Equations with Rational Expressions. Integers in the denominators. x 2. 1 Original equation. 1 2 x 2
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1 0 (-8) Chapter Rational Epressions In this section Equations with Rational Epressions Etraneous s. SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS Many problems in algebra can be solved by using equations involving rational epressions. In this section you will learn how to solve equations that involve rational epressions, and in Section.7 and.8 you will solve problems using these equations. Equations with Rational Epressions We solved some equations involving fractions in Section.. In that section the equations had only integers in the denominators. Our first step in solving those equations was to multiply by the LCD to eliminate all of the denominators. E X A M P L E Integers in the denominators Solve. helpful hint Note that it is not necessary to convert each fraction into an equivalent fraction with a common denominator here. Since we can multiply both sides of an equation by any epression we choose, we choose to multiply by the LCD. This tactic eliminates the fractions in one step. The LCD for,, and is. Multiply each side of the equation by : Original equation Multiply each side by. Distributive property ( ) Simplify. Distributive property Subtract 7 from each side. Divide each side by. Check in the original equation: The solution to the equation is. CAUTION When a numerator contains a binomial, as in Eample, the numerator must be enclosed in parentheses when the denominator is eliminated. To solve an equation involving rational epressions, we usually multiply each side of the equation by the LCD for all the denominators involved, just as we do for an equation with fractions.
2 . Solving Equations with Rational Epressions (-9) E X A M P L E Variables in the denominators Solve. We multiply each side of the equation by, the LCD for,, and : Original equation Multiply each side by. Distributive property Simplify. Subtract from each side. Check that satisfies the original equation: E X A M P L E study tip Your mood for studying should match the mood in which you are tested. Being too relaed during studying will not match the increased level of activation you attain during a test. Likewise, if you get too tensed-up during a test, you will not do well because your test-taking mood will not match your studying mood. An equation with two solutions Solve the equation The LCD for the denominators and 5 is ( 5): ( 5) ( 5) ( 5)9 5 ( 5)00 (00) ( 5) (9 5)( 0) or or 0 Original equation Multiply each side by ( 5). All denominators are eliminated. Simplify. Get 0 on one side. Factor. Zero factor property A check will show that both 5 and 0 satisfy the original equation. 9 Etraneous s In a rational epression we can replace the variable only by real numbers that do not cause the denominator to be 0. When solving equations involving rational epressions, we must check every solution to see whether it causes 0 to appear in a denominator. If a number causes the denominator to be 0, then it cannot be a solution to the equation. A number that appears to be a solution but causes 0 in a denominator is called an etraneous solution.
3 (-0) Chapter Rational Epressions E X A M P L E An equation with an etraneous solution Solve the equation. Because the denominator factors as ( ), the LCD is ( ). ( ) ( ) ( ) Multiply each side of the ( ) Simplify. Check in the original equation: original equation by ( ). The denominator is 0. So does not satisfy the equation, and it is an etraneous solution. The equation has no solutions. E X A M P L E 5 Another etraneous solution Solve the equation. The LCD for the denominators and is ( ): Original equation ( ) ( ) ( ) Multiply each side by ( ). ( ) 0 0 ( )( ) 0 or 0 or If, then the denominator has a value of 0. If, the original equation is satisfied. The only solution to the equation is. CAUTION Always be sure to check your answers in the original equation to determine whether they are etraneous solutions.
4 . Solving Equations with Rational Epressions (-) WARM-UPS True or false? Eplain your answers.. The LCD is not used in solving equations with rational epressions. False. To solve the equation 8, we divide each side by. False. An etraneous solution is an irrational number. False Use the following equations for Questions 0. a) 5 b) c). To solve Eq. (a), we must add the epressions on the left-hand side. False 5. Both 0 and satisfy Eq. (a). False. To solve Eq. (a), we multiply each side by. True 7. The only solution to Eq. (b) is. True 8. Equation (b) is equivalent to. True 9. To solve Eq. (c), we multiply each side by. True 0. The numbers and do not satisfy Eq. (c). True. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the typical first step for solving an equation involving rational epressions? The first step is usually to multiply each side by the LCD.. What is the difference in procedure for solving an equation involving rational epressions and adding rational epressions? In adding rational epressions we build up each epression to get the LCD as the common denominator.. What is an etraneous solution? An etraneous solution is a number that appears when we solve an equation, but it does not check in the original equation.. Why do etraneous solutions sometimes occur for equations with rational epressions? Etraneous solutions occur because we multiply by an epression involving a variable and that variable epression might have a value of zero. Solve each equation. See Eample y 7. 5 y 0 8. z 5 z 9. t t v v w w. q 5 q q 0 Solve each equation. See Eample a a a a 8. b 5 b 5b 0 9. k k k k k 0. p p p 5 p 5 Solve each equation. See Eample. 5. 5,.,.,. w w 0 5, 5. a a a a,. b 7 b b b b, 5 Solve each equation. Watch for etraneous solutions. See Eamples and 5. 7.
5 (-) Chapter Rational Epressions y y y 8 y 8 y. y y 5 y 7. y y z z 5. z z z z z 7. z z 5 z z 0 In Eercises 5 5, solve each equation. 5. a 5 0 y w w 0 8. m m , m m m 55. t t t t t t 5. w w w w w Solve each problem. 57. Lens equation. The focal length f for a camera lens is related to the object distance o and the image distance i by the formula f o i. See the accompanying figure. The image is in focus at distance i from the lens. For an object that is 00 mm from a 50-mm lens, use f 50 mm and o 00 mm to find i. 5 mm 0.,., 5 5. a a a,. w. w w a a a 8 8. a a a c c c o i FIGURE FOR EXERCISE Telephoto lens. Use the formula from Eercise 57 to find the image distance i for an object that is,000,000 mm from a 50-mm telephoto lens mm FIGURE FOR EXERCISE 58
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