Equations Involving Fractions


 Homer Reynolds
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1 . Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation will be equivalent unless a solution results that makes a denominator in the original equation 0. More about this later! In Chapter, you learned how to solve a variety of equations. We now want to etend that work to the solution of fractional equations, which are equations that involve algebraic fractions as one or more of their terms. To solve a fractional equation, we multiply each term of the equation by the LCD of any fractions. The resulting equation should be equivalent to the original equation and be cleared of all fractions. Eample Solving Fractional Equations Solve NOTE This equation has three terms: The,, and. 6 sign of the term is not used to find the LCD. 6 The LCD for is 6. Multiply both sides of the equation by 6. Using the,, and 6 distributive property, we multiply each term by 6. () NOTE By the multiplication property of equality, this equation is equivalent to the original equation, labeled () Solving as before, we have or To check, substitute for in the original equation. 6 or () 6 6 (True) 00 McGrawHill Companies CAUTION Be Careful! Many students have difficulty because they don t distinguish between adding or subtracting epressions (as we did in Sections. and.) and solving equations (illustrated in the above eample). In the epression we want to add the two fractions to form a single fraction. In the equation we want to solve for.
2 6 CHAPTER ALGEBRAIC FRACTIONS CHECK YOURSELF Solve and check. 6 Recall that, for any fraction, the denominator must not be equal to zero. When a fraction has a variable in the denominator, we must eclude any value for the variable that would result in division by zero. Eample Finding Ecluded Values for In the following algebraic fractions, what values for must be ecluded? (a) (b) Here can have any value, so none need to be ecluded... If 0, then is undefined; 0 is the ecluded value. (c) which is undefined, so is the ecluded. If, then 0, value. CHECK YOURSELF What values for, if any, must be ecluded? (a) (b) (c) If the denominator of an algebraic fraction contains a product of two or more variable factors, the zeroproduct principle must be used to determine the ecluded values for the variable. In some cases, you will have to factor the denominator to see the restrictions on the values for the variable. Eample Finding Ecluded Values for What values for must be ecluded in each fraction? (a) 6 6 Factoring the denominator, we have 6 6 ( 8)( ) Letting 8 0or 0, we see that 8 and make the denominator 0 so both 8 and must be ecluded. 00 McGrawHill Companies
3 EQUATIONS INVOLVING FRACTIONS SECTION. (b) 8 The denominator is zero when 8 0 Factoring, we find ( 6)( 8) 0 The denominator is zero when 6 or 8 CHECK YOURSELF What values for must be ecluded in the following fractions? (a) 0 (b) The steps for solving an equation involving fractions are summarized in the following rule. Step by Step: To Solve a Fractional Equation NOTE The equation that is formed in step can be solved by the methods of Sections. and.. Step Step Step Remove the fractions in the equation by multiplying each term by the LCD of all the fractions. Solve the equation resulting from step as before. Check your solution in the original equation. We can also solve fractional equations with variables in the denominator by using the above algorithm. Eample illustrates this approach. Eample Solving Fractional Equations Solve NOTE The factor appears twice in the LCD. The LCD of the three terms in the equation is, and so we multiply both sides of the equation by. 00 McGrawHill Companies Simplifying, we have We ll leave the check to you. Be sure to return to the original equation.
4 8 CHAPTER ALGEBRAIC FRACTIONS CHECK YOURSELF Solve and check. The process of solving fractional equations is eactly the same when binomials are involved in the denominators. Eample Solving Fractional Equations (a) Solve NOTE There are three terms. NOTE Each of the terms is multiplied by. CAUTION Be careful of the signs! The LCD is, and so we multiply each side (every term) by. ( ) ( ) ( ) Simplifying, we have ( ) 6 To check, substitute for in the original equation. (b) Solve NOTE Recall that 9 ( )( ) 9 In factored form, the three denominators are,, and ( )( ). This means that the LCD is ( )( ), and so we multiply: ( )( ) ( Simplifying, we have ( ) ( ) )( ) ( )( ) 9 00 McGrawHill Companies
5 EQUATIONS INVOLVING FRACTIONS SECTION. 9 CHECK YOURSELF Solve and check. (a) (b) You should be aware that some fractional equations have no solutions. Eample 6 shows that possibility. Eample 6 Solving Fractional Equations Solve The LCD is, and so we multiply each side (every term) by. ( ) ( ) ( ) Simplifying, we have 6 Now, when we try to check our result, we have NOTE is substituted for in the original equation. or 0 0 These terms are undefined. What went wrong? Remember that two of the terms in our original equation were The variable cannot have the value because is an ecluded value and. (it makes the denominator 0). So our original equation has no solution. 00 McGrawHill Companies CHECK YOURSELF 6 Solve, if possible. 6 Equations involving fractions may also lead to quadratic equations, as Eample illustrates.
6 0 CHAPTER ALGEBRAIC FRACTIONS Solving Fractional Equations Solve Eample The LCD is ( )( ). Multiply each side (every term) by ( )( ). ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) Simplifying, we have ( ) ( ) Multiply to clear of parentheses: 60 NOTE Notice that this equation is quadratic. It can be solved by the methods of Section.. In standard form, the equation is or ( 6)( 0) 0 Setting the factors to 0, we have 6 0 or So 6 and 0 are possible solutions. We will leave the check of each solution to you. CHECK YOURSELF Solve and check. 6 6 The following equation is a special kind of equation involving fractions: t 80 t a An equation of the form is said to be in proportion form, or more simply it is b c d called a proportion. This type of equation occurs often enough in algebra that it is worth developing some special methods for its solution. First, we will need some definitions. A ratio is a means of comparing two quantities. A ratio can be written as a fraction. For instance, the ratio of to can be written as. A statement that two ratios are equal is called a proportion. A proportion has the form a b c d 00 McGrawHill Companies
7 EQUATIONS INVOLVING FRACTIONS SECTION. NOTE bd is the LCD of the denominators. In the proportion above, a and d are called the etremes of the proportion, and b and c are called the means. A useful property of proportions is easily developed. If a b c d and we multiply both sides by b d, then a b bd d c bd or ad bc Rules and Properties: Proportions a If b c then ad bc d In words: In any proportion, the product of the etremes (ad) is equal to the product of the means (bc). Because a proportion is a special kind of fractional equation, this rule gives us an alternative approach to solving equations that are in the proportion form. Eample 8 Solving a Proportion 00 McGrawHill Companies NOTE The etremes are and. The means are and. Solve the equations for. (a) Set the product of the etremes equal to the product of the means. 60 Our solution is. You can check as before, by substituting in the original proportion. (b) 0 Set the product of the etremes equal to the product of the means. Be certain to use parentheses with a numerator with more than one term. ( ) 0 0 We will leave the checking of this result to the reader.
8 CHAPTER ALGEBRAIC FRACTIONS CHECK YOURSELF 8 Solve for. (a) (b) 8 9 CHECK YOURSELF ANSWERS.. (a) none; (b) 0; (c). (a), ; (b),.. (a) 8; (b) 6. No solution. or 8 8. (a) 6; (b) As the eamples of this section illustrated, whenever an equation involves algebraic fractions, the first step of the solution is to clear the equation of fractions by multiplication. The following algorithm summarizes our work in solving equations that involve algebraic fractions. Step by Step: To Solve an Equation Involving Fractions Step Step Step Remove the fractions appearing in the equation by multiplying each side (every term) by the LCD of all the fractions. Solve the equation resulting from step. If the equation is linear, use the methods of Section. for the solution. If the equation is quadratic, use the methods of Section.. Check all solutions by substitution in the original equation. Be sure to discard any etraneous solutions, that is, solutions that would result in a zero denominator in the original equation. 00 McGrawHill Companies
9 Name. Eercises Section Date What values for, if any, must be ecluded in each of the following algebraic fractions? ANSWERS ( )( ).. ( )( ) ( )( ) ( )( ) McGrawHill Companies Solve each of the following equations for
10 ANSWERS McGrawHill Companies
11 ANSWERS Solve each of the following equations for McGrawHill Companies
12 ANSWERS a b. c. d. e. f. Getting Ready for Section.6 [Section.] Write each of the following phrases using symbols. Use the variable to represent the number in each case. (a) Onefourth of a number added to fourfifths of the same number (b) 6 times a number, decreased by (c) The quotient when more than a number is divided by 6 (d) Three times the length of a side of a rectangle decreased by (e) A distance traveled divided by (f) The speed of a truck that is mi/h slower than a car Answers. None None.,.,.,., 9., No solution No solution , 6, 6. 8, a. b. 6 c. d. 6 e. f. 00 McGrawHill Companies 6
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