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Solving Rational Equations in Intermediate Algebra

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Terms in this set (27)

  • What is the first step in solving a rational equation?
    Factor each denominator.
  • What are non-permissible values (NPVs) in rational equations?
    Values that make any denominator zero and must be excluded from solutions.
  • What is the purpose of multiplying both sides of a rational equation by the LCD?
    To eliminate denominators and simplify the equation.
  • How do you check for extraneous solutions in rational equations?
    Verify that solutions are not NPVs; if they are, reject them as extraneous.
  • In the equation \(\frac{x}{2} + 3 = 2 + \frac{3x}{4}\), what is the LCD?
    4
  • What is the solution to \(\frac{x}{2} + 3 = 2 + \frac{3x}{4}\) after multiplying by the LCD and solving?
    x = 4
  • For the equation \(\frac{2x + 3}{x + 5} + \frac{1}{2} = \frac{7}{2x + 10}\), what are the NPVs?
    x ≠ -5
  • What is the LCD for \(\frac{2x + 3}{x + 5} + \frac{1}{2} = \frac{7}{2x + 10}\)?
    2(x + 5)
  • What is the solution to \(\frac{2x + 3}{x + 5} + \frac{1}{2} = \frac{7}{2x + 10}\) after multiplying by the LCD and simplifying?
    x = -\(\frac{4}{5}\)
  • How do you factor the denominator \(x^2 - 4\)?
    As (x - 2)(x + 2)
  • What are the NPVs for the equation with denominators \(x^2 - 4\) and \(6x + 12\)?
    x ≠ 2, -2
  • What is the LCD for denominators (x - 2)(x + 2) and 6(x + 2)?
    6(x + 2)(x - 2)
  • What is the solution to \(\frac{2}{x^2 - 4} + \frac{10}{6x + 12} = \frac{1}{x - 2}\) after multiplying by the LCD and simplifying?
    x = 5
  • How do you factor \(k^2 - 4\)?
    As (k + 2)(k - 2)
  • What are the NPVs for the equation with denominator \(k^2 - 4\)?
    k ≠ -2, 2
  • What is the LCD for denominators (k + 2), (k - 2), and (k^2 - 4)?
    (k + 2)(k - 2)
  • After multiplying by the LCD and simplifying, what quadratic equation results from \(\frac{4k - 1}{k + 2} - \frac{k + 1}{k - 2} = \frac{k^2 - 4k + 24}{k^2 - 4}\)?
    2k^2 - 8k - 24 = 0
  • What are the possible solutions to 2(k + 2)(k - 6) = 0?
    k = -2 or k = 6
  • Which solution must be rejected for the equation with NPVs k ≠ -2, 2?
    k = -2
  • What is the final solution for the equation \(\frac{4k - 1}{k + 2} - \frac{k + 1}{k - 2} = \frac{k^2 - 4k + 24}{k^2 - 4}\)?
    k = 6
  • How do you factor \(x^2 - x - 6\)?
    As (x + 2)(x - 3)
  • What are the NPVs for the equation with denominator \(x^2 - x - 6\)?
    x ≠ -2, 3
  • What is the LCD for denominators (x + 2) and (x - 3)?
    (x + 2)(x - 3)
  • What quadratic equation results from multiplying and simplifying \(\frac{3x}{x + 2} - \frac{5}{x - 3} = \frac{-25}{x^2 - x - 6}\)?
    3x^2 - 14x - 15 = 0
  • What are the possible solutions to (3x - 5)(x - 3) = 0?
    x = \(\frac{5}{3}\) or x = 3
  • Which solution must be rejected due to NPVs in the equation with denominators (x + 2)(x - 3)?
    x = 3
  • What is the final solution to \(\frac{3x}{x + 2} - \frac{5}{x - 3} = \frac{-25}{x^2 - x - 6}\)?
    x = \(\frac{5}{3}\)