A. Real numbers greater than 2 B. Real numbers less than or equal to 2. C. Real numbers between 1 and 3 D. Real numbers greater than or equal to 2
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1 39 CHAPTER 9 DAY 0 DAY 0 Opportunities To Learn You are what ou are when nobod Is looking. - Ann Landers 6. Match the graph with its description. A. Real numbers greater than B. Real numbers less than or equal to C. Real numbers between and 3 D. Real numbers greater than or equal to E. Real numbers less than F. Real numbers between and 3, inclusive 7. List ALL of the following values of n that make this inequalit a true statement: 3n 8 6 a) n = 0 b) n = 9 c) n = 8 d) n = 7 e) n = COPY and multipl (epand): 3 a) 3 4 b) 6 6 c) If 7 4, find the corresponding values for if: a) = b) = 0 c) = 4 d) =
2 Graph Of The Da (G.O.T.D.) In this chapter, ou will have a G.O.T.D. each da. These will all go on the same piece of graph paper (until ou run out of space, of course). Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 0 s equation: ( equals the absolute value of ) a) Label this as Chapter 9 DAY 0 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) How would ou describe the shape of this curve?. ACT If 8c 5d 3, then d 8 8 8c 3 8c 3 A) c 3 B) c 3 C) 3c 3 D) E) DAY SOLVING LINEAR INEQUALITIES When we solved linear equations, usuall there was just one solution, ie. one number that made the equation true. Sometimes there was no solution and sometimes all Real numbers was the solution. When solving linear inequalities, most of the time there are an infinite number of solutions but not all real numbers, necessaril. So to illustrate the numbers that are solutions, we draw a number line representation of all the solutions (a graph). Eample. a) This is the graph of all real numbers that are less than, Or written as an inequalit: b) This is the graph of all real numbers that are greater than or equal to 5. Or written as an inequalit: 5
3 33 c) This is the graph of all real numbers that are between and 3, including 3. Or written as an inequalit: n 3 To solve an inequalit means to find all values for the variables that make the inequalit true. We will alwas write our solutions BOTH as inequalities and as graphs. Also we will alwas show that our answers do, in fact, check. Eample. Solve, graph, and check: 8z z 6 Solution: 8z z 6 6z 8 6z z 3 This is the algebraic solution This is the graphical solution: CHECK? z = 5 is a point that is definitel on the graph. It should satisf the original inequalit. Go back and substitute into the original to see that it does make the inequalit true. 8( 5)? ( 5) 6 40? 0 6 8? 6 Keep in mind that the sign means is to the left of on the number line. Since 8 is to the left of 6 on the number line, THIS IS TRUE!!! So most likel our solution is correct. At least the graph is pointing in the correct direction. ALTERNATE WAY TO CHECK: The easiest number to substitute is usuall zero, 0. In this case 0 is not part of the solution set and it should not make the original inequalit true. Let s tr it. Substituting 0 in for z: 8(0)? (0) 6? 6 This is false. So 0 does not belong as part of our solution -- and it isn t! So again, most likel, our answer is correct. At least the graph is pointing in the correct direction.
4 33 Eample 3. Solve, graph, and check: 3n In the solution to the above eample, ou had to remember the rule: When ou multipl or divide each side of an inequalit b a NEGATIVE number, ou must reverse (flip) the inequalit sign. Eample 4. Solve, graph, and check: DAY Opportunities To Learn REVIEW BLOCK. COPY and simplif (no calculators allowed): a) 7 7 b) c) 6 ( 5). COPY and rewrite: 3? decimal?% 4 3. COPY and simplif: 3(4 p q) ( q p) 4. Use a calculator to evaluate. a) Write our answer as an eact fraction, if possible. b) Write our answer as a decimal rounded to four decimal places. 3 c d e for c 3; d 4; e 5 c d 5 0. a) Describe in words what numbers are graphed. b) Write as an inequalit.
5 Solve, graph, and show that our solution does check. 5. 7n 3n 3 6. a a a 5 a c 7 c n n Graph Of The Da (G.O.T.D.) In this chapter, ou will have a G.O.T.D. each da. These will all go on the same piece of graph paper (until ou run out of space, of course). Clearl label the da number, the equation, the table, and the graph on the graph paper. Put this on the same graph paper as Da 0 s G.O.T.D. Da s equation: a) Label this as Chapter 9 DAY b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) How would ou describe the shape of this curve?. ACT If 9 5 4, then? A) B) C) 4 D) 4 E) OGT Which of the following is a correct statement? A) B) C) D)
6 334 DAY AND & OR REVISITED You will recall that in mathematics and means intersection (what is in common ) and or means union (which means to combine ). We are going to use these ideas when dealing with compound inequalities. Eample. Graph: or 5 Eample. Graph: and 5. Also rewrite as a single inequalit. Eample 3. Rewrite in words in two different was: 3 z 9 Eample 4. Solve, graph and check. Write as a single inequalit if possible. 3 5 or 3 5 Eample 5. Solve, graph and check. Write as a single inequalit if possible. n 5 9 and n 5 Eample 6. Solve, graph and check. Write as a single inequalit if possible. 3 4a Finall: we will be discussing absolute value in the net section. Let s at least introduce this important, but often misunderstood, definition. ABSOLUTE VALUE This is a simple concept but it is difficult to put into words The formal definition: if 0 if 0 This is read: The absolute value of a number ( ) is equal to itself if is positive or zero. But the absolute value of a number ( ) is equal to its opposite if is negative. This definition often takes a while to understand. But stick with it.
7 335 DAY Opportunities To Learn If it weren t for the last minute, a lot REVIEW BLOCK of things wouldn t get done. - Michael S. Tralor. COPY and simplif (no calculators allowed): 6 3 a) 6 ( 6) b) c) 8 ( 9). COPY and multipl (epand): a) ( 4)( 5) b) ( c )( c ) 3. COPY and simplif: ( 3 ) (4 ) 4. Use a calculator to evaluate. a) Write our answer as an eact fraction, if possible. b) Write our answer as a decimal rounded to four decimal places. 3 c d e for c ; d ; e 3 c d 5. COPY and complete: In mathematics, the word and means while the word or means. 6. Rewrite in words two different was: < < 8 7. Graph and check. 7a) or b) Can this be written as a single inequalit? If so, do it. 8a) or b) Can this be written as a single inequalit? If so, do it. 9a) and b) Can this be written as a single inequalit? If so, do it. 0a) 3 and 5 b) Can this be written as a single inequalit? If so, do it.. 5 a. 4 a Everthing is funn as long as it is happening to somebod else. - Will Rogers
8 Solve, graph and show that our solution does check. Write our answer as a single inequalit if possible or and or a and a 9. a) Cop the definition of absolute value onto our paper. b) Write out clearl in words how that definition is read. 0. Graph Of The Da (G.O.T.D.) In this chapter, ou will have a G.O.T.D. each da. These will all go on the same piece of graph paper (until ou run out of space, of course). Clearl label the da number, the equation, the table, and the graph on the graph paper. Da s equation: a) Label this as Chapter 9 DAY b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) How would ou describe the shape of this curve?. ACT Which of the following statements BEST describes the solution set for 3( 4) 3? A) = 3 onl B) = 0 onl C) = - onl D) There are no solutions for E) All real numbers are the solutions for. OGT Which of the following is a correct statement? A) B) C) D) The entire sum of eistence is the magic of being needed b just one person. - VII Putnam
9 337 DAY 3 ABSOLUTE VALUE: SOLVING EQUATIONS The absolute value of a number is the number s distance from the origin on a number line. It is alwas positive or zero never negative Eample. a) Solve b guess and check : 3 5 b) Solve the same equation using the definition of absolute value. c) Check graphicall. Eample. Solve and check: n 6 0 Eample 3. Solve and check: 5 7 DAY 3 Opportunities To Learn REVIEW BLOCK. COPY and simplif (no calculators allowed): 4 9 a) 8 ( 8) b) c) 6 5 ( 5). COPY and multipl (epand): a) ( a )( a 7) b) ( 6)( 6) 3. COPY and simplif: ( ) ( ) 4. Use a calculator to evaluate. a) Write our answer as an eact fraction, if possible. b) Write our answer as a decimal rounded to four decimal places. 3 c d e for c 0; d 4; e c d 5. Rewrite what this means without using absolute value notation: 3 Do not solve. 6. How man solutions do absolute value equations usuall have?
10 Solve and show that our solutions do check n c c n G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 3 s equation: a) Label this as Chapter 9 DAY 3 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) How would ou describe the shape of this curve? 8. ACT Which of the following statements BEST describes the solution set for 6 ( 5)? A) = 0 onl B) = onl C) = onl D) There is no solution E) All real numbers 9. OGT Erika knows that 3 pints of paint will cover an area of 00 square feet. She needs to paint an area of 800 square feet. Which of the following proportions could she use to find out how man pints of paint she should bu? 00? ? 3 00 A) B) C) D) ? ?
11 339 DAY 4 ABSOLUTE VALUE: INEQUALITIES Not onl should ou be able to understand absolute value equations, ou should also be able to solve and graph absolute value inequalities. Eample. a) Graph all values of that make the following inequalit true. b) Rewrite this set of numbers without using absolute value notation. Eample. a) Graph all values of that make the following inequalit true. b) Rewrite this set of numbers without using absolute value notation. SUMMARY: Cop and complete: 3 z a means? Its graph would be? b means? Its graph would be? c means? Its graph would be? You need to know these, like, reall well. Help? Courtes of Mr. Iacone: EG or LESS and = > < Eample 3. Solve, graph and check: 3 7 Eample 4. Solve, graph and check: 3 5 z Eample 5. Rewrite as an equivalent statement with OUT using absolute value. Do not solve. a) 4a 3 0 b) 3 j 5 c) 6k 7 I usuall don t take ver long to stud for these tests because I alwas complete and correct m homework. When I do this, I am learning more than if I would cram the night before the test. - Kevin Burnworth, former student, now an engineer for Ohio Edison
12 340 DAY 4 Opportunities To Learn REVIEW BLOCK. COPY and simplif (no calculators allowed): 3 3 a) b) c) COPY the figures and find the EXACT values for w,,, z. a) b) 3. COPY and epand (multipl): a) ( c 8)( c ) b) ( 7)( 7) 4. Solve and check: 0 n 5 8. COPY, rewrite as an equivalent statement without absolute value notation, and GRAPH z 8. a 5 9. COPY and rewrite as an equivalent statement without absolute value notation. Do NOT solve or graph w n Solve, graph, and check. Write our answer as a single inequalit, when possible. 3. 3n 6 4. n 5 5. a c
13 34 9. G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 4 s equation: a) Label this as Chapter 9 DAY 4 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these nine points and connect them with a smooth curve. 0. ACT The area of a triangle is given b the formula: A b h. If the base is cm and the area of the triangle is 0 sq cm, find the length, in cm, of the height? A) 5 B) 0 C) 0 D) 48 E) 08. OGT Leticia knows that 5 pints of paint will cover an area of 70 square feet. She needs to paint an area of 900 square feet. Which of the following proportions could she use to find out how man pints of paint she should bu?? 70 70?? A) B) C) D) ? 5 DAY 5 RELATIONS & FUNCTIONS A relation is an set of ordered pairs. Eample. C = { (, ), (3, 5), ( 6, 0) } is a relation with three ordered pairs. E = { (, ): = 3 } is a relation with an infinite number of ordered pairs. This is read: The set of all ordered pairs, such that the -coordinate is three times the -coordinate.
14 34 The set of all first coordinates in a relation is called its domain. The set of all second coordinates in a relation is called its range. Eample. In relation C on the previous page: Domain = {, 3, 6 } Range = {, 5, 0 } In relation E on the previous page: Domain = (all real numbers) Range = (all real numbers) A function is a relation with the requirement that no first coordinate can be paired with more than one second coordinate. The concept of function is EXTREMELY IMPORTANT in the stud of mathematics. We need to get a good foundation here. Eample 3. a) State the domain and range for each. b) Which of the following is a function? Eplain. P = { (, 4), ( 3, 6), (8, 5) } Q = { (3, 7), (4, 6), (3, 8), (, ) } R = { (, 8), (3, 8), (5, 8) } FUNCTION NOTATION f( ) The smbol: f( ) is read f of or f at Eample 4. f ( ) is called the function s rule. It sas that for ever number put in for, we are to multipl that number b 3 and then add 5. f (4)? f (4) f ( ) f(4) 7 f( )
15 343 DAY 5 Opportunities To Learn REVIEW BLOCK. COPY and simplif (no calculators allowed): a) b) c) COPY the figures and find the EXACT values for w,,, z. a) b) 3. COPY and epand (multipl): a) ( 5)( 3) b) ( 3)( 3) 4. Solve and check: k COPY and complete: 5. A relation is. 7. The range of a relation is. 6. The domain of a relation is. 8. A function is a) State the domain b) State the range c) Is this a function? Eplain. 9. { ( 3, 4), ( 0, ), ( 6, ), ( 9, 5) } 0. { ( 4, ), ( 6, 0 ), ( 9, ) }. { ( 4, 7), (6, 5), ( 3, 7), ( 0, 5) }. { (, 3), (, 5), ( 4, 3), (7, 5) } 3. { (4, ), (5, 6), (4, ) } 4. { (, 8), (5, 3), (, 0) }
16 Evaluate the functions. Show each step. 5. f ( ) 4 a) f (3) b) f (0) c) f ( ) 6. g( ) 5 a) g(3) b) g(0) c) g( ) 7. h( ) 3 a) h(5) b) h(0) c) h( 3) 8. f ( ) a) f (4) b) f (0) c) f ( 5) 9. g ( ) 4 a) g() b) g(0) c) g( 4) 0. h ( ) 5 a) h() b) h(0) c) h( 4). G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 5 s equation: a) Label this as Chapter 9 DAY 5 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) What is the shape of this curve?. ACT The area of a triangle is given b the formula: A b h. If the height is 6 cm and the area of the triangle is 48 sq cm, find the length, in cm, of the base? A) 4 B) 8 C) 6 D) 4 E) 45
17 OGT In esterda s basketball game, Barbara scored seven of her team s 56 points. What percent of the total points did Barbara make? (to the nearest tenth of a %) A) 8.0% B).% C).5% D) 4.3% DAY 6 FACTORING *** To factor means to write an epression as a product, that is, as a multiplication problem. Make sure ou understand what it means to factor!!! Eample. Factor. (That is, write as a product) We will be factoring polnomials, that is, rewriting polnomials as products as epressions that are multiplied. Eample. Factor: 6 We wish to write 6 as a multiplication problem. Right now it is an addition problem. Actuall there are several was to write 6 as a product, just as could be factored in man was. Some possibilities include: 6 = (3 6 ) 3( 4 ) (6 ) None of the above epressions factors factorization: 6 6 ( ) 6 completel. The complete 6 is called the G.C.F., Greatest Common Factor NOTE: You can alwas check our factorization b multipling our answer. Eample 3. Factor completel: Eample 4. Factor completel: Eample 5. Factor completel: 5 AND CHECK! a 7 AND CHECK! a 8 AND CHECK! Parents often talk about the ounger generation as if the didn t have anthing to do with it. - Anonmous
18 346 DAY 6 Opportunities To Learn We must accept finite disappointment, but REVIEW BLOCK never lose infinite hope. - Martin Luther King, Jr.. COPY and simplif (no calculators allowed): 3 a) 0 b) c) COPY the figures and find the values for w,,, z. a) b) 3. A triangle with all sides equal is called a(n) triangle. 4. Solve and check: 3 a 5. COPY and complete: To factor an epression means to. 6. Factor 4 in at least three different was a) What is the G.C.F.? b) Factor completel. 8a 7 a a) What is the G.C.F.? b) Factor completel Factor completel. Double underline our final answer. Show that it checks n 7n 0. p 8p 5 3. r 9 4. q a a 4 8. b 4b 4 9. d d 8 0. c 7c 30 In spite of the cost of living, it s still popular. - Kathleen Norris
19 347. G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 6 s equation: a) Label this as Chapter 9 DAY 6 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) What is the shape of this curve?. ACT Compute the tan P : A) B) C) D) E ) OGT Doug scored 4 of his team s 5 points in a basketball game. What percent of the total points did Doug make? (to the nearest tenth of a percent) A) 4% B) 7.7% C) 3% D) 76.9% DAY 7 FACTORING II, The Sequel As ou saw during the previous assignment, factoring is fun b itself. But we are going to need to know how to factor in order to solve quadratic equations like: 4. But that is for another da (like tomorrow ). We still need to further review how to write an epression as a product, that is, how to factor. Eamples. Factor completel and check n n c c a a a
20 348 DAY 7 Opportunities To Learn The qualit of a person s life is in direct proportion REVIEW BLOCK to their commitment to ecellence, regardless of their chosen field of endeavor. - Vince Lombardi. COPY and simplif (no calculators allowed): (Football coach) a) 8 b) c) COPY the figures and find the values for w,,, z. a) b) 3. A polgon with 8 sides is called a(n). 4. Solve and check: 40 4 a 5 6. Factor completel. Double underline our final answer. Show that it checks z z 7. n 8n p p. a 7a 5. c 9c c c c d d d 7. Solve, graph and check: Graph: 4 9. Solve, graph and check: 3 0. g ( ) 3 5 Compute: a) g(4) b) g(0) c) g( )
21 349. G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. Da 7 s equation: a) Label this as Chapter 9 DAY 7 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) What is the shape of this curve?. ACT Compute the cos P : A) B) C) D) E ) OGT A computer password consists of four characters. The characters can be an one of the 6 letters of the alphabet. Each character can be used more than once. How man different passwords are possible? A) 04 B) 4,950 C) 358,800 D) 456,976 DAY 8 SOLVING QUADRATIC EQUATIONS First of all: What is a quadratic equation? It is an equation such that the highest power on the variable is. So far we have solved linear equations the highest power on the variable is. Let s progress to solving second power equations. Before we get to solving quadratic equations, we must consider one basic fact about numbers. Answer the following question: If two quantities are multiplied and the result is zero, then what can ou conclude about one of the two quantities?
22 350 This idea is summarized b the following theorem from algebra: If a b 0, then a 0 or b 0 ( or both). Eample. Solve and check: First get one side to be = 0 legall: 0 Then factor COMPLETELY: ( 4)( 3) 0 Set each factor = 0: 4 0 or 3 0 If a b 0, then a 0 or b 0 ( or both). Solve each equation: 4 or 3 CHECK back into the original equation: The solution set is 4, 3 4 4? 6 4 YES! ( 3) ( 3)? 9 3 YES! Eample. Solve and check: 5 Eample 3. Solve and check: Eample 4. Solve and check: NOTE: Since ou are able to check our solutions, ou MUST do so. If our solutions do in fact check, great! But if our solutions do not check, then ou are epected to go back, find our error, and make the appropriate corrections. Then check those solutions. If ou are unable to find the error, then write M solutions are incorrect but I can t find the error. Otherwise ou will be penalized. Do not let what ou cannot do interfere with what ou CAN do. - John Wooden, former UCLA basketball coach
23 35 DAY 8 Opportunities To Learn REVIEW BLOCK Being right half the time beats being half-right all the time. - Malcom S. Forbes. COPY and simplif (no calculators allowed): a) b) c) COPY the figures and find the values for w,,, z. a) a = 6 b) a = 6 3. An polgon with 4 sides is called a(n). 4. Solve and check: 0 5 a 5. Eplain what makes an equation a quadratic equation. 6. A linear equation usuall has one solution. How man solutions does a quadratic equation usuall have? 7 0. Solve and check. Remember that there is a penalt if ou sa that our solutions check and the don t. 7. n 7n p p c 49. d 8 3. a 4a 4. b 6b n n a a c c
24 35. G.O.T.D. Clearl label the da number, the equation, the table, and the graph on the graph paper. ( ) Da 8 s equation: a) Label this as Chapter 9 DAY 8 b) COPY and complete the table: c) Neatl draw a suitable pair of aes and label the units. Plot these seven points and connect them with a smooth curve. d) What is the shape of this curve?. ACT Which of the following is equal to 0? A) 5 B) 0 C) 4 5 D) 0 E ) 0 3. OGT A computer password consists of three letters of the alphabet. No letter ma be repeated. How man different passwords are possible? A) 78 B) 5,600 C) 6,50 D) 7,576 DAY 9 IT S TIME TO FIND OUT JUST HOW MUCH YOU LEARNED RECENTLY Chapter 9 Sample Test 6. NO CALCULATOR ALLOWED FOR THESE PROBLEMS!. Which of the following is correct? A) B) C) D) Simplif.. 3 ( ) f ( ) 4 Compute: a) f (3) b) f (0) c) f ( )
25 a) Make a table on graph paper with at least 7 ordered pairs that satisf this equation. b) Neatl graph on graph paper. c) What is the shape of this curve? *** For the remainder of this practice test, a calculator is allowed. 7. Cop and complete: a) b means? Its graph would be? b) z c means? Its graph would be? 8. a) Describe the graph in words in two different was. b) Describe the graph using a single inequalit Solve, graph and check n n 7 or n a Solve and check: 3n 9 7. a) Write an eample of a relation that is not a function. b) Eplain wh it is not a function. 8. R = { (4, ), (6, 3 ), (0, ) } a) Domain =? b) Range =?
26 Factor completel and check: a a a Solve and check. NOTE: Since ou are able to check our solutions, ou MUST do so. If our solutions do not check and ou are unable to find the error, then write M solutions are incorrect but I can t find the error. Otherwise ou will be penalized n n c c 8. If 7 6 4, then? 9. Simplif: (3 c d) ( d 4 c) 30. Evaluate: a) As an eact fraction b) As a decimal rounded to 4 decimal places ac b a ; b ; c 3 b a 3. Write in simplest form: 4 A) B) 4 6 C) D) 6 E ) ( ) a) Make a table on graph paper with at least 7 ordered pairs that satisf this equation. b) Neatl graph on graph paper. c) What is the shape of this curve? 33. g ( ) 7 Compute: a) g(5) b) g( ) c) g( 3)
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