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9.6 Solving Rational Equations Presentation by Amelia Tajik and Lydia Bunker

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What is a Rational Expression? A rational expression is a fraction in which the numerator and/or the denominator are polynomials. Examples:

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Reducing a rational expression to lowest terms A rational expression has been reduced to lowest terms if all common factors from the numerator and denominator have been canceled. This process applies to both regular fractions and rational expressions.

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Finding the Least Common Denominator The main difficulty in solving rational equations is finding the least common denominator. However, there is a really simple process for finding the least common denominator for rational expressions. Here is it! 1. Factor all the denominators. 2. Write down each factor that appears at least once in any of the denominators. DO NOT write down the power that is on each factor, only write down the factor 3. Now, for each factor written down in the previous step and write down the largest power that occurs in all the denominators containing that factor. 4. The product all the factors from the previous step is the least common denominator.

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2, x, x, x Example 2, 3, x, x3, x, x, x, x, x2, x, x, x 2 3 3, x, x, x, x, x2, 3, x, x { 6 { x5x5 = 6x 5 x, x, x, x, x

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Example, continued… 6x 5 24x 5 - 6x 5 – 30x 5 6x 2 3x 5 2x 3 4x 3 – 2 – 15x 2 4x 3 – 15x 2 - 2

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Cross Multiplying a c -- = -- b d ad = bc (x+2) x = (x-4) (x+3) (x+2)(x+3) = x(x-4) Variables:Example:

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Using the Least Common Denominator and Cross Multiplying to Solve Rational Equations 15x – (5x + 10) = x x – 10 = x + 2 9x = 12 x = 12/9 = 4/3 x = 4/3

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0.7 L of an acid solution whose acid concentration is 9 mol/L. You want to dilute the solution with water so that its acid concentration is 4 mol/L. How much water should you add to the solution? Concentration of new solution = 12 Moles of acid in original solution = 9(0.7) Volume of original solution = 0.7 Volume of water added = x 4 = [9(0.7) / (0.7+ x)] 4(0.7 + x) = x = 6.3 4x = 3.5 x = You should add L of water to the solution. Example of Rational Equations in Real Life

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Links Section 10.4 PowerPoint presentation Section 10.4 PowerPoint presentation

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