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Transcript
Rational Equations MATH 017 Intermediate Algebra S. Rook 1 Overview • Section 6.6 in the textbook – Solving rational equations – Extraneous solutions – More examples 2 Solving Rational Equations 3 Solving Rational Equations • So far, we have been working with rational expressions – No = • Find the LCD of all factored denominators for both rational expressions and rational equations • For rational expressions – Simplify by combining over one denominator • Section 6.2 • For rational equations – Multiply both sides of the equation by the LCD to clear out the denominators 4 Solving Rational Equations (Example) Ex 1: Solve 5x 1 x 6 2 3 5 Solving Rational Equations (Example) 3x 6 7 x 2x 2 Ex 2: Solve 5 6 Extraneous Solutions 7 Extraneous Solutions • Extraneous solution: a solution to a rational equation which causes at least one of the denominators in the original equation to equal zero Ex: x = 1 is an extraneous solution to 1 / (x – 1) = 1 • Must EXCLUDE an extraneous solution from a solution set • What If all proposed solutions are extraneous? – The answer is the empty set { } 8 Extraneous Solutions (Example) 2x 1 x 3 x 3 Ex 3: Solve x 5 9 Extraneous Solutions (Example) Ex 4: Solve 3x 1 1 3x 1 x 3x 1 10 Additional Examples 11 Additional Examples Ex 5: Solve 2 x 10 5 x x 2 x 4 x 16 x 4 12 Additional Examples Ex 6: Solve 3z 1 4 2 2 3z 7 z 6 z 3z 2 z 13 Additional Examples Ex 7: Solve x 5 3 4 x 14 Additional Examples Ex 8: Solve 64 2a 1 2 a 16 a4 15 Summary • After studying these slides, you should know how to do the following: – Solve rational equations – Identify extraneous solutions and exclude them from solution sets 16