Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category of algebraic expressions, rational expressions, also referred to as algebraic fractions. Similarly as in arithmetic, where a rational number is a quotient of two integers with a denominator that is different than zero, a rational expression is a quotient of two polynomials, also with a denominator that is different than zero. We start with introducing the related topic of integral exponents, including scientific notation. Then, we discuss operations on algebraic fractions, solving rational equations, and properties and graphs of rational functions with an emphasis on such features as domain, range, and asymptotes. In the end of this chapter, we show examples of applied problems, including work problems, that require solving rational equations. RT. Integral Exponents and Scientific Notation Integral Exponents In section P., we discussed the following power rules, using whole numbers for the exponents. product rule aa mm aa nn aa mm+nn (aaaa) nn aa nn bb nn quotient rule power rule aa mm aamm nn aann (aa mm ) nn aa mmmm aa bb nn aann bb nn aa 00 for aa 00 00 00 is undefined Observe that these rules gives us the following result. aann aann aann+ aa nn aa aa quotient rule product rule aa aa nn (nn+) Consequantly, aa nn (aa nn ) aa nn. power rule Since aa nn aa nn, then the expression aann is meaningful for any integral exponent nn and a nonzero real base aa. So, the above rules of exponents can be extended to include integral exponents. In practice, to work out the negative sign of an exponent, take the reciprocal of the base, or equivalently, change the level of the power. For example, 3 3 3 9 and 3 3 3 3 3 8.
Attention! Exponent refers to the immediate number, letter, or expression in a bracket. For example,, ( ) ( ), bbbbbb. Evaluating Expressions with Integral Exponents Evaluate each expression. a. 3 + b. 5 5 c. d. 7 3 3 a. 3 + 3 + 6 + 3 6 55 66 Caution! 3 + (3 + ), because the value of 3 + is 5, as shown in the 6 example, while the value of (3 + ) is. 5 b. 5 5 3333 5 5 Note: To work out the negative exponent, move the power from the numerator to the denominator or vice versa. c. 7 7 99 Attention! The role of a negative sign in front of a base number or in front of an exponent is different. To work out the negative in 7, we either take the reciprocal of the base, or we change the position of the power to a different level in the fraction. So, 7 7 or 7 7. However, the negative sign in just means that the number is negative. So, 4. Caution! 4 d. Note: 3 3 3 3 3 33 Exponential expressions can be simplified in many ways. For example, to simplify 3, we can work out the negative exponents first by moving the powers to a different level,, and then reduce the common factors as shown in the example; or we can employ the quotient rule of powers to obtain 3 ( 3) +3.
3 Simplifying Exponential Expressions Involving Negative Exponents Simplify the given expression. Leave the answer with only positive exponents. a. 4 5 b. ( + yy) c. + yy d. ( 3 ) e. 4 yy f. nn 3 yy 5 4mm5 4mmnn 6 a. 4 5 4 5 b. ( + yy) +yy c. + yy + yy exponent 5 refers to only! these expressions are NOT equivalent! d. ( 3 ) 3 3 work out the negative exponents inside the bracket work out the negative exponents outside the bracket 4 ( ) ( 3 ) 6 a sign can be treated as a factor of power rule multiply exponents e. 4 yy yy yy 5 yy7 yy 5 4 6 product rule add exponents f. 4mm5 nn 3 mm 4mmnn 4 nn 3 nn 6 6 ( )mm4 nn 9 6 6 6 4 6 ( )mm 4 nn 9 36 mm 8 nn 8 ( ) Scientific Notation Integral exponents allow us to record numbers with a very large or very small absolute value in a shorter, more convenient form. For example, the average distance from the Sun to the Saturn is,430,000,000 km, which can be recorded as.43 0,000,000 or more concisely as.43 0 9. Similarly, the mass of an electron is 0.0000000000000000000000000009 grams, which can be recorded as 9 0.000000000000000000000000000, or more concisely as 9 0 8. This more concise representation of numbers is called scientific notation and it is frequently used in sciences and engineering. Definition. A real number is written in scientific notation iff aa nn, where the coefficient aa is such that aa [, ), and the exponent nn is an integer.
4 Converting Numbers to Scientific Notation Convert each number to scientific notation. a. 50,000 b. 0.0000 c..5 0 3 an integer has its decimal dot after the last digit a. To represent 50,000 in scientific notation, we place a decimal point after the first nonzero digit, 5. 0 0 0 0 and then count the number of decimal places needed for the decimal point to move to its original position, which by default was after the last digit. In our example the number of places we need to move the decimal place is 5. This means that 5. needs to be multiplied by 0 5 in order to represent the value of 50,000. So, 555555, 000000 55. 55. Note: To comply with the scientific notation format, we always place the decimal point after the first nonzero digit of the given number. This will guarantee that the coefficient aa satisfies the condition aa <. b. As in the previous example, to represent 0.0000 in scientific notation, we place a decimal point after the first nonzero digit, 0. 0 0 0. 0 and then count the number of decimal places needed for the decimal point to move to its original position. In this example, we move the decimal 4 places to the left. So the number.0 needs to be divided by 0 4, or equivalently, multiplied by 0 4 in order to represent the value of 0.0000. So, 00. 000000000000. 0000 44. Observation: Notice that moving the decimal to the right corresponds to using a positive exponent, as in Example 3a, while moving the decimal to the left corresponds to using a negative exponent, as in Example 3b. c. Notice that.5 0 3 is not in scientific notation as the coefficient.5 is not smaller than 0. To convert.5 0 3 to scientific notation, first, convert.5 to scientific notation and then multiply the powers of 0. So,.5 0 3.5 0 0 3. 44 multiply powers by adding exponents
5 Converting from Scientific to Decimal Notation Convert each number to decimal notation. a. 6.57 0 6 b. 4.6 0 7 a. The exponent 6 indicates that the decimal point needs to be moved 6 places to the right. So, 6.57 0 6 6. 5 7. 66, 555555, 000000 fill the empty places by zeros b. The exponent 7 indicates that the decimal point needs to be moved 7 places to the left. So, 4.6 0 7 0. 4. 6 00. 0000000000000000 fill the empty places by zeros Using Scientific Notation in Computations Evaluate. Leave the answer in scientific notation. a. 6.5 0 7 3 0 5 b. 3.6 0 3 9 0 4 a. Since the product of the coefficients 6.5 3 9.5 is larger than 0, we convert it to scientific notation and then multiply the remaining powers of 0. So, 6.5 0 7 3 0 5 9.5 0 7 0 5.95 0 0. 9999 b. Similarly as in the previous example, since the quotient 3.6 0.4 is smaller than, 9 we convert it to scientific notation and then work out the remaining powers of 0. So, 3.6 0 3 9 0 4 0.4 0 4 0 0 44 divide powers by subtracting exponents Using Scientific Notation to Solve Problems The average distance from Earth to the sun is.5 0 8 km. How long would it take a rocket, traveling at 4.7 0 3 km/h, to reach the sun?
6 To find time TT needed for the rocket traveling at the rate RR 4.7 0 3 km/h to reach the sun that is at the distance DD.5 0 8 km from Earth, first, we solve the motion formula RR TT DD for TT. Since TT DD, we calculate, RR TT.5 08 4.7 0 3 0.3 05 3. 0 4 So, it will take approximately 33. 44 hours for the rocket to reach the sun. RT. Exercises Vocabulary Check Complete each blank with the most appropriate term or phrase from the given list: numerator, opposite, reciprocal, scientific.. To raise a base to a negative exponent, raise the of the base to the opposite exponent.. A power in a numerator of a fraction can be written equivalently as a power with the exponent in the denominator of this fraction. Similarly, a power in a denominator of a fraction can be written equivalently as a power with the opposite exponent in the of this fraction. 3. A number with a very large or very small absolute value is often written in the notation. Concept Check True or false. 4. 3 4 4 3 5. 0 4 0.0000 6. (0.5) 4 7. 4 5 4 5 8. ( ) 0 4 5 9. 8 0. 3 3. 4. 5 0 5 5 3. The number 0.68 0 5 is written in scientific notation. 4. 98.6 0 7 9.86 0 6 Concept Check 5. Match the expression in Row I with its equivalent expression in Row II. Choices may be used once, more than once, or not at all. a. 5 b. 5 c. ( 5) d. ( 5) e. 5 5 A. 5 B. 5 C. 5 D. 5 E. 5 Concept Check Evaluate each expression. 6. 4 6 4 3 7. 9 3 9 5 8. 3 7 9. 6 5
7 0. 3 4 5 3. 3. + 3 3. ( 3 ) Concept Check Simplify each expression, if possible. Leave the answer with only positive exponents. Assume that all variables represent nonzero real numbers. Keep large numerical coefficients as powers of prime numbers, if possible. 4. ( 3 )(7 8 ) 5. (5 yy 3 )( 4 7 yy ) 6. (9 4nn )( 4 8nn ) 7. ( 3yy 4aa )( 5yy 3aa ) 8. 4 3 9. 4nn 6nn 30. 3nn 5 nnnn 3. 4aa 4 bb 3 8aa 8 bb 3. 8 3 yy 3 5 5 yy 5 33. ( pp 7 qq) 4 34. ( 3aa bb 5 ) 3 35. 5 yy 3 3 36. 3 yy 3yy 3 3 37. 4 3 5 yy 4 4 38. 5 yy 3 5 4 yy 5 39. 003 yy 5 4 8 5 yy 7 40. [( 4 yy ) 3 ] 4. aa aa 3 6aa 7 4. ( kk) mm 5 (kkkk) 3 43. pp qq 3 3pp4 qq 4 44. 34 yy 6 5 6 yy 7 3 45. 4aa3 bb aa 6 bb 5 3 46. 9 4 yy 3 3 3 yy 8 47. (4 ) yy 48. (5 aa ) aa 49. aa aa 50. 9nn 3nn 5. aa+ 4 aa 5. bb 3 bb 4 53. 5aa+bb yy bb aa 5 aa bb yy bb aa Convert each number to scientific notation. 54. 6,000,000,000 55. 0.0003 56. 0.000000005 57. 705.6 Convert each number to decimal notation. 58. 6.7 0 8 59. 5.07 0 5 60. 0 6. 9.05 0 9 6. One gigabyte of computer memory equals 30 bytes. Write the number of bytes in gigabyte, using decimal notation. Then, using scientific notation, approximate this number by rounding the scientific notation coefficient to two decimals places. Evaluate. State your answer in scientific notation. 63. (6.5 0 3 )(5. 0 8 ) 64. (.34 0 5 )(5.7 0 6 ) 65. (3.6 0 6 )(5. 0 8 ) 66. 4 0 7 8 0 3
8 67. 7.5 09 4 0 7.5 04 68. 8 0 3 69. 0.05 6000 0.0004 70. 0.003 40,000 0.000 600 Analytic Skills Solve each problem. 7. A light-year is the distance that light travels in one year. Find the number of kilometers in a light-year if light travels approximately 3 0 5 kilometers per second. 7. In 07, the national debt in Canada was about 6.4 0 dollars. If Canadian population in 07 was approximately 3.56 0 7, what was the share of this debt per person? 73. The brightest star in the night sky, Sirius, is about 4.704 0 3 miles from Earth. If one light-year is approximately 5.88 0 miles, how many light-years is it from Earth to Sirius? 74. The average discharge at the mouth of the Amazon River is 4,00,000 cubic feet per second. How much water is discharged from the Amazon River in one hour? in one year? 75. If current trends continue, world population PP in billions may be modeled by the equation PP 6(.04), where is in years and 0 corresponds to the year 000. Estimate the world population in 00 and 05. 76. The mass of the sun is.989 0 30 kg and the mass of the earth is 5.976 0 4 kg. How many times larger in mass is the sun than the earth? Discussion Point 77. Many calculators can only handle scientific notation for powers of 0 between 99 and 99. How can we compute (4 0 0 )?
9 RT. Rational Expressions and Functions; Multiplication and Division of Rational Expressions Rational Expressions and Functions In arithmetic, a rational number is a quotient of two integers with denominator different than zero. In algebra, a rational expression, offten called an algebraic fraction, is a quotient of two polynomials, also with denominator different than zero. In this section, we will examine rational expressions and functions, paying attention to their domains. Then, we will simplify, multiply, and divide rational expressions, employing the factoring skills developed in Chapter P. Here are some examples of rational expressions:,, 4, 8 +6 5 4 +5, 3 3, 5, 3( ) Definition. A rational expression (algebraic fraction) is a quotient PP() of two polynomials PP() and QQ() QQ(), where QQ() 0. Since division by zero is not permitted, a rational expression is defined only for the -values that make the denominator of the expression different than zero. The set of such -values is referred to as the domain of the expression. Note : Negative exponents indicate hidden fractions and therefore represent rational expressions. For instance,. Note : A single polynomial can also be seen as a rational expression because it can be considered as a fraction with a denominator of. For instance, 5 5. Definition. A rational function is a function defined by a rational expression, ff() PP() QQ(). The domain of such function consists of all real numbers except for the -values that make the denominator QQ() equal to 0. So, the domain DD R { QQ() 00} ff() Figure 3 For example, the domain of the rational function ff() 3 is the set of all real numbers except for 3 because 3 would make the denominator equal to 0. So, we write DD R {3}. Sometimes, to make it clear that we refer to function ff, we might denote the domain of ff by DD ff, rather than just DD. Figure shows a graph of the function ff(). Notice that the graph does not cross 3 the dashed vertical line whose equation is 3. This is because ff(3) is not defined. A closer look at the graphs of rational functions will be given in Section RT.5.
0 Evaluating Rational Expressions or Functions Evaluate the given expression or function for, 0,. If the value cannot be calculated, write undefined. a. 3( ) b. ff() + a. If, then 3( ) 3( )( ) 3( ) 3 ( ) 33 44. If 0, then 3( ) 3(0)(0 ) 00. If, then 3( ) 3()( ) 3 0 uuuuuuuuuuuuuuuuuu, as division by zero is not permitted. Note: Since the expression 3( ) cannot be evaluated at, the number does not belong to its domain. b. ff( ) ff(0) ff() uuuuuuuuuuuuuuuuuu. ( ) +( ) 0 (0) +(0) 0 0 uuuuuuuuuuuuuuuuuu. () +(). Observation: Function ff() is undefined at 0 and. This is because + the denominator + ( + ) becomes zero when the -value is 0 or. Finding Domains of Rational Expressions or Functions Find the domain of each expression or function. a. 4 +5 b. c. ff() 4 +4 d. gg() 4 5 4 a. The domain of consists of all real numbers except for those that would make +5 the denominator + 5 equal to zero. To find these numbers, we solve the equation + 5 0 5 55
So, the domain of 4 +5 is the set of all real numbers except for 5. This can be recorded in set notation as R 55, or in set-builder notation as 55, or in interval notation as, 55 55,. b. To find the domain of, we want to exclude from the set of real numbers all the -values that would make the denominator equal to zero. After solving the equation 0 via factoring ( ) 0 and zero-product property 00 or, we conclude that the domain is the set of all real numbers except for 0 and, which can be recorded as R {00, }. This is because the -values of 0 or make the denominator of the expression equal to zero. c. To find the domain of the function ff() 4, we first look for all the -values that +4 make the denominator + 4 equal to zero. However, + 4, as a sum of squares, is never equal to 0. So, the domain of function ff is the set of all real numbers R. d. To find the domain of the function gg(), we first solve the equation 4 5 4 5 0 to find which -values make the denominator equal to zero. After factoring, we obtain ( 5)( + ) 0 which results in 5 and Thus, the domain of gg equals to DD gg R {, 55}. Equivalent Expressions Definition.3 Two expressions are equivalent in the common domain iff (if and only if) they produce the same values for every input from the domain. Consider the expression simplified to ( ) from Example b. Notice that this expression can be by reducing common factors in the numerator and the denominator. However, the domain of the simplified fraction,, is the set R {0}, which is different than the domain of the original fraction, R {0,}. Notice that for, the expression is undefined while the value of the expression is. So, the two expressions are not equivalent in the set of real numbers. However, if the domain of is
ff() gg() resticted to the set R {0,}, then the two expressions produce the same values and as such, they are equivalent. We say that the two expressions are equivalent in the common domain. The above situation can be illustrated by graphing the related functions, ff() and gg(), as in Figure. The graphs of both functions are exactly the same except for the hole in the graph of ff at the point,. Figure So, from now on, when writing statements like, we keep in mind that they apply only to real numbers which make both denominators different than zero. Thus, by saying in short that two expressions are equivalent, we really mean that they are equivalent in the common domain. Note: The domain of ff() ( ) term was simplified. is still R {00, }, even though the ( ) The process of simplifying expressions involve creating equivalent expressions. In the case of rational expressions, equivalent expressions can be obtained by multiplying or dividing the numerator and denominator of the expression by the same nonzero polynomial. For example, 33 5555 ( 3) ( ) + 33 ( 5) ( ) 5555 33 ( 3) 33 ( 3) To simplify a rational expression: Factor the numerator and denominator completely. Eliminate all common factors by following the property of multiplicative identity. Do not eliminate common terms - they must be factors! Simplifying Rational Expressions Simplify each expression. a. 7aa bb aa 3 bb 4aa 3 bb b. 9 6+9 c. 0 5 5 3 5 0 a. First, we factor the denominator and then reduce the common factors. So, 7aa bb aa 3 bb 4aa 3 bb 7aa bb 7aa 3 bb(3aa aaaa) bb aa(33 )
3 b. As before, we factor and then reduce. So, 9 6 + 9 ( 3)( + 3) ( 3) + 33 33 Neither nor 3 can be reduced, as they are NOT factors! c. Factoring and reducing the numerator and denominator gives us 0 5 5 3 5 0 5(4 3) 5(3 4) 4 3 (3 4)( + ) Since 4 3 (3 4), the above expression can be reduced further to 3 4 3 4 4 3 (3 4)( + ) + Notice: An opposite expression in the numerator and denominator can be reduced to. For example, since aa bb is opposite to bb aa, then aa bb bb aa, as long as aa bb. Caution: Note that aa bb is NOT opposite to aa + bb! Multiplication and Division of Rational Expressions Recall that to multiply common fractions, we multiply their numerators and denominators, and then simplify the resulting fraction. Multiplication of algebraic fractions is performed in a similar way. To multiply rational expressions: factor each numerator and denominator completely, reduce all common factors in any of the numerators and denominators, multiply the remaining expressions by writing the product of their numerators over the product of their denominators. For instance, 3( + 5) 3 3 + 5 + 5 6 3 ( + 5) 6 3 Multiplying Algebraic Fractions Multiply and simplify. Assume non-zero denominators. a. yy 3 3yy 3 yy () b. 3 yy 3 3+3yy 3 +yy yy
4 Recall: 33 yy 33 ( yy) + + yy yy ( + yy)( yy) a. To multiply the two algebraic fractions, we use appropriate rules of powers to simplify each fraction, and then reduce all the remaining common factors. So, 3 yy 3 3yy (3 yy) () 3 3 46 yy 3 3 yy3 3 yy b. After factoring and simplifying, we have 3 yy 3 3 + 3yy + yy yy ( yy)( + + yy ) + yy 4444 33 44 33 44 equivalent answers 3( + yy) ( yy)( + yy) 33 + + yy + yy To divide rational expressions, multiply the first, the dividend, by the reciprocal of the second, the divisor. For instance, 5 0 3 6 5 0 3 3 3 6 5( ) 3 3( ) 0 9 multiply by the reciprocal follow multiplication rules Dividing Algebraic Fractions Perform operations and simplify. Assume non-zero denominators. a. + ( + ) b. 5 0+5 + +5+4 +8 4 a. To divide by ( + ) we multiply by the reciprocal (+). So, + ( + ) ( + ) ( + ) b. The order of operations indicates to perform the division first. To do this, we convert the division into multiplication by the reciprocal of the middle expression. Therefore, 5 + 5 + 4 0 + 5 + 8 + 4 Reduction of common factors can be done gradually, especially if there is many common factors to divide out. ( 5)( + 5) ( + 4)( + ) + 8 ( + ) 0 + 5 4 ( 5)( + 5) ( + 4) ( + 4) ( 5) 4 ( + 55) ( 55)
5 RT. Exercises Vocabulary Check Complete each blank with the most appropriate term or phrase from the given list: rational, zero, common, factor, reciprocal.. If PP() and QQ() are polynomials, then ff() PP() is a function provided that QQ() is not the QQ() zero polynomial.. The domain of a rational function consists of all real numbers except for the -values that will make the denominator equal to. 3. Two rational expressions are equivalent if they produce the same values for any inputs from their domain. 4. To simplify, multiply, or divide rational expressions, we first each numerator and denominator. 5. To divide by a rational expression, we multiply by its. Concept Check True or false. 6. ff() 4 4 is a rational function. 7. The domain of ff() is the set of all real numbers. 4 8. 3 4 is equivalent to 3 4. 9. nn + nn is equivalent to nn+ nn. Concept Check Given the rational function f, find ff( ), ff(0), and ff(). 0. ff(). ff() 5 3. ff() + 6 Concept Check For each rational function, find all numbers that are not in the domain. Then give the domain, using both set notation and interval notation. 3. ff() + 6. ff() 3+ 5 4 4. gg() 6 7. gg() + 4 5. h() 3+7 8. h() +4 9. ff() 5 3 0. gg() + 6 ++35. h() 7 4 3 Discussion Point. Is there a rational function ff such that is not in its domain and ff(0) 5? If yes, how many such functions are there? If not, explain why not.
6 Concept Check 3. Which rational expressions are equivalent and what is the simplest expression that they are equivalent to? a. +3 3 b. 3 3 c. +3 3+ d. +3 3 e. 3 3 Concept Check 4. Which rational expressions can be simplified? a. + b. + c. d. yy yy e. Simplify each expression, if possible. 5. 4aa3 bb 3aabb 3 6. 8 yy 3 8 3 yy 7. 7 7 8. + 9. aa 5 5+aa 30. (3 yy)(+) (yy 3)( ) 3. 5 3. 8aa 33. 4yy 4yy+ 34. 7+4 7 4 35. 6mm+8 7mm+ 36. 3zz +zz 8zz+6 37. mm 5 0 4mm 38. 9nn 3 4 nn 39. tt 5 tt 0tt+5 40. pp 36 pp +tt+36 4. 9+8 +3 4 4. pp +8pp 9 pp 5pp+4 43. 3 yy 3 44. bb aa yy aa 3 bb 3 Perform operations and simplify. Assume non-zero denominators. 45. 8aa4 5bb 5bb4 8 9aa3 46. 633 47. (7) yy 49(yy ) 3 8 48. + 3 3 49. 0aa 0aa 40 6aa 30aa 50. aa 3 4aa aa 5. yy 5 4yy 5 yy 54. 3nn 9 nn 9 (nn3 + 7) 5. (8 6) 3 6 0 55. 6 4 53. (yy 4) yy 8yy 56. yy +0yy+5 yy 9 yy 3yy yy+5 57. bb 3 bb bb bb 4bb+3 bb 58. 6+9 +3 9 59. 3 5 9 4 9 +4 60. tt 49 tt +8tt+5 tt +4tt tt tt 35 6. aa3 bb 3 aa bb aa bb aa+bb 6. 64 3 + 4+0 4 00 64 6+4
7 63. 3 yy 64yy yy 6yy 3 yy+64yy yy 4yy +6yy 64. pp 3 7qq 3 pp pppp 4qq pp +pppp qq pp 5pppp 6qq 65. 67. 4 9yy 4yy +6+9yy 8 3 7yy 3 4yy ++9yy 6 3+6 4 5+6 4 49 +7 6 66. + 6 + +5+3 6 +3+6 68. 4yy yy+36 7 3yy (yy 3 + 7) 69. 3yy yy yy 5 70. + yy + yy yy 7. aa 4bb aa+bb 73. 5 4 (aa + bb) bb aa bb 5 0+4 +3 +0+5 7. 74. 9 4yy 6yy +4 3 yy 3 yy 6 yy +3yy 8 yy 8yy+6 yy+4 yy +yy+30 Given ff() and gg(), find ff() gg() and ff() gg(). 75. ff() 4 + and gg() + 76. ff() 3 3 +5 and gg() 4 3 77. ff() 7+ +3 and gg() 9 4 78. ff() +6 4 and gg() +8+
8 RT.3 Addition and Subtraction of Rational Expressions Many real-world applications involve adding or subtracting algebraic fractions. Similarly as in the case of common fractions, to add or subtract algebraic fractions, we first need to change them equivalently to fractions with the same denominator. Thus, we begin by discussing the techniques of finding the least common denominator. Least Common Denominator The least common denominator (LCD) for fractions with given denominators is the same as the least common multiple (LCM) of these denominators. The methods of finding the LCD for fractions with numerical denominators were reviewed in section R3. For example, LLLLLL(4,6,8) 4, because 4 is a multiple of 4, 6, and 8, and there is no smaller natural number that would be divisible by all three numbers, 4, 6, and 8. Suppose the denominators of three algebraic fractions are 4( yy ), 6( + yy), and 8. The numerical factor of the least common multiple is 4. The variable part of the LCM is built by taking the product of all the different variable factors from each expression, with each factor raised to the greatest exponent that occurs in any of the expressions. In our example, since 4( yy ) 4( + yy)( yy), then LLLLLL( 4( + yy)( yy), 6( + yy), 8 ) ( + yy) ( yy) Notice that we do not worry about the negative sign of the middle expression. This is because a negative sign can always be written in front of a fraction or in the numerator rather than in the denominator. For example, 6( + yy) 6( + yy) 6( + yy) In summary, to find the LCD for algebraic fractions, follow the steps: Factor each denominator completely. Build the LCD for the denominators by including the following as factors: o LCD of all numerical coefficients, o all of the different factors from each denominator, with each factor raised to the greatest exponent that occurs in any of the denominators. Note: Disregard any factor of. Determining the LCM for the Given Expressions Find the LCM for the given expressions. a. 3 yy and 5yy ( ) b. 8 and + 3 + c. yy,, and + + yy
9 a. Notice that both expressions, 3 yy and 5yy ( ), are already in factored form. The LLLLLL(,5) 60, as divide by 3 33 4 5 5 6666 no more common factors, so we multiply the numbers in the letter L The highest power of is 3, the highest power of yy is, and ( ) appears in the first power. Therefore, LLLLLL 3 yy, 5yy ( ) 6666 33 yy ( ) b. To find the LCM of 8 and + 3 +, we factor each expression first: 8 ( 4)( + ) + 3 + ( + )( + ) There are three different factors in these expressions, ( 4), ( + ), and ( + ). All of these factors appear in the first power, so LLLLLL( 8, + 3 + ) ( 44)( + )( + ) notice that ( + ) is taken only ones! c. As before, to find the LCM of yy,, and + + yy, we factor each expression first: yy (yy + )(yy ) ( + yy)( yy) ( yy) + + yy ( + yy) as yy ( yy) and yy + + yy Since the factor of can be disregarded when finding the LCM, the opposite factors can be treated as the same by factoring the out of one of the expressions. So, there are four different factors to consider,,, ( + yy), and ( yy). The highest power of ( + yy) is and the other factors appear in the first power. Therefore, LLLLLL( yy,, + + yy ) ( yy)( + yy) Addition and Subtraction of Rational Expressions Observe addition and subtraction of common fractions, as review in section R3. work out the numerator + 3 5 6 3 + 5 3 + 4 5 6 6 6 33 convert fractions to the lowest common denominator simplify, if possible
0 To add or subtract algebraic fractions, follow the steps: Step : Step : Step 3: Step 4: Factor the denominators of all algebraic fractions completely. Find the LCD of all the denominators. Convert each algebraic fraction to the lowest common denominator found in Step and write the sum (or difference) as a single fraction. Simplify the numerator and the whole fraction, if possible. Adding and Subtracting Rational Expressions Perform the operations and simplify if possible. a. aa 3bb 5 aa b. yy + yy yy c. 3 +3 yy 3 yy d. yy+ yy 7yy+6 + yy+ yy 5yy 6 e. + 5 4 + f. ( ) + ( ) Multiplying the numerator and denominator of a fraction by the same factor is equivalent to multiplying the whole fraction by, which does not change the value of the fraction. a. Since LLLLLL(5, aa) 0aa, we would like to rewrite expressions, aa 3bb and, so that they 5 aa have a denominator of 0a. This can be done by multiplying the numerator and denominator of each expression by the factors of 0a that are missing in each denominator. So, we obtain aa 5 3bb aa aa 5 aa aa 3bb aa 5 5 aa b. Notice that the two denominators, yy and yy, are opposite expressions. If we write yy as ( yy), then yy + yy yy yy + yy ( yy) yy yy yy combine the signs yy yy c. To find the LCD, we begin by factoring yy ( yy)( + yy). Since this expression includes the second denominator as a factor, the LCD of the two fractions is ( yy)( + yy). So, we calculate keep the bracket after a sign 3 + 3 yy 3 yy (3 + 3) + ( + 3) ( + yy) ( yy)( + yy) 3 + 3 ( + yy + 3 + 3) ( yy)( + yy) yy ( yy)( + yy) 3 + 3 yy 3 3 ( yy)( + yy) ( + yy) ( yy)( + yy) ( yy)
d. To find the LCD, we first factor each denominator. Since yy 7yy + 6 (yy 6)(yy ) and yy 5yy 6 (yy 6)(yy + ), then LLLLLL (yy 6)(yy )(yy + ) and we calculate multiply by the missing bracket yy + yy 7yy + 6 + yy yy 5yy 6 yy + (yy 6)(yy ) + yy (yy 6)(yy + ) (yy + ) (yy + ) + (yy ) (yy ) (yy 6)(yy )(yy + ) yy + yy + + (yy ) (yy 6)(yy )(yy + ) yy + yy (yy 6)(yy )(yy + ) yy(yy + ) (yy 6)(yy )(yy + ) (yy 66)(yy ) e. As in the previous examples, we first factor the denominators, including factoring out a negative from any opposite expression. So, 4 + 5 + ( )( + ) + 5 ( ) + LLLLLL ( )( + ) 5( + ) ( ) ( )( + ) 5 0 + ( )( + ) 4 8 4( + ) ( )( + ) ( )( + ) 44 ( ) e. Recall that a negative exponent really represents a hidden fraction. So, we may choose to rewrite the negative powers as fractions, and then add them using techniques as shown in previous examples. 3( ) + ( ) ( ) + + ( ) ( ) 3 + ( ) + ( + ) ( ) ( ) nothing to simplify this time Note: Since addition (or subtraction) of rational expressions results in a rational expression, from now on the term rational expression will include sums of rational expressions as well. Adding Rational Expressions in Application Problems An airplane flies mm miles with a windspeed of ww mph. On the return flight, the airplane flies against the same wind. The expression mm + mm, where ss is the speed of the airplane ss+ww ss ww in still air, represents the total time in hours that it takes to make the round-trip flight. Write a single rational expression representing the total time.
To find a single rational expression representing the total time, we perform the addition using (ss + ww)(ss ww) as the lowest common denominator. So, mm ss + ww + mm ss ww mm(ss ww) + mm(ss + ww) (ss + ww)(ss ww) mmss mmmm + mmmm + mmmm (ss + ww)(ss ww) ss ss ww Adding and Subtracting Rational Functions Given ff() +0+4 and gg() +4, find a. (ff + gg)() b. (ff gg)(). a. (ff + gg)() ff() + gg() + 0 + 4 + + 4 ( + 6)( + 4) + + ( + 6) + + ( + 4) ( + 6)( + 4) ( + 6)( + 4) 3 + ( + 6)( + 4) 3( + 4) ( + 6)( + 4) 33 ( + 66) b. ((ff gg)() ff() gg() + 0 + 4 + 4 ( + 6)( + 4) ( + 6) ( + 4) ( + 6)( + 4) ( + 6)( + 4) ( + 66)( + 44) RT.3 Exercises Vocabulary Check Complete each blank with the most appropriate term from the given list: denominator, different, factor, highest, rational.. To add (or subtract) rational expressions with the same denominator, add (or subtract) the numerators and keep the same.. To find the LCD of rational expressions, first each denominator completely. 3. To add (or subtract) rational expressions with denominators, first find the LCD for all the involved expressions.
3 4. The LCD of rational expressions is the product of the different factors in the denominators, where the power of each factor is the number of times that it occurs in any single denominator. 5. A polynomial expression raised to a negative exponent represents a expression. Concept Check 6. a. What is the LCM for 6 and 9? b. What is the LCD for 6 and 9? 7. a. What is the LCM for 5 and + 5? b. What is the LCD for and? 5 +5 Concept Check Find the LCD and then perform the indicated operations. Simplify the resulting fraction. 8. 5 + 3 8 9. 9 30 75 0. 3 4 + 7 30 6. 5 8 7 + 40 Concept Check Find the least common multiple (LCM) for each group of expressions.. 4aa 3 bb 4, 8aa 5 bb 3. 6 yy, 9 3 yy, 5yy 3 4. 4, + 5. 0, 5( ) 6. ( ), 7. yy 5, 5 yy 8. yy, + yy 9. 5aa 5, aa 6aa + 9 0. + +, 4. nn 7nn + 0, nn 8nn + 5. 5 3,, 6 + 9 3., +, 4 4. 5 4 4 + 4 3, 3, + 4 Concept Check True or false? If true, explain why. If false, correct it. 5. + 3 5 6. + 0 7. + 3 3 yy +yy 8. 3 4 + 5 3+ 0 Perform the indicated operations and simplify if possible. 9. yy +yy + 3yy +yy 30. aa+3 aa 5 aa+ aa+ 3. 4aa+3 aa 3 3. nn+ nn + 33. yy + yy yy 35. yy 3 yy 4 yy yy 36. aa 3 aa bb + bb3 bb aa 34. 37. 4aa + 5+3aa aa 49 49 aa +h h 38. + + +3 4 39. + 3+ 3 40. 4 + yy yy +yy 4. +3 3+5 5+5 4. yy yy+6 4yy+8 5yy+0 43. 4 4 + 44. + 5 + yy+3 yy+ yy yy 4 45. yy + yy yy 0 yy+4 46. 5 3 6+8 47. 9+ + 7 3 8 3 + 4 48. 3yy+ + 8 yy yy 0 yy 7yy+5 49. 6 + 5 yy +6yy+9 yy 9 50. 3 +4 + 3 9 5. + + + 5. yy+3 yy yy + yy + yy +yy 3
4 53. 4 + 3 4 54. 5yy yy + 3 yy yy+ 4yy 55. +5 3 + + 6+0 3 Discussion Point 56. Consider the following calculation 4 ( + ) 4 4 + 4 4 4 4 Is this correct? If yes, check if the result can be simplified. If no, correct it. Perform the indicated operations and simplify if possible. 57. 3 + (3) 58. ( 9) + ( 3) 59. + 3 4 60. aa 3 aa 3 aa 9 aa 9 3aa 6. 4+4 + 3 + 3 4 + 6. 3 3 6 Given ff() and gg(), find (ff + gg)() and (ff gg)(). Leave the answer in simplified single fraction form. 63. ff() +, gg() 4 3 65. ff() 3 + 3, gg() + 64. ff() 4, gg() +4+4 66. ff() +, gg() + Solve each problem. 67. Two friends work part-time at a store. The first person works every sixth day and the second person works every tenth day. If they are both working today, how many days pass before they both work on the same day again? 68. Suppose a cylindrical water tank is being drained. The change in the water level can be found using the expression VV VV ππrr ππrr, where VV and VV represent the original and new volume, respectively, and rr is the radius of the tank. Write the change in water level as a single algebraic fraction. 69. To determine the percent growth in sales from the previous year, the owner of a company uses the expression 00 SS, where SS SS represents the current year s sales and SS 0 represents last year s sales. Write this 0 expression as a single algebraic fraction. 70. A boat travels dd mi against the current whose rate is cc mph. On the return trip, the boat travels with the same current. The expression dd + dd, where rr is the speed rr cc rr+cc of the boat in calm water, represents the total amount of time in hours it takes for the entire boating trip. Represent this amount of time as a single algebraic fraction.
5 RT.4 Complex Fractions When working with algebraic expressions, sometimes we come across needing to simplify expressions like these: 9 +, + 3 +, yy + + h +, h aa bb Simplifying Complex Fractions A complex fraction is a quotient of rational expressions (including sums of rational expressions) where at least one of these expressions contains a fraction itself. In this section, we will examine two methods of simplifying such fractions. Definition 4. A complex fraction is a quotient of rational expressions (including their sums) that result in a fraction with more than two levels. For example, has three levels while 3 3 has four 4 levels. Such fractions can be simplified to a single fraction with only two levels. For example, 3 3 6, oooo 3 4 3 3 4 There are two common methods of simplifying complex fractions. Method I (multiplying by the reciprocal of the denominator) Replace the main division in the complex fraction with a multiplication of the numerator fraction by the reciprocal of the denominator fraction. We then simplify the resulting fraction if possible. Both examples given in Definition 4. were simplified using this strategy. Method I is the most convenient to use when both the numerator and the denominator of a complex fraction consist of single fractions. However, if either the numerator or the denominator of a complex fraction contains addition or subtraction of fractions, it is usually easier to use the method shown below. Method II (multiplying by LCD) Multiply the numerator and denominator of a complex fraction by the least common denominator of all the fractions appearing in the numerator or in the denominator of the complex fraction. Then, simplify the resulting fraction if possible. For example, to simplify yy+ + yy, multiply the numerator yy + and the denominator + by the LLLLLL yy,. So, yy yy + + yy yy + yy yy( + ) yy + ( + ) yy
6 Simplifying Complex Fractions Use a method of your choice to simplify each complex fraction. a. 5 +8+ 5 4 b. aa+bb aa 3 + bb 3 c. + 5 3 d. 6 4 5 + 7 4 4 a. Since the expression 5 +8+ 5 4 denominator, we will simplify it using method I, as below. contains a single fraction in both the numerator and 8 5 + 8 + 4 ( 4)( + ) ( 7)( + 3) ( 44)( 77) ( 5)( + 3) ( + 6)( + ) ( 55)( + 66) factor and multiply by the reciprocal b. aa+bb aa 3 + can be simplified in the following two ways: bb 3 Method I Method II aa+bb aa 3 + aa+bb bb 3 bb 3 +aa 3 (aa+bb)aa3bb3 aa 3 +bb 3 aa 3 bb 3 (aa+bb)aa 3 bb 3 (aa+bb)(aa aaaa+bb ) aa33 bb 33 aa aaaa+bb aa+bb aa 3 + aa3bb3 (aa+bb)aa3 bb 3 bb 3 aa 3 bb 3 bb 3 +aa 3 (aa+bb)aa 3 bb 3 (aa+bb)(aa aaaa+bb ) aa33 bb 33 aa aaaa+bb Caution: In Method II, the factor that we multiply the complex fraction by must be equal to. This means that the numerator and denominator of this factor must be exactly the same. c. To simplify + 5, we will use method II. Multiplying the numerator and denominator 3 by the LLLLLL, 5, we obtain 5 3 + 5 3 5 + 33 5 55
d. Again, to simplify 6 4 5 + 7 4 4 7, we will use method II. Notice that the lowest common multiple of the denominators in blue is ( + )( ). So, after multiplying the numerator and denominator of the whole expression by the LCD, we obtain 6 4 5 + 7 4 4 5 + 6 4 ( + )( ) 6 5( ) 6 5 + 0 ( + )( ) 7 4( + ) 7 4 8 5555 4444 + Simplifying Rational Expressions with Negative Exponents Simplify each expression. Leave the answer with only positive exponents. a. yy yy b. aa 3 aa bb a. If we write the expression with no negative exponents, it becomes a complex fraction, which can be simplified as in Example. So, yy yy yy yy yy (yy ) b. As above, first, we rewrite the expression with only positive exponents and then simplify as any other complex fraction. aa 3 aa bb aa 3 aa aa3 bb aa 3 bb bb Remember! This factor must be bb aa bb aa 3 bb aa (bb aa) Simplifying the Difference Quotient for a Rational Function Find and simplify the expression ff(aa+h) ff(aa) h for the function ff() +. Since ff(aa + h) aa+h+ and ff(aa) aa+, then ff(aa + h) ff(aa) h aa + h + aa + h
To simplify this expression, we can multiply the numerator and denominator by the lowest common denominator, which is (aa + h + )(aa + ). Thus, aa + h + h aa + aa + aa h h(aa + h + )(aa + ) keep the denominator in a factored form (aa + h + )(aa + ) aa + (aa + h + ) (aa + h + )(aa + ) h(aa + h + )(aa + ) h h(aa + h + )(aa + ) This bracket is essential! (aa + hh + )(aa + ) 8 RT.4 Exercises Vocabulary Check Complete each blank with the most appropriate term from the given list: complex, LCD, multiply, reciprocal, same, single.. A quotient of two rational expressions that results in a fraction with more than two levels is called a algebraic fraction.. To simplify a complex fraction means to find an equivalent fraction PP, where PP and QQ are QQ polynomials with no essential common factors. 3. To simplify a complex rational expression using the method, first write both the numerator and the denominator as single fractions in simplified form. 4. To simplify a complex fraction using the method, first find the LCD of all rational expressions within the complex fraction. Then, the numerator and denominator by the LCD expression. Concept Check Simplify each complex fraction. 5. 3 3 + 7 3 6. 5 3 4 4 + 7. 3 8 5 3 + 6 8. 3 + 4 5 3 4 Simplify each complex rational expression. 9. 3 yy yy 3 0. nn 5 6nn nn 5 8nn. aa 4 + aa. nn + 3 5 nn 6 3. 9 3 4 + 3 6 4 4. 9 yy 5 yy 6 5. 4 yy 4 + yy 6. 3 aa + 4 bb 4 aa 3 bb
9 7. aa 3aa bb bb bb aa 8. yy yy 9. 4 yy yy yy 0. 5 pp qq 5qq 5 pp. nn nn +nn nn + 4. tt 3tt + tt tt 3. aa h aa h 4. (+h) h 5. 4 + 3 5 5 + 3 6. + 3 + 6 + 7. bb aa bb aa 8. yy + yy 9. +3 4 + 30. 3 +6+9 + 3 +3 6 9 + 6 3 3. aa bb aa 3 + 3. bb 3 4pp pp+9 pp +7pp 5 pp 5pp+8 pp pp 30 Discussion Point 33. Are the expressions +yy +yy and +yy +yy equivalent? Explain why or why not. Simplify each expression. Leave your answer with only positive exponents. 34. aa bb 35. + 3 36. yy 3 3 (nn+) 37. + (nn+) Analytic Skills 38. ff() 5 Find and simplify the difference quotient ff(aa+h) ff(aa) h 39. ff() 40. ff() for the given function. 4. ff() Analytic Skills 4. aa aa aa aa Simplify each continued fraction. 43. 3 3 44. aa + aa + aa
30 RT.5 Rational Equations and Graphs Rational Equations In previous sections of this chapter, we worked with rational expressions. If two rational expressions are equated, a rational equation arises. Such equations often appear when solving application problems that involve rates of work or amounts of time considered in motion problems. In this section, we will discuss how to solve rational equations, with close attention to their domains. We will also take a look at the graphs of reciprocal functions, their properties and transformations. Definition 5. A rational equation is an equation involving only rational expressions and containing at least one fractional expression. Here are some examples of rational equations:, 5 5 5, 3 6 8 3 Attention! A rational equation contains an equals sign, while a rational expression does not. An equation can be solved for a given variable, while an expression can only be simplified or evaluated. For example, is an expression to simplify, while is an equation to solve. When working with algebraic structures, it is essential to identify whether they are equations or expressions before applying appropriate strategies. By Definition 5., rational equations contain one or more denominators. Since division by zero is not allowed, we need to pay special attention to the variable values that would make any of these denominators equal to zero. Such values would have to be excluded from the set of possible solutions. For example, neither 0 nor 3 can be solutions to the equation 3 6 8 3, as it is impossible to evaluate either of its sides for 0 or 3. So, when solving a rational equation, it is important to find its domain first. Definition 5. The domain of the variable(s) of a rational equation (in short, the domain of a rational equation) is the intersection of the domains of all rational expressions within the equation. As stated in Definition., the domain of each single algebraic fraction is the set of all real numbers except for the zeros of the denominator (the variable values that would make the denominator equal to zero). Therefore, the domain of a rational equation is the set of all real numbers except for the zeros of all the denominators appearing in this equation.
3 Determining Domains of Rational Equations Find the domain of the variable in each of the given equations. a. c. b. 3 + 4 4 yy yy 8 yy +6yy+8 yy 6 a. The equation contains two denominators, and. is never equal to zero and becomes zero when 0. Thus, the domain of this equation is R {00}. b. The equation 3 + 4 contains two types of denominators, and. The becomes zero when, and becomes zero when 0. Thus, the domain of this equation is R {00, }. c. The equation 4 contains three different denominators. yy yy 8 yy +6yy+8 yy 6 To find the zeros of these denominators, we solve the following equations by factoring: yy yy 8 0 yy + 6yy + 8 0 yy 6 0 (yy 4)(yy + ) 0 (yy + 4)(yy + ) 0 (yy 4)(yy + 4) 0 yy 4 or yy yy 4 or yy yy 4 or yy 4 So, 4,, and 4 must be excluded from the domain of this equation. Therefore, the domain DD R { 44,, 44}. To solve a rational equation, it is convenient to clear all the fractions first and then solve the resulting polynomial equation. This can be achieved by multiplying all the terms of the equation by the least common denominator. Caution! Only equations, not expressions, can be changed equivalently by multiplying both of their sides by the LCD. Multiplying expressions by any number other than creates expressions that are NOT equivalent to the original ones. So, avoid multiplying rational expressions by the LCD. Solving Rational Equations Solve each equation. a. c. b. 3 + 4 4 yy yy 8 yy +6yy+8 yy 6 d. 3 3
3 a. The domain of the equation is the set R {0}, as discussed in Example a. The LLLLLL(, ), so we calculate 4 / multiply each term by the LCD + 4 0 ( + 6)( 4) 0 6 or 4 factor to find the possible roots Since both of these numbers belong to the domain, the solution set of the original equation is { 66, 44}. b. The domain of the equation 3 + 4 is the set R {0, }, as discussed in Example b. The LLLLLL(, ) ( ), so we calculate 3 + 4 / ( ) ( ) 3 ( ) + 4 ( ) 3( ) + 4 3 + 6 + 4 + 6 0 expand the bracket, collect like terms, and bring the terms over to one side ( + 3)( ) 0 factor to find the possible roots 3 or Since is excluded from the domain, there is only one solution to the original equation, 33. c. The domain of the equation 4 is the set R { 4,, 4}, yy yy 8 yy +6yy+8 yy 6 as discussed in Example c. To find the LCD, it is useful to factor the denominators first. Since yy yy 8 (yy 4)(yy + ), yy + 6yy + 8 (yy + 4)(yy + ), and yy 6 (yy 4)(yy + 4), then the LCD needed to clear the fractions in the original equation is (yy 4)(yy + 4)(yy + ). So, we calculate (yy 4)(yy + ) 4 (yy + 4)(yy + ) (yy 4)(yy + 4) / (yy 4)(yy + 4)(yy + )
33 (yy 4)(yy + 4)(yy + ) (yy 4)(yy + ) (yy 4)(yy + 4)(yy + ) 4 (yy + 4)(yy + ) (yy + 4) 4(yy 4) (yy + ) yy + 8 4yy + 6 yy + 4 0 4yy yy 5 Since 5 is in the domain, this is the true solution. d. First, we notice that the domain of the equation (yy 4)(yy + 4)(yy + ) (yy 4)(yy + 4) 3 3 is the set R {3}. To solve this equation, we can multiply it by the LLLLLL 3, as in the previous examples, or we can apply the method of cross-multiplication, as the equation is a proportion. Here, we show both methods. Use the method of your choice either one is fine. Multiplication by LCD: 3 3 / ( 3) Cross-multiplication: 3 3 ( )( 3) ( 3) this multiplication is permitted as 3 0 3 3 / ( 3) this division is permitted as 3 0 Since 3 is excluded from the domain, there is no solution to the original equation. Summary of Solving Rational Equations in One Variable. Determine the domain of the variable.. Clear all the fractions by multiplying both sides of the equation by the LCD of these fractions. 3. Find possible solutions by solving the resulting equation. 4. Check the possible solutions against the domain. The solution set consists of only these possible solutions that belong to the domain. Graphs of Basic Rational Functions So far, we discussed operations on rational expressions and solving rational equations. Now, we will look at rational functions, such as
34 ff() 3, gg(), oooo h() + 3. Definition 5.3 A rational function is any function that can be written in the form ff() PP() QQ(), where PP and QQ are polynomials and QQ is not a zero polynomial. The domain DD ff of such function ff includes all -values for which QQ() 0. Finding the Domain of a Rational Function Find the domain of each function. a. gg() +3 b. h() 3 a. Since + 3 0 for 3, the domain of gg is the set of all real numbers except for 3. So, the domain DD gg R { 33}. b. Since 0 for, the domain of h is the set of all real numbers except for. So, the domain DD hh R {}. Note: The subindex ff in the notation DD ff indicates that the domain is of function ff. To graph a rational function, we usually start by making a table of values. Because the graphs of rational functions are typically nonlinear, it is a good idea to plot at least 3 points on each side of each -value where the function is undefined. For example, to graph the ff() 00 undefined basic rational function, ff(), called the reciprocal function, we evaluate ff for a few points to the right of zero and to the left of zero. This is because ff is undefined at 0, which means that the graph of ff does not cross the yy-axis. After ff() plotting the obtained points, we connect them within each group, to the right of zero and to the left of zero, creating two disjoint curves. To see the shape of each curve clearly, we might need to evaluate ff at some additional points. The domain of the reciprocal function ff() is R {00}, as the denominator must be different than zero. Projecting the graph of this function onto the yy-axis helps us determine the range, which is also R {00}.
35 There is another interesting feature of the graph of the reciprocal function ff(). Vertical Asymptote ff() aa Observe that the graph approaches two lines, yy 0, the -axis, and 0, the yy-axis. These lines are called asymptotes. They effect the shape of the graph, but they themselves do not belong to the graph. To indicate the fact that asymptotes do not belong to the graph, we use a dashed line when graphing them. In general, if the yy-values of a rational function approach or as the -values approach a real number aa, the vertical line aa is a vertical asymptote of the graph. This can be recorded with the use of arrows, as follows: read: approaches aa is a vertical asymptote yy (or ) when aa. ff() bb Horizontal Asymptote Also, if the yy-values approach a real number bb as -values approach or, the horizontal line yy bb is a horizontal asymptote of the graph. Again, using arrows, we can record this statement as: yy aa is a horizontal asymptote yy bb when (or ). Graphing and Analysing the Graphs of Basic Rational Functions For each function, state its domain and the equation of the vertical asymptote, graph it, and then state its range and the equation of the horizontal asymptote. a. gg() +3 b. h() 3 a. The domain of function gg() +3 is DD gg R { 33}, as discussed in Example 3a. Since 3 is excluded from the domain, we expect the vertical asymptote to be at 33. 55 gg() 4 33 77 4 44 55 66 undefined 3 gg() To graph function gg, we evaluate it at some points to the right and to the left of 3. The reader is encouraged to check the values given in the table. 3 Then, we draw the vertical asymptote 3 and plot and join the obtained points on each side of this asymptote. The graph suggests that the horizontal asymptote is the -axis. Indeed, the value of zero cannot be attained by the function gg(), as in order +3 for a fraction to become zero, its numerator would have to be zero. So, the range of function gg is R {00} and yy 00 is the equation of the horizontal asymptote. b. The domain of function h() 3 is DD hh R {}, as discussed in Example 3b. Since is excluded from the domain, we expect the vertical asymptote to be at.
36 hh() 4 3 3 00 3 33 55 33 0 44 66 undefined 3 4 As before, to graph function h, we evaluate it at some points to the right and to the left of. Then, we draw the vertical asymptote and plot and join the obtained points on each side of this asymptote. The graph suggests that the horizontal asymptote is the line yy. Thus, the range of function h is R {}. h() Notice that 3. Since is never equal to zero than is never equal to. This confirms the range and the horizontal asymptote stated above. Connecting the Algebraic and Graphical s of Rational Equations Given that ff() +, find all the -values for which ff(). Illustrate the situation with a graph. To find all the -values for which ff(), we replace ff() in the equation ff() + with and solve the resulting equation. So, we have + + / ( ) /, + 4 Thus, ff() for 44. yy ff() ff() + 44 The geometrical connection can be observed by graphing the function ff() + +3 + 3 and the line yy on the same grid, as illustrated by the accompanying graph. The -coordinate of the intersection of the two graphs is the solution to the equation +. This also means that ff(4) 4+. So, we can say that ff(4). 4 Graphing the Reciprocal of a Linear Function Suppose ff() 3. a. Determine the reciprocal function gg() ff() and its domain DD gg.
37 b. Determine the equation of the vertical asymptote of the reciprocal function gg. c. Graph the function ff and its reciprocal function gg on the same grid. Then, describe the relations between the two graphs. a. The reciprocal of ff() 3 is the function gg(). Since 3 0 for 33 3, then the domain DD gg R 33. b. A vertical asymptote of a rational function in simplified form is a vertical line passing through any of the -values that are excluded from the domain of such a function. So, the equation of the vertical asymptote of function gg() 3 is 33. c. To graph functions ff and gg, we can use a table of values as below. ff() gg() 4 4 55 44 33 77 44 0 undefined 55 77 4 4 gg() 3 Notice that the vertical asymptote of the reciprocal function comes through the zero of the linear function. Also, the values of both functions are positive to the right of 3 and negative to the left of 3. In addition, ff() gg() and ff() gg(). This is because the reciprocal of is and the reciprocal of is. For the rest of the values, observe that the values of the linear function that are very close to zero become very large in the reciprocal function and conversely, the values of the linear function that are very far from zero become very close to zero in the reciprocal function. This suggests the horizontal asymptote at zero. ff() 3 Using Properties of a Rational Function in an Application Problem elevation When curves are designed for train tracks, the outer rail is usually elevated so that a locomotive and its cars can safely take the curve at a higher speed than if the tracks were at the same level. Suppose that a circular curve with a radius of rr feet is being designed for a train traveling 60 miles per hour. The function ff(rr) 540 calculates the proper elevation rr yy ff(rr), in inches, for the outer rail.
38 a. Evaluate ff(300) and interpret the result. b. Suppose that the outer rail for a curve is elevated 6 inches. What should the radius of the curve be? c. Observe the accompanying graph of the function ff and discuss how the elevation of the outer rail changes as the radius rr increases. a. ff(300) 540 8.5. Thus, the outer rail on a curve with 300 radius 300 ft should be elevated about 8.5 inches for a train to safely travel through it at 60 miles per hour. elevation (inches) ff(rr) 50 40 30 0 0 ff(rr) 540 rr 00 300 500 radius (feet) rr b. Since the elevation yy ff(rr) 6 inches, to find the corresponding value of rr, we need to solve the equation 6 540 rr. After multiplying this equation by rr and dividing it by 6, we obtain rr 540 6 43 So, the radius of the curve should be about 43 feet. c. As the radius increases, the outer rail needs less elevation. RT.5 Exercises Vocabulary Check Complete each blank with one of the suggested words, or with the most appropriate term from the given list: asymptote, cross-product, domain, equations, LCD, numerator, rational.. A equation involves fractional expressions.. To solve a rational equation, first find the of all the rational expressions. 3. Multiplication by any quantity other than zero can be used to create equivalent. 4. When simplifying rational expressions, we multiply by LCD. cccccc / cccccccccccc 5. The of a rational function is the set of all real numbers except for the variable values that would make the denominator of this function equal to zero. 6. An is a line that a given graph approaches in a long run. 7. A fractional expression is equal to zero when its is equal to zero. 8. To solve a rational proportion we can use either the LCD method or the method.
39 Concept Check State the domain for each equation. There is no need to solve it. 9. +5 4 +3 3 6 0. 5 aa 8 6aa 4 aa. 3 +4 9. 4 + 9 3 5 4+7 3. 4 yy 5 yy+5 yy 7 4. 3 6 6+9 3 9 Solve each equation. 5. 3 8 + 3 6. 4 5 6 yy 7. + 8 9 8. 4 3 0 3aa aa 3 9. rr 8 + rr 4 rr 4 0. nn nn 6 4nn 9. 5 3 rr+0 rr. 5 3 aa+4 aa 3. yy+ yy 5 3 4. 4 +3 +6 5. 5 6. 3 5 7. 3 3 8. + +0 +0 9. yy + 5 3yy 3 30. 7 6+3 3 + 3. 8 8 3kk+9 5 5kk+5 3. 6 mm 4 + 5 mm 0 mm 4mm 33. 3 + yy 5 yy 4 yy yy+ 34. + +3 4 + 35. 8 +0 5 5 36. 5 + yy+3 4yy 36 yy 3 37. 6 4+3 3 4 4 38. 7 8 +5 +6 0 39. 5 3 4 5+4 40. yy + 3yy+5 yy+ yy +4yy+3 yy+3 4. 3 + 7 4 + 3 +8 +4 4. 4 + 7 08 +3 3+9 3 +7 43. 9 3 0 9 44. 5 4 + 3 3 3 3+9 For the given rational function ff, find all values of for which ff() has the indicated value. 45. ff() 5 ; ff() 5 46. ff() ; ff() 3 + 5 47. gg() 3 +3 + ; gg() 4 48. gg() 4 + ; gg() 3
40 Graph each rational function. State its domain, range and the equations of the vertical and horizontal asymptotes. 49. ff() 5. ff() + 50. gg() 53. gg() + 5. h() 3 54. h() + 3 Analytic Skills For each function ff, find its reciprocal function gg() and graph both functions on the ff() same grid. Then, state the equations of the vertical and horizontal asymptotes of function gg. 55. ff() + 56. ff() + 57. ff() 3 Analytic Skills Solve each equation. 58. + + 3 59. 4 Solve each problem. 60. The average number of vehicles waiting in line to enter a parking area is modeled by the function defined by ww() ( ), where is a quantity between 0 and known as the traffic intensity. a. For each traffic intensity, find the average number of vehicles waiting. Round the answer to the nearest one. i. 0. ii. 0.8 iii. 0.9 b. What happens to the number of vehicles waiting in line as traffic intensity increases? 6. The percent of deaths caused by smoking, called the incidence rate, is modeled by the rational function pp(), where is the number of times a smoker is more likely to die of lung cancer than a non-smoker is. For example, 0 means that a smoker is 0 times more likely than a non-smoker to die from lung cancer. a. Find pp() if is 0. b. For what values of is pp() 80%? (Hint: Change 80% to a decimal.) c. Can the incidence rate equal 0? Explain. d. Can the incidence rate equal? Explain.
4 RT.6 Applications of Rational Equations In previous sections of this chapter, we studied operations on rational expressions, simplifying complex fractions, and solving rational equations. These skills are needed when working with real-word problems that lead to a rational equation. The common types of such problems are motion or work problems. In this section, we first discuss how to solve a rational formula for a given variable, and then present several examples of application problems involving rational equations. Formulas Containing Rational Expressions Solving application problems often involves working with formulas. We might need to form a formula, evaluate it, or solve it for a desired variable. The basic strategies used to solve a formula for a variable were shown in section L and F4. Recall the guidelines that we used to isolate the desired variable: Reverse operations to clear unwanted factors or addends; Example: To solve AA+BB CC for AA, we multiply by and then subtract BB. Multiply by the LCD to keep the desired variable in the numerator; AA Example: To solve PP for rr, first, we multiply by ( + rr). +rr Take the reciprocal of both sides of the equation to keep the desired variable in the numerator (this applies to proportions only); Example: To solve CC AA+BB AAAA obtain CC AAAA AA+BB. for CC, we can take the reciprocal of both sides to Factor to keep the desired variable in one place. Example: To solve PP + PPPPPP AA for PP, we first factor PP out. Below we show how to solve formulas containing rational expressions, using a combination of the above strategies. Solving Rational Formulas for a Given Variable Solve each formula for the indicated variable. a. + dddd dddd, for pp b. LL, for DD c. LL, for dd ff pp qq DD dd DD dd a. I: First, we isolate the term containing pp, by moving qq to the other side of the equation. So, ff pp + qq / qq rewrite from the right to the left, ff qq pp qq ff pp ffff and perform the subtraction to leave this side as a single fraction
Then, to bring pp to the numerator, we can take the reciprocal of both sides of the equation, obtaining pp ffff qq ff Caution! This method can be applied only to a proportion (an equation with a single fraction on each side). 4 II: The same result can be achieved by multiplying the original equation by the LLLLLL ffffff, as shown below ff pp + / ffffff qq factor pp out ppqq ffff + ffpp ppqq ffpp ffff pp(qq ff) ffff pp ffff qq ff / ffff / (qq ff) b. To solve LL dddd DD dd denominator to bring the variable DD to the numerator. So, This can be done in one step by interchanging LL with DD dd. The movement of the expressions resembles that of a teeter-totter. for DD, we may start with multiplying the equation by the LL dddd DD dd LL(DD dd) dd DD dd dddd LL / (DD dd) / LL /+dd DD dddd LL + dd dddd + dddd LL Both forms are correct answers. c. When solving LL dddd for dd, we first observe that the variable dd appears in both the DD dd numerator and denominator. Similarly as in the previous example, we bring the dd from the denominator to the numerator by multiplying the formula by the denominator DD dd. Thus, LL ddrr DD dd LL(DD dd) ddrr. / (DD dd) Then, to keep the dd in one place, we need to expand the bracket, collect terms with dd, and finally factor the dd out. So, we have
43 LLLL LLdd ddrr /+LLLL LLLL ddrr + LLdd LLLL dd(rr + LL) LLLL RR + LL dd Obviously, the final formula can be written starting with dd, dd LLLL RR + LL. / (RR + LL) Forming and Evaluating a Rational Formula Suppose a trip consists of two parts, each of length. a. Find a formula for determining the average speed vv for the whole trip, given the speed vv for the first part of the trip and vv for the second part of the trip. b. Find the average speed vv for the whole trip, if the speed for the first part of the trip was 60 km/h and the speed for the second part of the trip was 90 km/h. a. The total distance, dd, for the whole trip is +. The total time, tt, for the whole trip is the sum of the times for the two parts of the trip, tt and tt. From the relation rrrrrrrr tttttttt dddddddddddddddd, we have Therefore, tt vv and tt vv. tt vv + vv, which after substituting to the formula for the average speed, vv dd, gives us tt vv +. vv vv Since the formula involves a complex fraction, it should be simplified. We can do this by multiplying the numerator and denominator by the LLLLLL vv vv. So, we have vv vv vv + vv vv vv vv vv vv vv vv vv vv + vv vv vv vv vv vv vv + vv factor the
44 vv vv vv (vv + vv ) vv vv vv vv + vv Note : The average speed in this formula does not depend on the distance travelled. Note : The average speed for the total trip is not the average (arithmetic mean) of the speeds for each part of the trip. In fact, this formula represents the harmonic mean of the two speeds. b. Since vv 60 km/h and vv 90km/h, using the formula developed in Example a, we calculate 60 90 vv 60 + 90 0800 7777 kkkk/hh 50 Observation: Notice that the average speed for the whole trip is lower than the average of the speeds for each part of the trip, which is 60+90 75 km/h. Applied Problems Proportion Problems Many types of application problems were already introduced in sections L3 and E. Some of these types, for example motion problems, may involve solving rational equations. Below we show examples of proportion and motion problems as well as introduce another type of problems, work problems. When forming a proportion, cccccccccccccccc II bbbbbbbbbbbb cccccccccccccccc II aaaaaaaaaa cccccccccccccccc IIII bbbbbbbbbbbb cccccccccccccccc IIII aaaaaaaaaa, it is essential that the same type of data are placed in the same row or the same column. Recall: To solve a proportion aa bb cc dd, for example, for aa, it is enough to multiply the equation by bb. This gives us aa bbbb dd.
Similarly, to solve aa bb cc dd for bb, we can use the cross-multiplication method, which eventually (we encourage the reader to check this) leads us to aa aaaa cc. Notice that in both cases the desired variable equals the product of the blue variables lying across each other, divided by the remaining purple variable. This is often referred to as the cross multiply and divide approach to solving a proportion. 45 In statistics, proportions are often used to estimate the population by analysing its sample in situations where the exact count of the population is too costly or not possible to obtain. Estimating Numbers of Wild Animals To estimate the number of wild horses in Utah, a forest ranger catches 60 wild horses, tags them, and releases them. Later, horses are caught and it is found that 3 of them are tagged. Assuming that the horses mix freely when they are released, estimate how many wild horses there are in Utah. Suppose there are wild horses in Utah. 60 of them were tagged, so the ratio of the tagged horses in the whole population of the wild horses in Utah is 60 The ratio of the tagged horses found in the sample of horses caught in the later time is 3 So, we form the proportion: 60 3 tagged horses all horses 3 60 x wild horses population sample After solving for, we have 60 3 So, we can estimate that over 400 wild horses live in Utah.
In geometry, proportions are the defining properties of similar figures. One frequently used theorem that involves proportions is the theorem about similar triangles, attributed to the Greek mathematician Thales. 46 Thales Theorem Two triangles are similar iff the ratios of the corresponding sides are the same. BB CC BB CC AA AAAAAA AABB CC AAAA AAAA AAAA AAAA BBBB BB CC Using Similar Triangles in an Application Problem A cross-section of a small storage room is in the shape of a right triangle with a height of meters and a base of. meters, as shown in Figure 6.. What is the largest cubic box that can fit in this room? Assume that the base of the box is positioned on the floor of the storage room. Suppose that the height of the box is meters. Since the height of the storage room is meters, the expression represents the height of the wall above the box, as shown in Figure 6.b.. Figure 6.a. Figure 6.b Since the blue and brown triangles are similar, we can use the Thales Theorem to form the proportion.. Employing cross-multiplication, we obtain.4..4 3..4 00. 7777 3. So, the dimensions of the largest cubic box fitting in this storage room are 75 cm by 75 cm by 75 cm. Motion Problems Motion problems in which we compare times usually involve solving rational equations. This is because when solving the motion formula rrrrrrrr RR tttttttt TT dddddddddddddddd DD for time, we create a fraction tttttttt TT dddddddddddddddd DD rrrrrrrr RR
47 Solving a Motion Problem Where Times are the Same The speed of one mountain biker is 3 km/h faster than the speed of another biker. The first biker travels 40 km in the same amount of time that it takes the second to travel 30 km. Find the speed of each biker. Let rr represent the speed of the slower biker. Then rr + 3 represents the speed of the faster biker. The slower biker travels 30 km, while the faster biker travels 40 km. Now, we can complete the table slower biker faster biker rr + 3 RR TT DD rr 30 rr 30 40 rr + 3 40 To complete the Time column, we divide the Distance by the Rate. Since the time of travel is the same for both bikers, we form and then solve the equation: 30 rr 40 rr + 3 3(rr + 3) 4rr 3rr + 9 4rr / 0 and cross-multiply / 3rr rr 9 Thus, the speed of the slower biker is rr 99 km/h and the speed of the faster biker is rr + 3 km/h. Solving a Motion Problem Where the Total Time is Given Kris and Pat are driving from Vancouver to Princeton, a distance of 97 km. Kris, whose average rate is 6 mph faster than Pat s, will drive the first 53 km of the trip, and then Pat will drive the rest of the way to their destination. If the total driving time is 3 hours, determine the average rate of each driver. Let rr represent Pat s average rate. Then rr + 6 represents Kris average rate. Kris travelled 53 km, while Pat travelled 97 53 44 km. Now, we can complete the table: RR TT DD Kris rr + 6 53 rr + 6 53 Pat rr 44 rr 44 total 3 97 Note: In motion problems we may add times or distances but we usually do not add rates!
48 The equation to solve comes from the Time column. 53 rr + 6 + 44 rr 3 53rr + 44(rr + 6) 3rr(rr + 6) 5rr + 48rr + 88 rr + 6rr 0 rr 93rr 88 (rr 96)(rr + 3) 0 / rr(rr + 6) / 3 and distribute; then collect like terms on one side factor rr 96 oooo rr 3 Since a rate cannot be negative, we discard the solution rr 3. Therefore, Pat s average rate was rr 9999 km/h and Kris average rate was rr + 6 km/h. Work Problems Notice the similarity to the formula RR TT DD used in motion problems. When solving work problems, refer to the formula RRRRRRRR oooo wwwwwwww TTTTTTTT aaaaaaaaaaaa oooo JJJJJJ cccccccccccccccccc and organize data in a table like this: worker I worker II together RR TT J Note: In work problems we usually add rates but do not add times! Solving a Work Problem Involving Addition of Rates Alex can trim the shrubs at Beecher Community College in 6 hr. Bruce can do the same job in 4 hr. How long would it take them to complete the same trimming job if they work together? Let tt be the time needed to trim the shrubs when Alex and Bruce work together. Since trimming the shrubs at Beecher Community College is considered to be the whole one job to complete, then the rate RR in which this work is done equals RR JJoooo TTiiiiii TTiiiiii. To organize the information, we can complete the table below.
49 Alex Bruce together RR TT JJ 66 6 44 4 tt tt The job column is often equal to, although sometimes other values might need to be used. To complete the Rate column, we divide the Job by the Time. Since the rate of work when both Alex and Bruce trim the shrubs is the sum of rates of individual workers, we form and solve the equation 6 + 4 tt / tt tt + 3tt 5tt / 5 tt 5.4 So, if both Alex and Bruce work together, the amount of time needed to complete the job if.4 hours hours 4 minutes. Note: The time needed for both workers is shorter than either of the individual times. Solving a Work Problem Involving Subtraction of Rates One pipe can fill a swimming pool in 0 hours, while another pipe can empty the pool in 5 hours. How long would it take to fill the pool if both pipes were left open? Suppose tt is the time needed to fill the pool when both pipes are left open. If filling the pool is the whole one job to complete, then emptying the pool corresponds to job. This is because when emptying the pool, we reverse the filling job. To organize the information given in the problem, we complete the following table. pipe I RR TT JJ 0 pipe II both pipes tt 5 tt
50 The equation to solve comes from the Rate column. 0 5 tt 3tt tt 30 / 30tt tt 30 So, it will take 30 hours to fill the pool when both pipes are left open. RT.6 Exercises Vocabulary Check Complete each blank with the most appropriate term or phrase from the given list: column, fraction, motion, numerator, proportions, row, similar, work.. When solving a formula for a specified variable, we want to keep the variable in the.. The strategy of taking reciprocal of each side of an equation is applicable only in solving. This means that each side of such equation must be in the form of a single. 3. When forming a proportion, it is essential that the same type of data are placed in the same or in the same. 4. The Thales Theorem states that for any two triangles the ratios of the corresponding sides are the same. 5. In problems, we usually add times or distances but not the rates. 6. In problems, we usually add rates but not times. Concept Check 7. Using the formula +, find bb if aa 8 and cc. aa bb cc 8. The gravitational force between two masses is given by the formula FF GGGGGG. dd Find MM if FF 0, GG 6.67 0, mm, and dd 3 0 6. Round your answer to two decimal places. Concept Check 9. What is the first step in solving the formula rrrr rrrr pp + qq for rr? 0. What is the first step in solving the formula mm aaaa for aa? aa bb
5 Solve each formula for the specified variable.. mm FF aa for aa. II EE RR for RR 3. WW WW dd dd for dd 4. FF GGGGGG dd for mm 5. ss (vv +vv )tt for tt 6. ss (vv +vv )tt for vv 7. RR rr + rr for RR 8. RR rr + rr for rr 9. pp + qq ff for qq 0. tt aa + tt bb for aa. PPPP TT pppp tt for vv. PPPP TT pppp tt for TT 3. AA h(aa+bb) for bb 4. aa VV vv tt for VV 5. RR gggg gg+ss for ss 6. II VV VV+rr nnnn for VV 7. II for nn 8. EE RR+rr EE+nnnn ee rr for ee 9. EE ee RR+rr rr for rr 30. SS HH mm(tt tt ) for tt 3. VV ππh (3RR h) 3 for RR 3. PP AA +rr for rr 33. VV RR gg RR+h for h 34. vv dd dd tt tt for tt Analytic Skills Solve each problem 35. The ratio of the weight of an object on the moon to the weight of an object on Earth is 0.6 to. How much will an 80-kg astronaut weigh on the moon? 36. A rope is 4 meters long. How can the rope be cut in such a way that the ratio of the resulting two segments is 3 to 5? 37. Walking 4 miles in hours will use up 650 calories. Walking at the same rate, how many miles would a person need to walk to lose lb? (Burning 3500 calories is equivalent to losing pound.) Round to the nearest hundredth. 38. On a map of Canada, the linear distance between Vancouver and Calgary is.8 cm. The airline distance between the two cities is about 675 kilometers. On this same map, what would be the linear distance between Calgary and Montreal if the airline distance between the two cities is approximately 3000 kilometers? 39. To estimate the deer population of a forest preserve, wildlife biologists caught, tagged, and then released 4 deer. A month later, they returned and caught a sample of 75 deer and found that 5 of them were tagged. Based on this experiment, approximately how many deer lived in the forest preserve? 40. Biologists tagged 500 fish in a lake on January. On February, they returned and collected a random sample of 400 fish, 8 of which had been previously tagged. On the basis of this experiment, approximately how many fish does the lake have? 4. Twenty-two bald eagles are tagged and released into the wilderness. Later, an observed sample of 56 bald eagles contains 7 eagles that are tagged. Estimate the bald eagle population in this wilderness area. 4. A 6 foot tall person casts a 4 foot long shadow. If a nearby tree casts a 44 foot long shadow, estimate the height of the tree.
43. Suppose the following triangles are similar. Find yy and the lengths of the unknown sides of each triangle. A 4 B 5yy 5 44. A rectangle has sides of 9 cm and 4 cm. In a similar rectangle the longer side is 8 cm. What is the length of the shorter side? 45. What is the average speed if Shane runs 8 kilometers per hour for the first half of a race and kilometers per hour for the second half of the race? 46. If you average kilometers per hour during the first half of a trip, find the speed yy in kilometers per hour needed during the second half of the trip to reach 80 kilometers per hour as an overall average. 47. Kellen s boat goes mph. Find the rate of the current of the river if she can go 6 mi upstream in the same amount of time she can go 0 mi downstream. 48. A plane averaged 500 mph on a trip going east, but only 350 mph on the return trip. The total flying time in both directions was 8.5 hr. What was the one-way distance? 49. A Boeing 747 flies 40 mi with the wind. In the same amount of time, it can fly 40 mi against the wind. The cruising speed (in still air) is 570 mph. Find the speed of the wind. 50. A moving sidewalk moves at a rate of.7 ft/sec. Walking on the moving sidewalk, Hunter can travel 0 ft forward in the same time it takes to travel 5 ft in the opposite direction. How fast would Hunter be walking on a non-moving sidewalk? 5. On his drive from Montpelier, Vermont, to Columbia, South Carolina, Victor Samuels averaged 5 mph. If he had been able to average 60 mph, he would have reached his destination 3 hr earlier. What is the driving distance between Montpelier and Columbia? 5. On the first part of a trip to Carmel traveling on the freeway, Marge averaged 60 mph. On the rest of the trip, which was 0 mi longer than the first part, she averaged 50 mph. Find the total distance to Carmel if the second part of the trip took 30 min more than the first part. 53. Cathy is a college professor who lives in an off-campus apartment. On days when she rides her bike to campus, she gets to her first class 36 min faster than when she walks. If her average walking rate is 3 mph and her average biking rate is mph, how far is it from her apartment to her first class? 54. Gina can file all of the daily invoices in 4 hr and Bert can do the same job in 6 hr. If they work together, then what portion of the invoices can they file in hr? 55. Melanie can paint the entire house in hours and Timothy can do the same job in yy hours. Write a rational expression that represents the portion of the house that they can paint in hr working together. 56. Walter and Helen are asked to paint a house. Walter can paint the house by himself in hours and Helen can paint the house by herself in 6 hours. How long would it take to paint the house if they worked together? C P 3 Q yy + R 5
57. Brian, Mark, and Jeff are painting a house. Working together they can paint the house is 6 hours. Working alone Brain can paint the house in 5 hours and Jeff can paint the house in 0 hours. How long would it take Mark to paint the house working alone? 58. An experienced carpenter can frame a house twice as fast as an apprentice. Working together, it takes the carpenters days. How long would it take the apprentice working alone? 59. A tank can be filled in 9 hr and drained in hr. How long will it take to fill the tank if the drain is left open? 60. A cold water faucet can fill the bath tub in minutes, and a hot water faucet can fill the bath tub in 8 minutes. The drain can empty the bath tub in 4 minutes. If both faucets are on and the drain is open, how long would it take to fill the bath tub? 6. Together, a 00-cm wide escalator and a 60-cm wide escalator can empty a 575-person auditorium in 4 min. The wider escalator moves twice as many people as the narrower one does. How many people per hour does the 60-cm wide escalator move? Discussion Point 6. At what time after 4:00 will the minute hand overlap the hour hand of a clock for the first time? 63. Michelle drives to work at 50 mph and arrives min late. When she drives to work at 60 mph, she arrives 5 min early. How far does Michelle live from work? 53