contains each denominator is $(x - 1) (x + 1) .$ Multiply the whole equation by the LCD, cancel out the denominators, and solve the remaining equation.

$\begin{matrix} (x - 1) (x + 1) (\frac{2}{x + 1} - \frac{1}{x - 1}) & = & (\frac{2 x}{(x - 1) (x + 1)}) (x - 1) (x + 1) \\ 2 (x - 1) - 1 (x + 1) & = & 2 x \\ 2 x - 2 - x - 1 & = & 2 x Distribute the negative sign . \\ −3 - x & = & 0 \\ −3 & = & x \end{matrix}$
The solution is $−3.$ The excluded values are $1$ and $−1.$

Your turn

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

Solve the rational equation: $\frac{2}{x - 2} + \frac{1}{x + 1} = \frac{1}{x^{2} - x - 2} .$

How did it go?

Solution

Your turn

Solution

Your turn