LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II A rational epression is a fraction that contains variables. A variable is very useful in mathematics. In most situations, any number can be substituted for a variable. In the case of rational epressions, certain numbers cannot be used, specifically when the denominator is equal to zero. These are known as restricted or undefined values. 6 Eercise #: Consider the rational function given by f. For what value of is this function undefined? Why is it undefined at this value? 5 Eercise #: Find all values of for which the rational function h is undefined. Verify by 6 using your calculator to evaluate this epression for these values using the STORE function. Eercise #: State the restrictions for each rational epression. a) 5 b) 7 c) + 5 d) 9 e) 70 The following evaluations reveal an important property when we simplify rational epressions. Eercise #: Simplify: 5 5 = 4 0 0 4 = 0 0 emathinstruction, RED HOOK, NY 57, 05
Simplifying a rational epression means cancelling or dividing out any common factors in the numerator and denominator. We ADD like terms. In rational epressions we cancel common factors NOT like terms. Rules for simplifying rational epressions Monomials (One term) Eercise #4: Since monomials contain multiplication only, and multiplication is the same operation as the division indicated by the fraction bar, you can cancel any common factors. 6 y y 4 Polynomials (More than one term) Eercise #5: Since polynomials contain addition/subtraction, they are not the same operation as division. Therefore, you must factor polynomials before you can find the common factors to cancel. Parenthesis around polynomials can help you remember this. 6 6 Eercise #6: Simplify each rational epression.. 5 5 6. 6 4 6. 7 4. 5y 0 4y y y 5. 4y 6y 5 6. r 5r 0 4r emathinstruction, RED HOOK, NY 57, 05
Eercise #7: Continue simplifying these rational epressions. 7. 9 5 4 8. 6 4 9. ac ab bc b a ab b 0. 9 y 9 6y y emathinstruction, RED HOOK, NY 57, 05
4 LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the following values of a restricted value for f () 7 () () 7 (4). g ()? 7 4 is undefined for which of the following values of? () 4 () (4) 4. Which values of, when substituted into the function y, would make it undefined? 8 () and 8 () 4 and 4 () 4 and 8 (4) 4 and 0 4. Which of the following is true for the rational epression, - 4 + 5-4? () ¹ ± () ¹ -4 and 4 () ¹ -7 and (4) ¹ -5 and 4 5. Write each of the following ratios in simplest form. 8 5 0 (b) y 8y (c) 6 y 5y 0 4 5 (d) 4y y 7 6 0 emathinstruction, RED HOOK, NY 57, 05
5 6. Which of the following is equivalent to the epression () 4 y () y 4y 6 4 6 y? () y (4) y 7. Simplify each of the following rational epressions. 5 4 0 (b) 4 9 (c) 4 50 (d) 9 4 44 (e) 7 4 48 (f) 5 5 4 8. Which of the following is equivalent to the fraction () () 6 5 5 98? 5 5 () 6 5 6 (4) 5 emathinstruction, RED HOOK, NY 57, 05
6 9. The rational epression () () 6 76 4 () (4) can be equivalently rewritten as 0. Written in simplest form, the fraction y 5 5y is equal to () 5y 5 () y 5 () y 5 (4) y 5 REASONING. When we simplify an algebraic fraction, we are producing equivalent epressions for most values of. 4 Consider the epressions and. 4 Show by simplifying the first epression that these two are equivalent. (b) Use your calculator to fill out the value for both of these epressions to show their equivalence. 0 4 4 (c) Clearly these two epressions are not equivalent for an input value of. Eplain why. 4 emathinstruction, RED HOOK, NY 57, 05
7 Eercise #: LESSON #5 - MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II Simplify the following problem without cross canceling: 9 5 = 5 8 Simplify the following problem by cross canceling first: 0 = 4 7 Imagine doing the problem, 5 4 0 5, without cross cancelling. You would have to use the distributive property in both the numerator & denominator. It would take forever. Q: Why can you cross cancel when multiplying rational epressions? A: Rational epressions are division. For eample, 5 0 is the same as 5 0. Since multiplication and division are really the same operation, you can cancel or divide out common factors before you multiply. ) Factor each numerator and denominator Steps for Multiplying Rational Epressions 5 4 0 5 ) Cancel any factors that appear in one of the numerators and one of the denominators. ) Multiply the remaining terms in the numerator. Multiply the remaining terms in the denominator. 4) Check to see if you can reduce any further. emathinstruction, RED HOOK, NY 57, 05
8 Eercise #: Simplify each of the following rational epressions. 6 y 0y (b) 5 5 7 4 y 9 y 4 5 6 6 (c) 9 4 4 6 Division of rational epressions continues to follow from what you have seen in previous courses. Since division by a fraction can always be thought of in terms of multiplying by it s reciprocal, these problems simply involve an additional step. Eercise #: Perform each of the following division problems. Epress all answers in simplest form. 8 5 5 (b) 5 7 6y y 0y 4y 0 8 6 (c) 8 6 86 0 (d) 9 5 5 7 5 4 emathinstruction, RED HOOK, NY 57, 05
9 SKILLS LESSON #5 - MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II HOMEWORK. Epress each of the following products in simplest form. 4 5 y (b) 8 5y 0 4a 0b (c) 9 6 5b a 4 0 y z 5 9z 0 8y 7. When 4 y 0 is divided by 6 8 6y the result is () 8 7 y () 8 y 7 () y 5 7 (4) 4 y 7. Epress each of the following problems in simplest form. 5 0 5 6 0 40 (b) 9 6 88 6 4 emathinstruction, RED HOOK, NY 57, 05
0 (c) 5 8 5 6 (d) 49 5 9 4 6 (e) 7 4 4 8 4 6 8 (f) 4 8 9 6 7 4 (g) 6 8 4 4 6 (h) 9 7 8 4 emathinstruction, RED HOOK, NY 57, 05
LESSON #6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II Addition and subtraction always involve combining like terms. Rational epressions are no different. Eercise #: To help with this, we will think about what a couple of fractions mean. 7 : 5 : As you can see, the denominator of the fraction tells you the size or the type of term, while the numerator tells you how many of that term you have. This is illustrated by the fact that. In words, three sevenths is 7 7 three one-sevenths. Therefore, two fractions must have the same denominator in order to add them. Otherwise they are not like terms. Steps for adding two fractions:. Find the LCD and write it on the side. 6 y. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! emathinstruction, RED HOOK, NY 57, 05
Eercise #: Combine each of the following fractions by first finding a common denominator. Epress your answers in simplest form. 5 4 (b) 4 6 4 5 (c) 5 0 4 (d) b a y (e) a b y y Each of the combinations in Eercise # should have been reasonably easy because each denominator was monomial in nature. If this is not the case, then it is wise to factor the denominators before trying to find a common denominator.. Find the LCD and write it on the side. 5. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! emathinstruction, RED HOOK, NY 57, 05
Eercise #: Combine each of the following fractions by factoring the denominators first. Then find a common denominator and add. 8 (b) 9 (c) ( ) (d) 9 6 6 7 (e) 4 4 5 6 (f) 8 emathinstruction, RED HOOK, NY 57, 05
4 LESSON #6 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Combine each of the following using addition. Simply your result whenever possible. 5 (b) 6 9 0 5 (c) 5 7 4. Combine and simplify each of the following. Note that each pair of fractions already has a common denominator. 7 (b) 5 8 4 4 (c) 6 8 4 5 5. Simplify each of the following problems. (b) 9 9 (b) y 0 y 6 y4 emathinstruction, RED HOOK, NY 57, 05
5 8 4 4 45 (b) 7 4. Which of the following represents the sum of () () () (4) and? 5. When the epressions () () 5 8 6 and 9 9 () 7 (4) are added the result can be written as 6. When 7 4 is subtracted from 6 the result can be simplified to () () 5 () 0 (4) 7 emathinstruction, RED HOOK, NY 57, 05
6 LESSON #7 MANIPULATING FRACTIONS COMMON CORE ALGEBRA II Throughout the year, there are a number of situations where it is important to be able to manipulate fractions. This lesson will introduce you to those concepts so that they are familiar when you need to use them in the future. Each of these concepts are tools can be used to complete different types of problems. CONCEPT #: A whole number can be written as a fraction over. Eercise #: Write each of the following numbers as a fraction. 5 (b) (c) CONCEPT #: Just as two fractions can be multiplied to create one fraction, a single fraction can be broken up into a product: a unit fraction containing the denominator multiplied by the numerator. Eercise #: Write each of the following fractions as an equivalent product. NOTE: This is not simplifying, but there may be times where it is helpful to solve a problem. t 0 (b) 5h (c) 6 (d) t 6 (e) 8 (f) 5 Eercise #: Write each of the following products as a single fraction. (b) 4 t (c) 5 (b) 8 (c) 6 emathinstruction, RED HOOK, NY 57, 05
CONCEPT #: A fraction over a fraction is a division problem. Together with Concept # (any whole number can be written as a fraction) and the rules for dividing fractions, these problems can be simplified. Eercise #4: Simplify each of the following fractions. 7 8 (b) 6 (c) 6 (d) (e) 5 (f) 4 5 5 CONCEPT #4: Just as two fractions can be added to create one fraction, a single fraction can be broken up into a sum. In this case, we will be splitting up fractions into the sum of a whole number and a fraction. Eercise #5: Write each of the following fractions as an equivalent sum. NOTE: This is not simplifying, but there may be times where it is helpful to solve a problem. (b) 6 5 (c) 68 (d) 6 5 (e) 7 (f) 5 emathinstruction, RED HOOK, NY 57, 05
8 LESSON #7 MANIPULATING FRACTIONS COMMON CORE ALGEBRA II HOMEWORK. Write each of the following numbers as a fraction. (b) 6 (c) 4. Write each of the following fractions as an equivalent product. (b) t 00 (c) (d) 4 (e) 6 (f) 6 5. Write each of the following products as a single fraction. (b) 4 y (c) 5 (b) (c) 5 emathinstruction, RED HOOK, NY 57, 05
9 4. Simplify each of the following fractions. 5 4 (b) 0 5 (c) 5 (d) (e) 5 4 (f) 8 9 0 5. Write each of the following fractions as an equivalent sum. NOTE: This is not simplifying, but there may be times where it is helpful to solve a problem. 5 (b) 7 (c) 57 55 (d) 8 6 4 (e) 0 (f) 7 emathinstruction, RED HOOK, NY 57, 05
0 LESSON #8 - SOLVING RATIONAL EQUATIONS COMMON CORE ALGEBRA II The most simple type of rational equations are those with a single fraction on each side. These rational equations can be solved by cross-multiplying. Remember to find restricted values because they cannot be the solution(s) to your equation. Eercise #: Use the technique of cross multiplication to solve each of the following equations. 4 5 (b) 5 6 Most rational equations have more than one fraction on at least one side of the equation. In these cases, we will need to use all three of the skills we practiced in the last lesson. Eercise #: Steps for Solving Rational Equations 4 5. State any restrictions for the rational epressions in the equations. These values cannot be solutions to your equations.. Find the LCD of all of the rational epressions in the equation.. Multiply both sides of the equation by the LCD to cancel all of the fractions. (Leaving the LCD in factored form often makes this step easier). 4. Solve the resulting equation. 5. Reject any solutions that are restricted values. emathinstruction, RED HOOK, NY 57, 05
Eercise #: Which of the following values of solves: () 4 () 8 4? 6 0 5 () 6 (4) These equations can involve quadratic as well as root epressions. The key, though, remains the same multiplying both sides of the equation by the same quantity. Eercise #4: Solve each of the following equations for all values of. 0 5 (b) 4 Because fractional equations often involve denominators containing variables, it is important that we check to see if any solutions to the equation make it undefined. These represent further eamples of etraneous roots. Eercise #5: Solve and reject any etraneous roots. 8 9 5 8 5 (b) 4 4 emathinstruction, RED HOOK, NY 57, 05
4 (c) 4 6 (d) 5 a a a emathinstruction, RED HOOK, NY 57, 05
FLUENCY LESSON #8 - SOLVING RATIONAL EQUATIONS COMMON CORE ALGEBRA II HOMEWORK. Solve each of the following fractional equations. After clearing the denominators you should have a linear equation to solve. (b) 6 4 (c) 5 6 5. Solve each of the fractional equations for all value(s) of. 8 (b) 4 (c) 7 5 8 (d) 0 emathinstruction, RED HOOK, NY 57, 05
4. Solve the following equation for all values of. Epress your answers in simplest a bi form. (Hint: These directions are telling you that you will probably get an equation where you have to use the quadratic formula). 9 4. Solve each of the following equations. Be sure to check for etraneous roots. 5 6 0 8 (b) 7 7 emathinstruction, RED HOOK, NY 57, 05
5 LESSON #9 WORKING WITH RATIONAL EQUATIONS COMMON CORE ALGEBRA II Rate of work is that part of a task that is completed in one unit of time. If a mason can build a retaining wall in hours, then in hour the mason can build / of the wall. The mason s rate of work is / of the wall each hour. If an apprentice can build the wall in hours, the rate of work for the apprentice is / of the wall each hour. In solving a work problem, the goal is to determine the time it takes to complete a task. The basic equation that is used to solve work problems is Rate of work Time worked = Part of task completed For eample, if a pump can fill an oil tank in 5 hours, then in hours the pump will fill /5 = /5 of the tank. In t hours, the pump will fill /5 t = t/5 of the tank. 0 t 5 Eercise #: A mason can build a wall in 0 hours. An apprentice can build a wall in 5 hours. How long would it take them to build the wall if they worked together? Let t be the unknown time to build the wall working together. Mason Rate Time = 0 t = Part of task completed Apprentice t = Determine how the parts of the task completed are related. Use the fact that the sum of the parts of the task completed must equal, the complete task. Eercise #: An electrician requires hours to wire a house. The electrician s helper can wire a house in 6 hours. After working alone on one job for 4 hours, the electrician quits, and the helper completes the task. How long does it take the helper to finish wiring the house? Electrician Rate Time Part Helper emathinstruction, RED HOOK, NY 57, 05
Eercise #: You can paint a room in 8 hours. Working together, you and your friend can paint the room in just 5 hours. a. Let t be the time (in hours) your friend would take to paint the room when working alone. You Friend Rate Time Part room 8 hours 5 hours 5 hours b. Eplain what the sum of the epressions represents in the last column. Write and solve an equation to find how long your friend would take to paint the room when working alone. 6 Eercise #4: So far in your baseball season, you have hits out of 60 times at-bat. Find the number of consecutive hits you need to raise your batting average to.60, by solving the equation. 60 = + 60+ Eercise #5: You have taken 5 quizzes in your history class, and your average score is 8 points. You think you can score 95 points on each remaining quiz. How many quizzes do you need to take to raise your average quiz score to 90 points? Justify your answer. Eercise #6: Consider the rational equation. R y a. Solve this equation for R and simplify. b. Solve this equation for y and simplify. emathinstruction, RED HOOK, NY 57, 05
Eercise #7: Anne and Maria play tennis almost every weekend. So far, Anne has won out of 0 matches. a. How many matches will Anne have to win in a row to improve her winning percentage to 75%? 7 b. How many matches will Anne have to win in a row to improve her winning percentage to 90%? c. Can Anne reach a winning percentage of 00%? Eplain. emathinstruction, RED HOOK, NY 57, 05
8 LESSON #9 WORKING WITH RATIONAL EQUATIONS COMMON CORE ALGEBRA II HOMEWORK. You can clean a park in hours. Working together, you and your friend can clean the park in. hours. a. Let t be the time (in hours) your friend would take to clean the park when working alone. Rate Time Part You park hours. hours Friend. hours b. Eplain what the sum of the epressions represents in the last column. Write and solve an equation to find how long your friend would take to clean the park when working alone.. Two water pumps can fill a tank with water in 6 hours. The larger pump working alone can fill the tank in 9 hours. How long would it take the smaller pump, working alone, to fill the tank? Rate Time Part. Bob can paint a fence in 5 hours, and working with Jen, the two of them painted the fence in hours. How long would it take Jen to paint the fence alone? Rate Time Part emathinstruction, RED HOOK, NY 57, 05
4. Working together, it take Sam, Jenna, and Fred two hours to paint one room. When Sam is working alone, he can paint one room in 6 hours. When Jenna works alone, she can paint one room in 4 hours. Determine how long it would take Fred to paint one room on his own. Rate Time Part 9 5. If two pumps can fill a pool in one hour and 0 mins., and one pump can fill the pool in two hours and 0 mins. on its own, how long would the other pump take to fill the pool on its own? Make your own table. 6. So far in your volleyball practice, you have put into play 7 of the 44 serves you have attempted. Find the number of consecutive serves you need to put into play in order to raise your serve percentage to 90%, by solving the equation 90 = 7+ 00 44+ 7. Solve for. Show your work. z y 8. John averaged 77 on the first four tests of the semester in his math class. If he scores 84 on each of the remaining tests, his average will be 80. Which equation could be used to determine how many tests, T, are left in the semester? 08+84T 4T = 80 ) 08+80T 4T = 84 ) 08+84T T+4 = 80 ) 08+80T T+4 = 80 emathinstruction, RED HOOK, NY 57, 05
0 LESSON #40 - SOLVING SQUARE ROOT EQUATIONS COMMON CORE ALGEBRA II Equations involving square roots arise in a variety of contets, both applied and purely mathematical. As always, the key to solving these equations lies in the applications of inverse operations. The key inverse relationship in these equations is that between taking a square root and squaring. Eercise #: Solve each of the following square root equations, which are arranged from less to more comple. Check each equation. 7 (b) 5 (c) 4 (d) 4 0 (e) 5 7 (f) 5 4 6 Eercise #: Which of the following is the solution to 5? ().5 () 50 () 5 (4) 4050 emathinstruction, RED HOOK, NY 57, 05
A more complicated scenario arises when a square root epression is equal to a linear epression. The net eercise will illustrate both the graphical and algebraic issues involved. Eercise #: Consider the system of equations shown below. y and y Solve this system graphically using the grid to the right. y (b) Solve this system algebraically for only the -values. (c) Why does your answer from part contradict what you found in part (b)? Oftentimes, roots are introduced by various algebraic techniques that for one reason or another are not valid solutions of the equations. These roots are known as etraneous and can always be found by checking within the original equation. Eercise #4: Find the solution set of each of the following equations. Be sure to check your work and reject any etraneous roots. (b) 6 emathinstruction, RED HOOK, NY 57, 05
Eercise #5: Solve each of the following equations for all values of. As in Eercise #, be sure to isolate the square root epression first before squaring both sides of the equation. Check your possible solutions in the original equation. Reject any etraneous roots. 6 6 (b) 5 5 emathinstruction, RED HOOK, NY 57, 05
FLUENCY LESSON #40 - SOLVING SQUARE ROOT EQUATIONS COMMON CORE ALGEBRA II HOMEWORK. Solve each of the following square root equations. As in the lesson, they are arranged from lesser to more comple. Check your answers. a) 5 b) 0 c) 6 d) e) 5 f) 4 4 g) 0 5 5 h) 7 emathinstruction, RED HOOK, NY 57, 05
4. Which of the following values solves the equation 4 9? () 9 () 4 () (4) 4. Solve each of the following equations for all values of. Check your possible solutions in the original equation. Reject any etraneous roots. (b) 4 6 6 (c) 6 47 8 (d) 6 4 4 emathinstruction, RED HOOK, NY 57, 05
LESSON #4 - REASONING ABOUT RADICAL AND RATIONAL EQUATIONS COMMON CORE ALGEBRA II In previous lessons we have looked at solutions to square root (radical) and rational (fractional) equations. Sometimes, the solutions to these equations introduced etraneous roots that needed to be rejected. In this lesson we will look to justify the steps in solving these types of equations and understand why etraneous roots are introduced. First, let s review two basic properties of equality. Eercise #: Give a reason or cite a property for each of the following lines in the solution of the square root equation 6. 6 6 6 6 6 6 6 0 0 0 or 0 or PROPERTIES OF EQUALITY. The Addition Property: If and then.. The Multiplication Property: If and then.. The Squaring Property: If then. Note, this is not true in reverse. Eercise #: Check each of the values of from Eercise #. Show your check. Which root is etraneous? Does the etraneous root satisfy the equation 6? 5 The etraneous root gets introduced because the operation of squaring is irreversible, meaning that once we ve squared, we cannot know the original quantity. Let s take a look at another eercise. Eercise #: Consider the relatively easy equation 6. Solve this equation by using the reversible (b) Solve this equation by first squaring both operation of division. sides. What etraneous root has been introduced? emathinstruction, RED HOOK, NY 57, 05
So, in the case of solving square root equations, because we square both sides at some point to remove the radical, we sometimes introduce solutions that do not make our original equation true. Let s investigate why it happens sometimes with rational equations. 6 Eercise #4: Consider the rational equation. Justify each step of the solution to this equation by citing a property or some other reason. Some steps of algebra have been omitted for the sake of space. 6 6 6 4 6 0 6 0 0 or 0 or Eercise #5: Which of these two solutions is etraneous? How can you tell? 6 The etraneous root is arising in this case because multiplying both sides of an equation by anything containing a variable, like the factors and, is irreversible due to the fact that we could be multiplying both sides by zero. Let s investigate this by solving another equation. Eercise #6: Consider the very simple equation 5 5 whose only solution is. Solve this equation by first multiplying both sides of this equation by. What etraneous root has been introduced? emathinstruction, RED HOOK, NY 57, 05
The following problems involve proving identities with rational epressions. Recall, identities are equations that are true for all values of the variable. Rational identities are true for all values of the variable EXCEPT for the restricted values. These restrictions will be stated in the problem. Remember, when proving any identity, you cannot use algebraic manipulation on both sides of the equation. You must work with each side of the equation separately. Eercise #7: Algebraically prove each of the following identities. 7 a a a a where, (b) 6 6 4 where,, The following identity can be proved in two different ways. See if you can do both of them. (c) 4 7 5 5 4 4 emathinstruction, RED HOOK, NY 57, 05
LESSON #4 - REASONING ABOUT RADICAL AND RATIONAL EQUATIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Solve the following equation involving a square root. Be sure to reject the etraneous solution (and there will be one). 0 8. Solve the following rational equation. Reject any etraneous roots. 7 4 REASONING. Consider the square root equation. Show that 4 is a solution to this equation. (b) The value is not a solution to the original equation. Show that after squaring both sides, is a solution to this new equation. emathinstruction, RED HOOK, NY 57, 05
9 4. Given the equation 7 answer the following. Solve this equation for the one and only value of that is a solution. (b) What etraneous root is introduced if the first step taken to solve the equation is squaring both sides? Show the work that leads to this etraneous root. 5. Consider the equation 4, for which is the only solution. If Dakota begins to solve the problem in the following way, what property could Dakota use to justify the unusual move of multiplying both sides by the epression 6? 4 6 6 4 4 7 0 4 7 0 8 (b) Solve the equation both sides? 0 8. What etraneous root was introduced by multiplying by 6 on 6. Squaring both sides of an equation is irreversible. Is cubing both sides of an equation reversible? Provide numerical eamples to help support your answer. emathinstruction, RED HOOK, NY 57, 05
40 7. Algebraically prove each of the following identities. y 5 y 5 5 y y 5 where y 5,5 (b) 5 7 6 6 where 6 emathinstruction, RED HOOK, NY 57, 05