Chapter 0: Algebraic Concepts

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1 Chapter 0: Algebraic Concepts Eercise 0.. {,, z, a}. {,,,,...}. {,,,, } 7. {, 7,,,,, } 9. {: is a natural number greater than and less than 8}. Yes. Ever element of A is an element of B.. No. c A but c B.. D C since ever element of D is an element of C. 7. D A since ever element of D is an element of A. 9. A B and B A. (Also A B.). A B and B A. Thus, A B.. D E because E and D.. A and B are disjoint since the have no elements in common. B and D are disjoint since the have no elements in common. C and D are disjoint. 7. A B {, } since and are elements of each set. 9. A B since the have no common elements.. A B {,,,, }. A B {,,, } or A B B. For problems -, we have U {,,,..., 9, 0}. A {,, 9, 0} since these are the onl elements in U that are not elements of A. 7. B {,,,, 7, 9} A B {,,, 7} 9. A B {,,,,, 7, 8, 0} ( A B) {, 9}. A {,, 9, 0} B {,,,, 7, 9} A B {, 790,,,,,, }. B {,,,, 7, 9}, C {,,, 7, 9} A B {,,,, 7, 8} {,,,, 7, 9} {,,, 7} (A B ) C {,,,, 7, 9}. B {,,,, 7, 9}, A B {,,,, 7, 8} {,,,, 7, 9} {,,, 7} (A B ) C {,,, 8, 9, 0} {,,, 8, 0} {,, 8, 0} 7. A B {,, 7, 9} {,, 8, 9} {, 7} 9. A B {,, } {,,,,, }. a. L {9, 9, 9, 97} H {9, 9, 9, 9, 9, 97} C {90, 9, 9, 9, 97} b. L H c. C is the ears when the percentage change (from low to high) was less than or equal to %. d. H {90, 9} C {9, 9, 9} H C {90, 9, 9, 9, 9} H C are the ears when the high was less than or equal to 00 or the percentage gain was less than or equal to %. e. L {90, 9, 9, 9} L C {90, 9} L C are the ears when the low was less than or equal to 00 and the percentage gain was more than %.. a. From the table, there are 00 white Republicans and 0 non-white Republicans who favor national health care, for a total of 0. b. From the table, there are Republicans, and Democrats who favor national health care, for a total of 80. c. From the table, there are 0 white Republicans, and 0 white Democrats and 0 non-whites who oppose national health care, for a total of 0.

2 Chapter 0: Algebraic Concepts. a. The ke to solving this problem is to work from "the inside out". There are 0 aides in E F. This leaves 0 aides who speak English but do not speak French. Also we have aides who speak French but do not speak English. Thus there are aides who speak English or French. This means there are aides who do not speak English or French. U E F 0 0 b. From the Venn diagram E F has 0 aides. c. From the Venn diagram E F has 8 aides. d. From the Venn diagram E F has aides. 7. Since students take M and E but not FA, and take M and E, take all three classes. Since 9 students take M and FA and we have alread counted, there are taking M and FA which are not taking E. Since students take E and FAand we have alread counted, there is onl taking E and FA but not taking M also. Since 0 students take E and we alread have enrolled in E, this leaves taking onl E. Since students take FA and we alread have 0 enrolled in FA, this leaves taking onl FA. Since 8 students take M and we alread have enrolled in M, this leaves 7 taking onl M. Math 7 U Fine Arts Economics a. In the union of the courses we have students enrolled. Thus, there are 00 7 students who are not enrolled in an of these courses. b. In M E we have enrolled. c. We have students enrolled in eactl one of the courses. 9. (a) and (b) A B A AB B A + AB+ B + O + Rh + O U Supplementar Eercises. If U {,,,, }, A {,, }, and B {,, }, then ( A B ) is equal to which of the following? a. {} b. {, } c. {,, } d. {,,, } e.. Records at Midtown Universit show the following about the enrollments of 00 students: 70 take Math 0 take Histor 70 take English take Math and Histor take Histor and English 0 take Math and English 0 take all three How man take none of these three courses? a. 90 b. 0 c. 0 d. 0 e. 0. U A B Regions and are represnted b which of the following? a. ( A B) b. A B c. A B d. ( A B ) ( A B) e. ( A B ) ( A B)

3 Eercise 0.. Question to be answered: Is there enough information to tell how man students are in this dorm wing? Suppose that each student in a college dorm wing is taking at least one of the three courses, A, B, and C. Enrollments from the dorm are given in the table below. Course A onl B onl C onl A and B onl B and C onl A and C onl Number of students 7 8 () Can ou answer the question if there eactl two students taking all three courses? (Yes/No) () Can ou answer the question if eactl students are taking two or more of the three courses? (Yes/No) Additional question: How man students are in this dorm wing? Solutions to Supplementar Eercises. A B {, } {, } {,, } Answer: c. Math Math Histor English Histor English Step : Adding all the numbers in step, we have n(m H E) 0. Thus, 0 take none of the three courses. Answer: b. A B Region A B Region Answer: d. A B 7 8 C Yes, we can answer the question with (). Yes, we can answer the question with (). There are students in this dorm wing. Eercise 0. π. a. Note that π, where π is 0 0 irrational and is rational. The product of 0 a rational number and an irrational number is an irrational number. b. 9 is rational and an integer. 9 c.. This is a natural number, an integer, and a rational number. d. Division b zero is meaningless.. a. Commutative b. Distributive c. Multiplicative identit. < < > ( ) + 0 8( ) ( ) The entire line. (, ] 7. (, 0) 9.

4 Chapter 0: Algebraic Concepts. >. (, ) (, ) (, ). > and 0 (, ) 0 7. [, 0 ) [, ] [, ) 9. (, 0) ( 7, ) [(. ) ] a. $ $788.9 $088.9 b. 0.[ (088.9)] $8. c. Retirement: 0. 0( ) $. State ta Retirement $. Local ta 0. 0( ) $ Federal ta 0. ( ) $ 8. Soc. Sec. ta 0. 07( ) $ 8. 0 Total Withholding $. 7 Take-home pa $ a. 99: C. 9( ) +. 9 $. 09 C 0. 0( ) ( ) $. 997: C. 9( 7) +. 9 $. C 0. 0( 7) ( 7) $. 0 The second formula is more accurate. b. C 0. 0( ) ( ) $ 0. 9 Supplementar Eercises. The value of + is which of the following? a. 8 7 e Choose one of the following to replace in the statement + 0. Do not use a calculator. a. > b. c. < d. Cannot be determined. ( ) ( )? a. b. c. 0 d. e.. ( )? a. b. c. 0 d. e. Solutions to Supplementar Eercises Answer: c. + 0 Answer: c. ( ) ( ) ( ) Answer: e. ( ) Answer: a b. 8 7 c. d. Eercise 0.. ( ) ( )( )( ).

5 Eercise ( ) ( ). 9 ( ) 7. ( ) 0 9. z ( 7) 7 9. ( ). ( ) ( ) + 7. ( 8a b )( a b ) a b a a b b a b c b ac a b c 0 ac b 8 a c b. a. ( ) ( ) b. ( ) ( ) ( ) ( ) 7. c. d. ( ) 8 ( ) 9. ( ) 8. ( ) P 00, i 0., n S P( + i) n 00( + 0. ) 00( ) $. 8 I S P.8 00 $9.8. P 000, i 0., n S P( + i) n 000( + 0. ) 000(. ) $ I S P $ S,000, n, i 0. n P S( + i), 000( + 0. ), 000(. ) $ t D (. 0) Year a. t-value 8 0 b. Dividend $

6 Chapter 0: Algebraic Concepts c. 0 D (. 0) $. 7. a. t 0 9 Debt in billions $ b. The end of WWar II was a factor. 9. a. t b. 0 H (. 09) $ 0. billion c. H (. 09) $ 87. billion d. H (. 09) $. billion Supplementar Eercises (no calculators). Choose one of the following to replace in the 7 statement a. > b. c. < d. Cannot be determined. Choose one of the following to replace in the statement + ( + ). a. > b. c. < d. Cannot be determined. Simplif: ba ( b) aa ( + b) b a ab ( ). Find the value of ( + ).. Find the value of ( ). Solutions to Supplementar Eercises ( + + ) > Answer: a Eercise 0.. Since we have ( + ) < ( + ) Answer: c (Note: This problem is preparing ou for the question, Is ( + ) +? ) ba ( b) aa ( + b). b a ab ( ) ab b a + ab b a ab a + abb ab b a a b b a a b ( b a ) a b. ( + ) ( ) (Note that different methods were required for problems and.). / ( ) ( ) ( ) 8 /. ( ) 8

7 Eercise 0. 7 ( ) / 7. ( ) The square root of a negative number is not real. / / /. a. 8 ( 8) / b. ( 8) / ( 8) 8 ( ) ( ). 9 / 9 (. ) (. ). 70 /. m m 7. m n ( m n ) / m / n / 9. 7/ 7. / /. / / ( / ) + ( / ) /. z / z ( z / ) + ( / ) z / 7. / ( / ) ( / ) / / 9. / (/) ( /) / /. / ( / ) ( / ) / ( / 0) + ( / 0) / 0 / 0. / / ( / )( / ) / / ( ). / ( ) / /. / /. / a b b b b b a a a. 9 9 ( A ) A 9 A A / 7 ( R) R / 7 R R m m m m m 9. m m m m m m 7. / / 9. a. 7 7/ 7 R 8. I 0 0 b. I , 7, 7

8 8 Chapter 0: Algebraic Concepts 8. I0 0. c. 0. I k, t 0, q 0 98 t/ k q q0( ) 0/ 98( ) / 98( ) 7 kg ht 7. P P0 (. ) 000. ( ) 0, 000(. ) 0. 0, 000(. ) 9, 9 7. a. N 00( 0. 0) ( 07. ) ; at t 0 we have ( 07. ) 0. Thus, N 00( 0. 0) 0. b. ( 0. 7) N 00( 0. 0) ( 0. 0) 9 Supplementar Eercises (No calculators). / / / Find the value of ( 8 7 ).. / / 9 8 Find the value of. + ( ).? ( ) a. b. c. 9 d. e.. ( ) ( )? a. b. c. d. e. t Solutions to Supplementar Eercises / / / /. ( 8 7 ) 8 ( ) 7 / ( ) 9 / / ( ) ( ) + ( ( ) ) ( ) ( ) Answer: b. ( ) ( ) Answer: b Eercise a. The largest eponent is. The degree of the polnomial is. b. The coefficient of is. c. The constant term is 0. d. It is a polnomial of one variable.. 7 z a. The sum of the eponents in each term is and, respectivel. The degree of the polnomial is. b. The coefficient of z is. c. The constant term is zero. d. It is a polnomial of three variables;,, and z.. a. an n means a. b. a 0 (Term is 0 ) c. a

9 Eercise 0. 9 d. a 0, the constant term. 7.. When, ( ) ( ) 8 9. When and, ( ) ( ) ( ) ( ) +. ( pq 7p ) + ( pq + p ) pq p. ( m n + ) ( m + n + 8) m n + m n 8 m 7n. [ 8 ( q+ ) + q] [ 8q 0+ q] [ q] + q 7. [ ( ) + ( )] + [ ] + [ + ] ( )( 7 ). ( 9 ) ( ) r s r s r s rs. a ( + a + ab) a + a + a b. ( + )( ) ( )( ) ( + ) + ( )( ) ( ) + +. ( + )( ) ( ). ( 0. )( 0. + ) ( 0. ) ( ) m n+ m n+ m n. m n 8m n mn m n + + m n m n m n + m+ m n ( + ) + ( )() + ( )() ( ) ( ) ( ) ( ) + ( )( ) ( 0. )( ) ) Quotient: + +

10 0 Chapter 0: Algebraic Concepts +. + ) Quotient: / ( / + / ) / + / + / / / / 7. ( + )( ) + / 9. ( + )( ) ( ) ( ) / / /. ( ) [( ) + + ( + ) ] ( + ) ( + ) a. ( ) ( ) b. ( ) ( )( ) + ( ) ( ) +. R 7. a. 000 b. 00. c ( 000 ) d ( 000 ) or V ( )( 0 ) 7. a. Lengths decrease b one. Thus ( A) A. b. Width 0 Length c. ( B) 0A d. ( C) AB e. Your spreadsheet will give Length, Width 7. Supplementar Eercise. ( + ) ( )( + )? a b c a. Year t Ta load b. From the spreadsheet the ear is 007. d e [ ( + ) ]? a. 8+ b. 8+

11 Eercise 0. c. + d e. none of these. If, then +? a. b. + c. + d. + e.. ( )[ ( + ) ]? (Compare with problem.). When is ( + ) +? >? <? Solutions to Supplementar Eercises. ( + ) ( )( + ) ( + ) Answer: a. [ ( + ) ] [ ( + + )] [ ] Answer: d. ( ) ( ) + ( ) Answer: a. ( )[ ( + ) ] ( )[ ( + + )] ( )( ) ( + )( ) ( + 8). occurs when 0. > occurs when and have the same sign. < occurs when and have opposite signs. (Note: The purpose of the question is for the student to know that algebraicall ( + ) + +.) Eercise 0.. 9ab a b + 8b b( a a + b) ( + + ). ( 7 ) + ( ) 7 ( ) + ( ) ( )( 7 + ) 7. m + m ( m) + ( m) ( m) + ( m) ( m)( + ) ( + )( + ). ( )( + ) The factors and + give a sum of ( ) + ( ) ( )( 7+ ) ( ) 7. 9a b ( 7a) ( b) ( 7a+ b)( 7ab) 9. a ( 8) 7 The factors and give a sum of ( + 8) ( + 8) ( + 8)( ) b The factors 8 and give a sum of ( + ) + ( + ) ( + )( 9+ ). ( ). + 0 ( + ) ( + ) ( + )( ). 8 ( ) ( 7)( + ) ( + ) ( + ) ( ) 9. + The factors and give a sum of. + + ( + ) ( + ) ( )( + ). + ( + ) ( + )( ). 8 ( ) ( + )( )

12 Chapter 0: Algebraic Concepts The factors and give a sum of ( + ) + ( + ) ( + )( + ) The factors and give a sum of ( ) ( ) ( )( 9) or ( )( 9) 9. ( ) ( ) ( )( + ) ( )( + )( + ). 8 + ( ) + ( ) [( )( + )] ( ) ( + ). + ( )( ) ( + )( )( + )( ) ( + ) ( ) + ( ) ( ) + ( ) ( ) 9. ( )( + + ) ( ) ( + )( 9 + ). / + / / ( / + ) / ( + )? ( + ) Solutions to Supplementar Eercises ( )( 0 97) ( ) when 0. In general, the conclusion for algebraic purposes is ( ) + ( ) ( ) [ + ( )] ( ) [ + ( )] ( ) ( 7 ). + ( + ) ( + )? + / / 7. ( + )( ) + ( ) / / ( ) [( + ) + ( ) ] / ( ) [ + + ] / ( ) [ 7 ]? 7 9. P+ Pr t P( + rt). S cm m m( cm). a. In the form p we have p( 0, p). 0,000 00p b. If p 8, then 0, Supplementar Eercises. Using factoring find the value of Using factoring choose one of the following to replace in the statement. 8 a. > b. c. < d. Cannot be determined. When is + +?. Factor: ( ) + ( ). Factor: 8 ( + ) + ( + ). 8 ( + ) + ( + ) ( + ) [ 8 + ( + ) ] ( + ) [ 8+ ( + ) ] ( + ) [ ] (Note: The factoring in problems and is necessar in the second course of the two course sequence.) Eercise z z z ( ). 9 ( ) +. + ( )( ) ( )( )

13 Eercise ( ) + 9 ( + ) ( ) ( ) 8 ( ) ( ) ( ) ( )( + ) ( ) ( + ) ( + ) ( )( ) ( )( + ) ( 9) ( )( + ) ( ) ( )( ) ( )( ) ( )( + ) ( )( + ) ( + ) ( )( + ) ( ) ( )( ) ( + )( + ) ( )( ) ac a ac b d 7bd b d 7bd a c b bc ( ) + ( )( ) 7 ( ) ( )( ) ( 9) ( )( + ) ( ) ( )( + ) ( )( + ) + + ( )( + ) ( )( + ) + a a a a a a a a a a a a a ( a a+ ) aa ( ) a aa ( ) ( a ) aa ( ) a a a a ( + ) ( + ) a a + ( + ) ( + ) a+ a ( + ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( )

14 Chapter 0: Algebraic Concepts.. + ( )( + ) ( )( ) + ( )( + ) ( )( ) + ( ) ( + ) ( ) ( + ) ( ) ( + ) ( + )( ) ( )( + ) ( + )( + ) ( )( ) ( )( ) ( )( ) ( + ) + ( + 8) ( + ) 9 + ( + )( + )( ) ( + )( + )( ) ( ) () () + ( + ) ( + ) ( + ) + + ( + ) ( + ) ( + ) ( + ) ( ) + ( )( + ) + ( ) ( )( + ) ( )( + ) + 9 ( )( + ) ( )( + ) + 9 ( ) + 9. a b a b a a a a b a b a a or a a a b aa ( b) 9. a. ( ). b. ( + ) + a b a b ab b a ( ab) ab ab or b a ( + ) ( + ) ( + )( + ) ( + )

15 Eercise h + h + h + 7. h h + h + ( + h) ( ) h( + h + ) h h( + h + ) + h + bc ac ab a b c a bc b ac c ab bc + ac + ab abc. a. Avg. cost b. Total cost (Avg. cost)(number of units) SV + t + ( t + ) ( t+ ) + ( t+ ) 8 ( t + ) t + t+ 9+ t+ 98 ( t + ) t + 9t ( t + ) Supplementar Eercises +.? + a. ( ) + 7 b. ( ) ( + ) + ( + ) + 9 ( + ) + + Answer: b. ( + ) +. You can divide common factors onl. You cannot divide terms.. c. 7 ( ) d. 7 ( ) e. + 7 ( ) + +? ( + ) a. b. + + c. + d. Both b and c e. None of the above. Find the value of ( + ).. Wh doesn t +? Solutions to Supplementar Eercises. + ( ) ( ) + ( ) ( ) ( ) ( ) ( ) + 7 ( ) Answer: e Review Eercises. B {,,,,,, 7, 8}. Since ever element of A is also an element of B, A is a subset of B.. No. {: > }. A and B are not disjoint since each set contains the element.. A {,,, 9} B {,, 9} A B {,,,, 9}. {,,, 7, 8, 0} {,,,, 7, 8, 0} {,, 7, 8, 0}. A {,,, 9} B {,,,, 7, 8, 0} A {,,, 7, 8, 0} A B {,, 7, 8, 0} ( A B ) {,,,, 9}

16 Chapter 0: Algebraic Concepts 7. {,,, 7, 8, 0} {,, 9} {,,,, 7, 8, 9, 0} ( A B ) {,,,, 7, 8, 9, 0 } {, } A B {,,, 9} {,,,, 7, 8, 0} {, } Yes. 8. a. + + illustrates the commutative propert of addition. b. ( ) ( ) illustrates the associative propert of multiplication. c. ( + 9) + illustrates the distributive propert. 9. a. irrational b. rational, integer c. meaningless 0. a. π >. b. 00 < 0. c. >. ( ) ( ) ( ) 9( ) 0 ()()( ) ()() ( )( 0) 0 0. [ ( )] + [ ( )] + [ ( )] + [ + ] ( ) ( 9. ) a. [0, ], closed 0 b. [, 7), half open c. (, 0) open 0. a. (, ) < < a. b. [, 8] 8 c. < 0 8 b. c. d ( 7). a. b. 8 8 ( ) 0 9 c. ( ) ( )( ) 8 d. ( ) 8 ( )( ) e. ( ) There are other correct methods of working problems 8.. ( ) ( )( ) ( ) ( ). ( ) ( ) 8 ( )( ) 9 8 ( ) ( z ). ( ) z ( ) ( ) ( z ) 8 z 8 z

17 Review Eercises z z ( ) ( ) z ( ) z 7 7 z ( ) 9. a. ( ) ( ) b c a. / / b. c. / / / /. a.. a. / b. / c. b. ( ) / / ( / ) + ( / ) /.. / / ( 7/ ) / 7/ / ( / ) + ( / ) 7/.. / / 7/ / / / ( / )( / ) / 7. ( ) / / 8 8. ( ) ( ) ( ) a b 8a b 8 8a b 8 a b a 8a b a ( + ) ( + 7) + 7. ( ) + [ ( + )] + ( ) 7. ( ) + ( + + ) ( )( ) 9. ( )( ) ( )( + ) + +. ( + )( ) ( 7)( + ) ( ) ( ) ( )( ) ( + )( ) 9 Difference of two squares ( ) 8 + Binomial cubed

18 8 Chapter 0: Algebraic Concepts ) Quotient is ) Quotient is / / / / / ( ) ( )( ) ( ) ( ). + a a a ( a) a+ a 70. Two epressions whose product is ( ) 88 and whose sum is are and 9. So, + 9 ( + ) 8( + ) ( + )( 8) ( ) ( 9) + 9 ( 9) [( + )( )] ( + ) ( ) / / / 7. + (?) / / / / + ( + ) /? + 7. a. b. + ( + ) + ( ) ( ) ( ) ( ) ( )( + ) ( ) ( )( + )( ) ( )( + ) ( + )( ) ( + ). ( ). ( + ) ( + ) ( + ) [ ( + )] ( + ) ( ) ( + ) ( ) ( + ) ( + )( ). + ( ) ( ) + ( ). 9 ( + )( ) 7. 8 ( ) ( + )( ) 8. ( 7)( + ) 9. ( + )( )

19 Review Eercises ( + )( + ) ( )( + ) ( )( ) ( + )( + ) + ( ) 9 ( ) + ( + )( ) ( )( ) ( + )( ) ( ) ( )( + ) ( ) ( )( ) ( + )( ) ( + ) + ( )( ) ( )( ) ( ) ( + + ) ( ) ( )( ) ( ) ( ) + + ( )( + ) ( )( + ) ( )( ) ( ) ( + )( ) + ( )( + )( ) ( )( + )( ) ( )( + )( ) ( )( + )( ) ( )( + )( ) ( ) ( )( + )( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) + ( ) ( )( + ) ( + ) ( ) + ( + ) ( + ) ( + ) +

20 0 Chapter 0: Algebraic Concepts 8. a. R: Recognized C: Involved E: Eercised R 0 0 C E U 9 R: recognized E: eercise C: communit involvement b. 00 ( ) 0 So, 0 eercised onl. c ( + 0) 00 So, 00 eercised or were involved in the communit. n (. 8. S ) (. ) a. S( ) $. 7 (. ) b. S( 0) $, a. R 0, 000 n (. 00) n n n. 00 0, n 0. 00(. 00) 0, 000 n. 00 n (. 00) n b. R 0, (. 00) 8 (. 00) R Chapter Test. a. A {, 8} B {,, } A B {,,, 8} b. {, }, {, }, and {, } are disjoint from B. / 88. S ka a. S k A b. Let S be the number of species on 0,000 acres. Then S k 0, 000 Let S be the number of species on,000 acres. Then S k , 000. k 0, 000. S S. S 0, a. C 00 p 0, ( 00 p) 00 p 00p 00 p 00( 0) 0 b. If p 0, C The cost of removing no pollution is zero. 00( 98) c. C $, d. The formula is not defined when p 00. We are dividing b zero. The cost increases as p approaches 00. It is cost prohibitive to remove all of the pollution. c. {}, and {8} are non-empt subsets of A.. ( ) ( ) a.

21 Chapter Test b. 0, if 0 c. 0 d. ( ) or 0 7 7( ) 0 e. a a a a f. / / / g. / h.. a. / / b. or or. a. b.. a. ( ) 8 b. a b a b a b ab a a b ab c a. Degree is. b. Constant is 8. c. Coefficient of is. 8. In interval notation, (, ) (, ] (, ] 9. a. 8 ( ) 0 b. 0 ( )( + ) c. + ( )( ) d. ( ) ( )( + ) 0. A quadratic polnomial has degree two. (c) is the quadratic. ( ) ( ) ) Quotient: + +. a. ( 9 ) b. t ( t t ) t + 9t c d. ( )( ) e. ( m 7) m 8m+ 9 f. g. h. i. 9 ( ) ( + )( ) ( + ) ( )( + ) ( ) ( ) ( + ) ( )( + ) ( )( + ) ( )( + ). + or + ( + )

22 Chapter 0: Algebraic Concepts. Construct a Venn diagram: B U L D a. 0 students ate onl breakfast. b students skipped breakfast.. S ( ) ( ) (. 0) 87. In 0 ears, the future value will be about $87.. Etended Applications I. Campaign Management. 0, , 000 0, , , 000, 000 7, 000 7,000 voters read the newspaper but do not watch the news.. 7,000 newspaper,000 cable news,000 both,000,000 read the newspaper or watch the cable news or both.. Number of Voters Reached Total Cost Cost per Voter Reached Pamphlet,000 $,00 $0.90 Television 0,000 $0,000 $0.80 Newspaper 90,000 $7,000 $0.0 II. Pricing for Maimum Profit a. A B Price per unit Number of units A B 99 A B 98 A B 97

23 Etended Applications b. C Profit ( A 8)* B, 00 ( A 8)* B, 707. ( A 8)* B, 8 ( A 8)* B, 99.

24 Chapter 0: Algebraic Concepts c. Abbreviated table: Price per unit Number of Units Income Profit d. Price per unit $, Maimum profit $8,

Chapter R REVIEW OF BASIC CONCEPTS. Section R.1: Sets

Chapter R REVIEW OF BASIC CONCEPTS Section R.: Sets. The elements of the set {,,,, 0} are all the natural numbers from to 0 inclusive. There are 9 elements in the set, {,,,,, 7,, 9, 0}.. Each element of