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Exercises Equations and inequalities

Example 11 from Algebra and Trigonometry, 2.6 Other Types of Equations

Solving a Rational Equation Leading to a Quadratic

Solve the following rational equation: $\frac{- 4 x}{x - 1} + \frac{4}{x + 1} = \frac{- 8}{x^{2} - 1} .$

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 want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However, $x^{2} −1 = (x + 1) (x - 1) .$ Then, the LCD is $(x + 1) (x - 1) .$ Next, we multiply the whole equation by the LCD.

$\begin{matrix} (x + 1) (x - 1) [\frac{−4 x}{x - 1} + \frac{4}{x + 1}] & = & [\frac{−8}{(x + 1) (x - 1)}] (x + 1) (x - 1) \\ −4 x (x + 1) + 4 (x - 1) & = & −8 \\ −4 x^{2} - 4 x + 4 x - 4 & = & −8 \\ −4 x^{2} + 4 & = & 0 \\ −4 (x^{2} - 1) & = & 0 \\ −4 (x + 1) (x - 1) & = & 0 \\ x & = & −1 \\ x & = & 1 \end{matrix}$

In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.

Your turn

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

Solve $\frac{3 x + 2}{x - 2} + \frac{1}{x} = \frac{- 2}{x^{2} - 2 x} .$

How did it go?

PreviousSolving an Equation in Quadratic Form Containing a Binomial Next Using Interval Notation to Express All Real Numbers Greater Than or Equal to a

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.