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Exercises Equations and inequalities

Example 6 from Algebra and Trigonometry, 2.2 Linear Equations in One Variable

Solving Rational Equations with a Binomial in the Denominator

Solve the following rational equations and state the excluded values:

ⓐ $\frac{3}{x - 6} = \frac{5}{x}$
ⓑ $\frac{x}{x - 3} = \frac{5}{x - 3} - \frac{1}{2}$
ⓒ $\frac{x}{x - 2} = \frac{5}{x - 2} - \frac{1}{2}$

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.

1
Solution
1. ⓐ
  The denominators $x$ and $x - 6$ have nothing in common. Therefore, the LCD is the product $x (x - 6) .$ However, for this problem, we can cross-multiply.
  
  $\begin{matrix} \frac{3}{x - 6} & = & \frac{5}{x} \\ 3 x & = & 5 (x - 6) & Distribute . \\ 3 x & = & 5 x - 30 \\ −2 x & = & −30 \\ x & = & 15 \end{matrix}$
  The solution is 15. The excluded values are $6$ and $0.$
2. ⓑ
  The LCD is $2 (x - 3) .$ Multiply both sides of the equation by $2 (x - 3) .$
  
  $\begin{matrix} 2 (x - 3) (\frac{x}{x - 3}) & = & (\frac{5}{x - 3} - \frac{1}{2}) 2 (x - 3) \\ \frac{2 (x - 3) x}{x - 3} & = & \frac{2 (x - 3) 5}{x - 3} - \frac{2 (x - 3)}{2} \\ 2 x & = & 10 - (x - 3) \\ 2 x & = & 10 - x + 3 \\ 2 x & = & 13 - x \\ 3 x & = & 13 \\ x & = & \frac{13}{3} \end{matrix}$
  The solution is $\frac{13}{3} .$ The excluded value is $3.$
3. ⓒ
  The least common denominator is $2 (x - 2) .$ Multiply both sides of the equation by $x (x - 2) .$
  
  $\begin{matrix} 2 (x - 2) (\frac{x}{x - 2}) & = & (\frac{5}{x - 2} - \frac{1}{2}) 2 (x - 2) \\ 2 x & = & 10 - (x - 2) \\ 2 x & = & 12 - x \\ 3 x & = & 12 \\ x & = & 4 \end{matrix}$
  The solution is 4. The excluded value is $2.$

Your turn

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

Solve $\frac{- 3}{2 x + 1} = \frac{4}{3 x + 1} .$ State the excluded values.

How did it go?

PreviousSolving a Rational Equation by Factoring the Denominator Next Solving a Rational Equation with Factored Denominators and Stating Excluded Values

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.2, 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.