Chapter P Prerequisites: Fundamental Concepts of Algebra

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1 Chapter P Prerequisites: Fundamental Concepts of Algebra Section P. Check Point Eercises. 8+ ( 8+ ( 8+ (0 8+ ( Since 00 is 0 ears after 000, substitute 0 for. T (0 + ( If trends continue, the tuition and fees will be $77. The elements common to {,,,, 7} and {, 7, 8, 9} are and 7. {,,,,7} {,7,8,9} {,7}. The union is the set containing all the elements of either set. {,,,,7} {,7,8,9} {,,,,7,8,9} π. 9,., 0, 0.,, 9, 0 a. Natural numbers: 9 because 9 b. Whole numbers: 0, 9 c. Integers: 9, 0, 9 d. Rational numbers: 9,., 0, 0., 9 π e. Irrational numbers:, 0 f. Real numbers: π 9,., 0, 0.,, 9, 0. a. Because., the number inside the absolute value bars is negative. The absolute value of when < 0 is. Thus, ( b. π Because π., the number inside the absolute value bars is positive. The absolute value of a positive number is the number itself. Thus, π π. c. Because > 0,. Thus, 7. ( 9 9 The distance between and is ( + + ( ( + + ( [7 ( ] + [7 + ] + [9 ] + Eercise Set P.. 7+ ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

2 Prerequisites: Fundamental Concepts of Algebra. 8 ( ( ( 7. 7 ( ( 9. + (9 7 + ( + ( ( 8 + ( + ( C ( F (8 ( C is equivalent to 8 F. h + 0t t + 0( ( + 0 ( Two seconds after it is kicked, the ball s height is 0 feet. h + 0tt + 0( ( + 80( Three seconds after it is kicked, the ball s height is 0 feet.. 8 (8 ( ( ( ( + (0 + (0 ( 0 7( 7(9 7( 7 (9 + ;, + ( + ( ; and ( ( C (0 ( C is equivalent to 0 F.. {,,, } {,, } {, }. {,, 7} {,, 8} { }. { set,, } { tes,, } { set,, }. { real,,, } { lear,,, } { real,,, }. {,,, 7} {,,, 8,0 } { } The empt set is also denoted b. or. {,,,7} {,, } { } 7. { abcd,,, } 8. { wz,, } 9. {,,, } {,,} {,,,, } 0. {,,7,8} {,,8} {,,,7,8}. {,,, 7} {,,, 8,0} {,,,,,, 7,8,0}. { 0,,, } {,,} { 0,,,,,,}. { aeiou,,,, } { aeiou,,,, }. { em,, pt,, } { em,, pt,, } Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

3 PreCalculus E Section P.. a. 00 b. 0, 00 c. 9, 0, 00 d. e. f.. a. 9 9,, 0, 0., 9., 00 9,, 0, 0.,, 9., 00 b. 0, 9 c. 7, 0, 9 d. 7, 0., 0, 9 e. 0 f. 7, 0., 0, 9, 0 7. a. b. 0, c., 0, d.,, 0, 0.7, e., π f. 8. a.,, 0, 0.7,, π, b. 0, c., 0, d., 0., 0, e. f., 0., 0,, Answers ma var. An eample is.. Answers ma var. An eample is.. Answers ma var. An eample is.. true; is to the left of on the number line.. false; is to the left of on the number line.. true; is to the right of 7 on the number line.. true; is to the left of on the number line. 7. true; π π 8. true; is to the right of on the number line. 9. true; 0 is to the right of on the number line. 0. true; 0 is to the right of on the number line π π. 7π 7π (. ( 7 7 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

4 Prerequisites: Fundamental Concepts of Algebra ; +, + inverse propert of multiplication 8. (. 8. ( ( ; inverse propert of addition ( 0 7. The distance is The distance is. 9. The distance is The distance is The distance is 9 ( The distance is ( The distance is. ( The distance is. ( ( ( + ; commutative propert of addition 7. ( ; distributive propert of multiplication over addition ( + 7 ( + + 7; associative propert of addition 78. ( (; commutative propert of multiplication 79. ( + + ( + ( + + ( + ; commutative propert of addition 80. 7(8 (87; commutative propert of multiplication 8. ( ; distributive propert of multiplication over addition 8. ( ( ( ( ( + ( ( + ( ( ( 8 9. ( ( ( 7[ + ] 7[ 8] ( + + ( 88 ; distributive propert of multiplication over addition Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

5 PreCalculus E Section P. 9. [8 ( ] [ 8 + ] [ ] 9. ( ( ( ( ( 98. ( ( ( ( + ( + ( [ ] ( ( 0. > Since >, > < 0 Since 0 < 0, 0 < Since 0. 0., Since.., < 0 Since 0<, < > Since > 0, > Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

6 Prerequisites: Fundamental Concepts of Algebra Since, > Since >, > [ ( (8 ] 8 [ ( (] 8[8] 8[ ] 8+. 8[ (7 ( ] 8[ ( (] 8[0] 8[ ] ( ( ( ( [ ( ] [9 ( ] 0 9 [9 + ] ( ( ( 7 ( ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

7 PreCalculus E Section P ( 0. ( 0 0. ( ( ( ( H ( ( The lower limit of the heart rate for a 0-earold with this eercise goal is 0 beats per minute. 9. a. H ( a 0 H ( 0 0 ( 00 0 The upper limit of the heart rate for a 0-earold with this eercise goal is 0 beats per minute. b. H ( a 0 H ( 0 0 ( 90 9 The lower limit of the heart rate for a 0-earold with this eercise goal is 9 beats per minute. 0. a. H ( a. a. 0 H ( 0 0 ( 90 The upper limit of the heart rate for a 0-earold with this eercise goal is beats per minute. b. H ( a T, , (7 (7, The formula estimates the cost to have been $, in 007. b. This underestimates the value in the graph b $. c.. a. T, , (0 (0,07 The formula projects the cost to be $,07 in 00. T, , ( (, The formula estimates the cost to have been $, in 00. b. This underestimates the value in the graph b $. c. T, , ( (,9 The formula projects the cost to be $,9 in 0.. a ( 0, b (000 $780 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

8 Prerequisites: Fundamental Concepts of Algebra. a. 0.0t + 0.(0 t 0.0t + 0.t 0.t b. 0.0(0 + 0.( (0. +. miles.. Answers ma var.. does not make sense; Eplanations will var. Sample eplanation: Models do not alwas accuratel predict future values.. does not make sense; Eplanations will var. Sample eplanation: To use the model, substitute 0 for. 7. makes sense 8. does not make sense; Eplanations will var. Sample eplanation: The commutative propert changes order and the associative propert changes groupings. 9. false; Changes to make the statement true will var. A sample change is: Some rational numbers are not integers. 0. false; Changes to make the statement true will var. A sample change is: All whole numbers are integers. true. false; Changes to make the statement true will var. A sample change is: Some irrational numbers are negative.. false; Changes to make the statement true will var. A sample change is: The term has a coefficient of.. false; Changes to make the statement true will var. A sample change is: + ( false; Changes to make the statement true will var. A sample change is:.. true 8. π > a...7 π.7.7 >.7. π > b.. a.. 7 b b ( b b b b( b b b b 0 b b ( b b b b b( b b b b b b c. add the eponents b. Section P. 7 b b 8 b b b b b b b b b b b b b b b b b b b b b b b c. subtract the eponents b It moves the decimal point places to the right. Check Point Eercises. a. ( ( ( ( b. ( c ( 7... <. <. 8 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

9 PreCalculus E Section P. d.. a. b.. a. b. ( ( ( ( ( ( ,00,000, , 0, 000, ( ( a. ( 7. 0 ( 0 7. b ( ( ( The average Pell grant was $00 in S (.7 0 [(. 0 r ] (.7 0 [(. 0 0 ]. The speed of the blood at the central ais of the arter is. centimeters per second. Eercise Set P ( 0 ( 7 ( ( ( ( ( ( 0 ( 0 ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

10 Prerequisites: Fundamental Concepts of Algebra ( (.... ( ( ( ( ( ( ( ( ( ( ( ( ( 9 ( ( 9( ( ( ( ( ( 7 7. ( ( ( ( ( (8 8 ( 8 ( ( ( a b a b a b a b a a b a b a b 7 7 7ab 7 a b a ab b 7 7 b b b 7 7 b b 7 7b 7 b b 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

11 PreCalculus E Section P b 0 b 0 0 0b 0 b b b 0 b 00 0 ( ( ( (0 0 ( ( ( ( ab b 0 0 a b a b a 7b 8 a 8 8 ( 0a b b 7 7 0a b a 0 b a 0 7b 9 a 0 a b ab 0 a b ab ,0, ,70, ,000,00, ,000,00,000, , Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

12 Prerequisites: Fundamental Concepts of Algebra ,000,000,000,000, ,000,000,000,000, ( ( ( 0 (. 0 (.( ( ( ( ( ( 90. ( ( ( ( ( 9. ( 8 (. 0 0 (. ( ( (. 0 (. 0 8 (..( ( ( ( ( ( , 000, 000, ( 0. 8, 000, 000, ( 0 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

13 PreCalculus E Section P ( 7. 0 ( (. 0 ( ( 0 ( ( ( 08. ( ( ( ( 8 7 z z z z z z z 7 7 z z 9z z z z z 8 0 ( z z ( z z 0 8 ( 7 8 z 0 z 0 8. ( ( ( ( 8 ( ( ( 0 (. 8 0 ( ( ( 9 ( 8 ( ( ( 8 (. a. b. c.. a. b. c $800 per American $77 per American Bo-office receipts were $9.7 0 in Bo-office receipts were $8.9 0 in 00. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

14 Prerequisites: Fundamental Concepts of Algebra , The mass is gram , The mass is. 0 9 gram Americans eat.0 0 chickens per ear.. das equals or 870 hours. 870 hours equals or,00 minutes.,00 min. equals,00 0 or,,000 seconds. 7 There are. 0 seconds in a ear.. 0. Answers ma var.. does not make sense; Eplanations will var. 9 7 Sample eplanation: ( not.. makes sense. does not make sense; Eplanations will var. Sample eplanation:. 0 represents over trillion. The entire world population is measured in 9 billions ( 0.. makes sense. false; Changes to make the statement true will var. A sample change is: >.. true 7. false; Changes to make the statement true will var. A sample change is: ( because. 8. false; Changes to make the statement true will var. A sample change is:. 9. false; Changes to make the statement true will var. A sample change is: false; Changes to make the statement true will var. A sample change is: ( false; Changes to make the statement true will var. A sample change is: (7 0 + ( 0 700, true. The doctor has gathered: So, is remaining. A C D. b MN, b M, b N A C D b b b A C+ D 70 bts 0 min hrs das. 80 rs min hr da r beats beats beats beats 9 The heartbeats approimatel.9 0 times over a lifetime of 80 ears.. Answers ma var. 7. a. 8 b. 8 c. 8. a b c a. + 0 b. + 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

15 PreCalculus E Section P. Section P. Check Point Eercises. a. 8 9 b. 9 c. d e a. 7 b.. a. b a ( b ( a ( + 7 b ( a. If we multipl numerator and denominator b, the denominator becomes 9. Therefore, multipl b, choosing for. 9 b. The smallest number that will produce a perfect square in the denominator of is because. So multipl b, choosing for. 7. Multipl b ( ( 8( 8( 8 or Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

16 Prerequisites: Fundamental Concepts of Algebra 8. a Eercise Set P. b c a (9..., The square root of a negative number is not real.., The square root of a negative number is not real. b c d. ( e ( 9. a. ( 7 7 ( 8. ( b. ( c. ( 8. a. ( / ( 8/ / 8/ ( / + (8/ / ( 0 0 b. ( ( 8 ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

17 PreCalculus E Section P ( ( ( ( ( ( ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

18 Prerequisites: Fundamental Concepts of Algebra ( ( ( ( ( ( 9 ( ( 7 ( 7 ( ( ( + ( 7( + 7( ( + ( ( + ( ( ( ( ( ( ( 8 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

19 PreCalculus E Section P ( 7 + ( 7 ( ( ( ( ( is not a real number is not a real number ( 7. (.. ( ( cannot be simplified further will not simplif / / / 8 8 / / 87. ( 8 8 / 88. ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

20 Prerequisites: Fundamental Concepts of Algebra / / / / ( / / / / /+ / 7/ ( ( 7 / / / / / / / 0 0 / / // / /+ / / / / / ( / / ( / / // / / / / / ( ( 9 / / 9/ / ( / ( / /0 /0 / /0 / /0 / / / / / / 0. / 0. / / / / ( / 9 ( / / / ( / 9 ( ( ( / / / / 7 7 ( ( / ( ( / / ( / + + ( / / / /. ( 8 (. ( ( / ( ( / ( /( ( /( / ( / / / ( / ( / / ( / / ( /( ( /( ( 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

21 PreCalculus E Section P.. / 7/ / / ( 7/ ( / ( / / ( /( ( /( (. a. In 00, we have According to the model, 7.% of was spam in 00. This underestimates the actual value shown in the bar graph b 0.%. b. In 0, we have According to the model, 9.% of will be spam in 0. This overestimates the value given in the bar graph b.%.. a. For 00: E For 00: E Difference: ( ( The difference is 0. b. 0.8 This underestimates the difference projected b the graph of b.7. This represents a difference of.7 million people. 8. R v a Rf c 0.9c R f c ( R 0.9 f R f R Ra 0.R 0.R 0.R R f f f f f If ou are gone for weeks, then 00 weeks will have passed for our friend. 9. Perimeter: P l+ w feet Area: A lw square feet + ( ( + +. About. to. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

22 Prerequisites: Fundamental Concepts of Algebra 0. Perimeter: P l+ w feet Area: A lw square feet. 8. Answers ma var. 9. does not make sense; Eplanations will var. Sample eplanation: The denominator is rationalized correctl. 0. makes sense. does not make sense; Eplanations will var. Sample eplanation: simplifies to + 0 and thus the radical terms are not common.. does not make sense; Eplanations will var. Sample eplanation: Finding the nth root first often gives smaller numbers on the middle step.. false; Changes to make the statement true will var. A sample change is: false; Changes to make the statement true will var. A sample change is: ( (. false; Changes to make the statement true will var. The cube root of 8 is the real number.. false; Changes to make the statement true will var. A sample change is: 7. ( + ( a > Calculator Check:.7 >. b > 7+ 8 Calculator Check:.888 > + 8. a. Her son is 8 ears old. b. Son s portion: ( ( Mom s portion: Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

23 PreCalculus E Section P (8 + (8 + ( + ( ( Section P. Check Point Eercises. a. b. ( ( ( 7 ( ( ( ( 9 7+ ( ( 9 7 ( ( 7 ( 9 ( 7 ( (( ( ( (7( ( + ( a. b. (7 ( (7 ( + (7 ( + ( ( + ( ( (+ ( + ( ( + ( + +. a. ( + + ( + ( + ( b. ( + + ( + + ( + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

24 Prerequisites: Fundamental Concepts of Algebra Eercise Set P.. es; + +. no; The term does not have a whole number eponent.. no; The form of a polnomial involves addition and subtraction, not division.. es; has degree has degree has degree 0 + has degree. has degree 7 has degree has degree has degree. has degree has degree 9 has degree has degree has degree has degree. has degree 8 has degree has degree 9 has degree has degree. ( (7 + ( ( + + ( 8 + (9 The degree is. + 7 ( ( ( ( + ( ( 7 The degree is. (7 + ( 9 8+ (7 + + ( The degree is. (7 + ( (+ 8 + ( + + ( (9 + 7 ( ( The degree is. (8 9 + ( + + ( 7+ + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

25 PreCalculus E Section P... ( ( + 7 ( ( ( ( + + The degree is ( ( 7 ( (8 + 7 ( ( + ( ( + + ( + The degree is (8 (7 ( ( + ( + ( + + ( ( ( ( ( ( ( (( + ( ( + ( ( + ( ( + ( ( + ( ( + ( ( (( + ( ( + ( ( + ( ( + ( ( + ( ( + ( ( ( + 7( ( + 8( ( ( + + ( ( (+ (+ + ( + ( (7+ (+ (7 ( + 7 ( + ( + ( ( (+ + ( ( + ( ( + ( ( ( (7+ ( (7 + ( ( + ( (7 + ( ( ( ( 7 ( ( + ( ( 7 + ( ( + ( ( (7 ( (7 ( + (7 ( + ( ( + ( ( ( 8 + ( ( 8 ( + ( 8 ( + ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

26 Prerequisites: Fundamental Concepts of Algebra ( + ( 9 ( + ( (+ ( ( 9 (+ ( ( ( 7 ( + 7 (7 9 ( ( + ( 9 ( + ( ( ( ( + ( ( ( 9 9. ( ( ( ( ( ( (+ ( + ( ( (+ ( + ( ( ( ( + 8+ ( ( ( ( ( ( ( ( (7 7 (7( + ( (9 9 (9( ( or ( ( (+ ( + ( + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

27 PreCalculus E Section P (+ ( + ( + ( ( ( + + ( ( ( + ( ( ( ( + ( ( + (7+ (7 + ( + ( (7 + ( ( ( + 9 (+ 7 ( + (7 + (9 ( + (9 ( ( (+ 7 ( + (7 + ( ( + ( ( ( (+ + ( ( + ( ( + ( ( + + ( ( + ( ( + ( ( + ( ( + ( ( + + (7 + ( (7 ( + (7 ( + ( + (( + 9 (7+ (7 + (7 ( + ( (9+ 7 (9 + (9 (7 + ( ( ( ( ( ( ( ( ( ( ( + + ( + ( + ( + ( ( + ( ( + ( ( + + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

28 Prerequisites: Fundamental Concepts of Algebra ( + ( + ( + ( + ( + ( + ( + ( (+ ( ( ( 9 (7+ (7 (7 ( 9 9 ( + + ( + ( ( + + ( + ( (+ 7 (+ 7 + (+ 7 ( (+ 7 (+ 7+ ( [ (+ ][ + (+ ] ( (+ ( [8 + (7 ][8 (7 ] (8 (7 ( ( + + ( + + ( ( + + ( + + ( + ( (+ + ( + + ( (+ + (+ + (+ ( + ( ( + ( ( + ( ( + ( ( ( ( + ( ( 9 ( ( + ( ( + ( ( + ( ( ( ( + ( ( 0 ( ( 7( ( ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

29 PreCalculus E Section P. 8. ( + ( 9 ( 7 ( ( 7 0 ( 7 7 ( 7 ( ( + ( ( + ( ( + ( ( + ( ( 88. ( + ( ( 9 + ( ( 9 + ( 9 ( 9 + ( 9 ( ( 7 ( 7 ( ( ( 7 ( 7 ( ( ( 7 ( ( ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

30 Prerequisites: Fundamental Concepts of Algebra 9. a. b. M M 77( + 88( + 707,99 The model estimates the median annual income for a man with ears of education to be $,99. The model underestimates the actual value of $7,0 shown in the bar graph b $. M W ( , ( ,7 M W , ,7 M W , +,7 M W + 7,99+,0 c. M W + 7,99+, 0 M W ( + 7(,99( +, 0,8 The difference in the median income between men and women with ears eperience is $,8. d., 0, 8 0,9 The actual difference displaed in the graph in the median income between men and women with ears eperience is $0,9. The model overestimates this difference b $,8 $0,9 $. 9. a. b. W 9+, W (8 9(8 +,,78 The model estimates the median annual income for a woman with 8 ears of education to be $,78. The model overestimates the actual value of $, shown in the bar graph b $. M W ( , ( ,7 M W , ,7 M W , +,7 M W + 7,99+, 0 c. M W + 7,99+,0 M W ( + 7(,99( +, 0, 8 The difference in the median income between men and women with ears eperience is $,8. d. 7, 0, 8, 9 The actual difference displaed in the graph in the median income between men and women with ears eperience is $,9. The model underestimates this difference b $,9 $, 8 $. 9. ( 8 ( 0 ( ( 8 ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

31 PreCalculus E Section P. 9. ( + 9( + ( + ( + ( ( + ( + ( + ( + ( Answers ma var. 0. makes sense 0. does not make sense; Eplanations will var. Sample eplanation: FOIL is used to multipl two binomials. 0. makes sense 0. makes sense, although answers ma var 07. ( + ( + (( + ( ( ( + ( + ( ( ( + ( ( + ( + ( ( + ( + ( + ( + ( + + ( ( n + ( n ( n n ( n n + 9 n n + n 9 n... ( + ( ( ( + (+ ( 8 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

32 Prerequisites: Fundamental Concepts of Algebra Section P. Check Point Eercises. a. 0 ( ( ( b. ( 7 + ( 7 ( 7( ( ( ( ( + ( (. Find two numbers whose product is 0 and whose sum is. The required integers are 8 and. Thus, ( + ( + 8 or ( + 8( +. Find two numbers whose product is and whose sum is. The required integers are 7 and. Thus, ( 7( + or ( + ( 7.. Find two First terms whose product is ( ( Find two Last terms whose product is 7. The possible factors are ( 7 and (7. Tr various combinations of these factors to find the factorization in which the sum of the Outside and Inside products is 9. Possible Factors of Sum of Outside and Inside Products (Should Equal 9 (+ ( 7 + ( 7( + 7 ( ( + 7 (+ 7( + 7 (+ ( ( 7(+ ( (+ 7 9 (+ 7( + Thus, ( (+ 7 or (+ 7(. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

33 PreCalculus E Section P.. Find two First terms whose product is + ( (. Find two Last terms whose product is. The possible factors are ( (, ( (, ( (, and ( (. Tr various combinations of these factors to find the factorization in which the sum of the Outside and Inside products is. + ( ( or ( (. 7. Epress each term as the square of some monomial. Then use the formula for factoring a. 8 9 ( + 9( 9 A B. b. ( (+ ( 8. Epress 8 as the difference of two squares and use the formula for factoring A 8 (9 (9 + (9 The factor 9 is the difference of two squares and can be factored. Epress squares and again use the formula for factoring A B. (9 + (9 (9 + ( (9 + (+ ( Thus, factored completel, 8 (9 + (+ (. B. 9 as the difference of two 9. a ( + 7 b. Since ( and 9 7, check to see if the middle term can be epressed as twice the product of and ( Since 7, + 9 is a perfect square trinomial. Thus, ( 7 0. a. + + ( + ( + ( + ( + b. 8 ( ( ( + ( ( + ( ( Factor out the greatest common factor ( Factor the perfect square trinomial. ( + ( 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

34 Prerequisites: Fundamental Concepts of Algebra. Reorder to write as a difference of squares. a a a ( ( 0 a ( 0 a( 0 a ( + ( + ( ( ( ( ( ( ( ( ( +. + ( + (. ( ( + + ( + ( + ( (. + ( + ( +. ( ( + + ( ( ( (. + ( ( 7. ( ( + + ( + ( + Eercise Set P (+. 8( + 8( 8( ( + 8 ( + ( ( ( + 9 ( 9 ( 9 ( ( ( ( ( 7. ( + + ( + ( + ( + 8. ( + + ( + ( + ( + 9. ( + ( ( ( + 0. ( + + 7( + ( + ( ( + ( + ( ( + ( ( ( ( + ( ( 9 ( + ( + ( ( + 8 ( 8( + ( ( + + ( ( 7+ (( + + (+ ( (8+ ( ((. + 0 ( + ( + ( ( (9 ( (+ 8( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

35 PreCalculus E Section P.. 9+ (( ( ( + ( ( + ( + (+ ( 7 ( ( ( + + ( ( + + (+ 9 + ( + ( ( + 0( 0. + (8 8 + ( ( + ( 9 ( 7 (+ 7( 7 8 (8 9 (8+ 9(8 9 9 ( ( (+ ( ( ( + + ( ( ( ( + + ( ( ( (7 (+ 7 (7 ( + ( + + ( ( + ( + + ( ( ( 8 ( 9 ( 9( 9 + ( 9[( ] + ( (9 (9 (9 + (9 [( ] + ( ( ( ( ( ( ( + + ( ( ( + + ( [( ( ( ] + + ( ( 7 ( ( [( ( ( ] + + ( ( ( + ( [( ( ( ] + + ( ( ( ( ( ( ( [( ( ( ] + + ( ( 0 ( ( + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

36 Prerequisites: Fundamental Concepts of Algebra ( 9 ( + ( ( ( + ( 8 0 ( 0 ( 8 ( + ( [( 9 ] ( 9( 9 + ( 9( ( ( [( ] 7 + 7( ( + ( ( 9( + ( 9( + ( ( ( ( + ( ( + ( ( ( + ( ( + ( ( ( ( + ( + ( ( 8( + 7 ( ( ( + ( ( ( ( ( 9 ( ( + + is prime. + is prime. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

37 PreCalculus E Section P ( + + ( 8 ( + ( + ( ( + ( ( + ( ( + ( + ( ( ( ( ( ( ( ( ( ( ( ( ( ( 8 [( 9 ] ( 9( 9 ( 9( ( ( [( ] ( ( ( ( 9 [( ] (+ ( 8 ( [( ] (+ ( + 9 ( 9 ( 9 ( 7 ( ( ( ( + ( 9 9 b + b ( 9b 9b ( 9b ( ( ( ( 9b ( ( b ( b a + a ( a a ( a ( ( ( ( a ( ( a ( a ( ( ( ( ( ( ( ( ( ( 7 ( 9 ( 9 ( 9 ( 9( ( ( ( a ( ( ( a a + a + a + + a a ( ( ( 9a a + 7 9a + 7 7a a 9. ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

38 Prerequisites: Fundamental Concepts of Algebra ( + + ( ( ( ( + ( ( + ( + ( ( ( ( ( ( ( + + ( + ( + + ( + ( + + ( + ( + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( + 0. ( ( ( ( ( ( ( 0. ( ( 8 0 ( ( ( ( ( ( ( + 7( + ( + ( + ( 0 7 ( + ( ( + 0. ( ( ( ( ( ( ( ( ( + ( ( 7 ( ( + ( + ( + ( + + ( + 8 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

39 PreCalculus E Section P ( ( ( ( ( ( ( ( + ( ( + ( ( ( ( + ( ( + ( ( + ( ( + (. ( ( ( (( ( (( (( ( ( ( + +. ( + 00( + ( + (( + 00 ( + ( + 0( ( (. + + ( + ( a. ( ( b. No, the computer is selling at % of its original price a. ( ( b. No, the computer is selling at 9% of its original price a. ( b. 9 ( + ( a. ( b. 9 ( 7+ ( 7 9. a. ( + ( + b. ( + ( + ( + ( 0. a b. ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

40 Prerequisites: Fundamental Concepts of Algebra.. V V V shaded outside inside a a ab b a a ab ( b a a a a+ b ab Vshaded Voutside Vinside a a ab b a a ab ( b a a a a+ b ab. 9. Answers ma var. 0. makes sense. makes sense. does not make sense; Eplanations will var. Sample eplanation: 00 ( ( + (. makes sense. false; Changes to make the statement true will var. A sample change is: ( + ( ( + ( + (. true. false; Changes to make the statement true will var. A sample change is: The binomial + is prime. 7. false; Changes to make the statement true will var. A sample change is: ( ( n + n + 8 ( n + ( n ( + ( + ( ( + ( 0. + ( ( ( ( ( ( ( ( ( ( ( ( ( + + ( ( 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

41 PreCalculus E Mid-Chapter P Check Point ( ( + ( ( + 0 ( ( + ( 0 ( ( +. ( ( + ( + ( ( ( + ( ( +. + b+, b,, 8 or 8. b 0,,, or c(c +, where c > 0 is an integer ( + ( + + ( + ( Mid-Chapter P Check Point. (+ ( 7 + ( ( 7 + (( + (( (+ ( ( [ ] [ ] [ ] (8 (8 (8 ( + ( ( ( ( ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

42 Prerequisites: Fundamental Concepts of Algebra 0.. (( + ( + ( ( ( (9a 0 b( a+ b (9 a( a + (9 a( b + ( 0 b( a + ( 0 b( b (9 a( a + (9 a( b + ( 0 b( a + ( 0 b( b 8a + 9ab0ab0b 8a ab0b. { acde,,, } { cd,, f, h} { acde,,,, f, h}. { acde,,, } { cd,, f, h} { cd, } ( (. ( ( 8. (+ + (+ ( (+ ( [ ][ ] 0. ( ( ( ( ( (.. ( + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

43 PreCalculus E Mid-Chapter P Check Point. ( + ( + ( ( + ( ( 7( is prime ( + + ( ( ( + + ( (. 7 ( 7( +. ( ( ( ( ( ( 9 ( + 7 ( 7 ( + 7( 7. ( ( + +. ( ( ( ( ( ( ( ( ( + 7.,, 0, 0., 7 8. Since < 0 then 9. Since < 0 then. Thus The total annual spending on ice cream is $ A human brain has times as man neurons as a gorilla brain. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

44 Prerequisites: Fundamental Concepts of Algebra. a. Model : N.8+ N.8(0 + N Model : N N (0 9.(0 N Model best describes the data in 000. b. N N ( 9.( N 0 Model underestimates the number of channels in 00 b. c. N.8+ N.8(0 + N Model predicts there will be channels in 00. Section P. Check Point Eercises. a. The denominator would equal zero if, so must be ecluded from the domain. b. ( + ( The denominator would equal zero if or, so and must both must be ecluded from the domain.. a.. + ( ( + +, b. ( ( ( + ( + +, ( ( + ( + ( ( + ( + + ( ( + ( + ( ( + ( +,,, ( ( + Because the denominator is +, Because the denominator is + +, Because the denominator has factors of +,, and +,,, and. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

45 PreCalculus E Section P ( ( ( + ( + + (, 0,, ( ( + +, + + ( + ( + ( + ( + + ( + ( 8 + ( + ( ( + ( + ( ( +, and. ( + ( 7. Factor each denominator completel. + 9 ( 9 ( + ( List the factors of the first denominator., Add an unlisted factors from the second denominator.,, + The least common denominator is the product of all factors in the final list. ( ( ( + or ( ( + is the least common denominator. 8. Find the least common denominator. 0+ ( 0 ( 9. The least common denominator is ( (. Write all rational epressions in terms of the least common denominator ( Add numerators, putting this sum over the least common denominator. ( , (, 0 + +, (, 0 and + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

46 Prerequisites: Fundamental Concepts of Algebra 0. Multipl each of the three terms,,, + 7 the least common denominator of + ( 7... ( + 7 ( ( + 7 ( ( ( + 7, 0, 7 ( ( + ( ( ( Eercise Set P... 7,, / + +.,, ( + ( , 7, 7 9 ( + 7( 7.,, ( + ( + 0 and 7 b., 9, + ( + 9( (, ( 8, + ( ( ( +. ( ( ( 8+, ( + 9( , +, ( ( +,, ( + +, ( + (, + 9 ( 7 9 ( 7( + 7 7, + 7 7, 7 + ( ( + (,, + 9 (+ + ( (+,, Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

47 PreCalculus E Section P ( ( + ( ( + ( ( ( +, 0,, ( ( ( + 9, 8 8. ( + ( ( + + ( +,,. + ( ( + + ( ; 0,, 9. + ( ( + + +,,,, +. + ( ( ( ( ( + ( + (,, ( + ( + ( ( ,,,, ,, 0, ( + ( +, ( ( + 7, ( + ( + ( ( + ( + ( ( + ( + ( +,, ( + ( ( + ( ( + ( + ( + ( + ( + ( + ( ( + ( ( + ( +,,,,, ( ( + ( + ( ( + ( ( ( + ( + ( ( ( +,, Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

48 Prerequisites: Fundamental Concepts of Algebra ( + ( ( + ( + ( + ( ( + ( + ( + ( ( + ( + ( + ( ( + ( +,,, ( + ( ( + ( ( + ( ( + ( + ( + ( +,,,, + ( ,,,, ( +, ( + + ( ( ( ( , 0, ( ( + ( ( + ( ( +,, 0 0 ( 0 + (, + + ( + + (, 8 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

49 PreCalculus E Section P ( ( + ( + (,,.. ( ( ( + ( + ( + ( + +, 0 + ( + + ( +,, ( + + ( ( +,, + ( ( ( + ( ( + ( +,, ( + ( 8 8 +, ,.. 7. ( ( + ( + ( + (,, ( + ( + ( + + +, ( + + ( ( + + +, ( ( ( ( + ( ( +,, Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

50 Prerequisites: Fundamental Concepts of Algebra ( + ( , ( ( ( ( ( + ( ( + ( + 9+ ( + ( ( + ( + + ( + ( + + ( + 9 ( ( ( , ( ( ( , ( + ( ( +, + ( ( + 0 ( ( +, ( + ( + ( + ( ( ( + ( ( + ( + ( ( +,,, ( ( + ( ( + ( ( ( + ( ( ( ( +,, ( + ( + ( + ( + ( + ( ( + ( ( + ( ( + ( + ( + ( + ( + (,, 0 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

51 PreCalculus E Section P ( + ( + ( + ( + ( + ( ( + ( ( + ( ( + ( ( + ( + ( + (, ( + ( + ( ( + ( + ( + ( + ( + ( ( ;, ( + ( ( + ( ( + ( ( + ( ( + ( + 0 ;, 9., [ ] (, + ( ( + +, 0, , + 0, + +, + [ + ] ( + 0, 0,, 0, ( ( ( + ( + ( + ( + ( + ( + ( + + ( [ ],, ( ( (,,, + + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

52 Prerequisites: Fundamental Concepts of Algebra ( ( + ( ( ( ( + + ( + ( , ( ( + + ( ( ( ( ( ( + ( + + ( ( + + ( ( ( + ( ( + ( + ( ( +, ( ( + +,,, ( + ( ( + ( + ( + ( ( + ( + ( + ( + +,, + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

53 PreCalculus E Section P ( + ( ( + ( ( + ( ( + ( ( + ( + ( + ( + ( + ( + ( + ( ,, ( + ( ( + h ( + h ( + h ( + h h h ( + h ( + h h ( + h ( + h+ h h ( + h hh h ( + h h h h ( + h h( + h h ( + h ( + h ( + h + h ( + h( + h+ ( + ( + h+ ( + + h+ + + h+ + h h( + h+ ( + ( + h( + ( + h+ h ( + h+ ( + h h h h ( + h+ ( + h h ( + h+ ( + ( + h+ ( + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

54 Prerequisites: Fundamental Concepts of Algebra ( 7. ( (, > 0 ( ( ( +, > ( + + ( + ( ( ( h + h + h h h + h + h, h 0 h ( + h ( + ( ( ( ( + 7 ( 7( ( ( ( ( + ( ( ( + ( ( ( + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

55 PreCalculus E Section P ( ( ( ( + ( + ( ( +, ( + ( + ( + ( ( ( + ( ( ( ( ( ( ( ( ( + ( ( + ( ( + ( ( + ( ( + ( ( ( ( ( ( ( ( ( + ( ( ( + ( + ( ( + ( ( ( LCD + ( ( + ( ( + ( + ( + Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

56 Prerequisites: Fundamental Concepts of Algebra ( ( LCD + ( ( + ( ( + ( + ( a( c d b( c d ac + ad bc bd c d + + c d a b a + ab+ b ( a b( a + ab+ b a + ab+ b ( ( c+ d ab c d c+ d cd ( a b ( a + ab+ b a + ab+ b a + ab+ b a + bd + b c+ d c+ d d a + ab+ b a + ab+ b ( c d ( a b ( c d ( a+ b ( ( ( ( ab ac ad bc + bd a b ab a cd b cd a + b + + a + ab+ b ac ad + bc bd a + b a + ab+ b a c d + b c d a b ( ( ab a+ b + a ab + b ab a ab + b + a + ab+ b ( a b ( a + ab+ b a + ab+ b a + ab+ b ab + a ab + b a + b a + ab+ b a + ab+ b 9. a. 0 is equal to , when , when , when 90 It costs $8,70,000 to inoculate 0% of the population against this strain of flu, and $0,000,000 to inoculate 80% of the population, and $,70,000,000 to inoculate 90% of the population. b. For 00, the function is not defined. c. As approaches 00, the value of the function increases rapidl. So it costs an astronomical amount of mone to inoculate almost all of the people, and it is impossible to inoculate 00% of the population. Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

57 PreCalculus E Section P. 9. d d d + r r LCD rr d d d + r r ( d rr d d rr + r r rr d rd+ rd rr d rr d r + r r + r ( If r 0 and r 0, the value of this epression will be Your average speed will be miles per hour a. Substitute for in the model. W W ( + ( + 00 W 078 According to the model, women between the ages of 9 and 0 with this lifestle need 078 calories per da. This underestimates the actual value shown in the bar graph b calories. b. Substitute for in the model. M M 0( + 998( + 90 M According to the model, men between the ages of 9 and 0 with this lifestle need calories per da. This underestimates the actual value shown in the bar graph b 8 calories. c. W M ( + + ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 7

58 Prerequisites: Fundamental Concepts of Algebra 9. R R + R + R R R R + + R R R R R R R R R R R + R R + R R 8 R(, 8, The parallel resistance is ohms P L+ W ( ( + ( + + ( + ( + P L+ W ( ( + ( + ( + ( + ( ( ( ( Answers ma var. 8 Copright 00 Pearson Education, Inc. Publishing as Prentice Hall.

59 PreCalculus E Section P. 09. does not make sense; Eplanations will var. Sample eplanation: ( 0 ( (( 0 which is undefined. 0. does not make sense; Eplanations will var. Sample eplanation: The numerator and denominator of share a common factor does not make sense; Eplanations will var. Sample eplanation: The first step is to invert the second fraction.. makes sense do not. false; Changes to make the statement true will var. A sample change is: ( + ( +. true. true. false; Changes to make the statement true will var. A sample change is: n n n n n n n n n + + n n n ( + ( + ( ( ( ( ( ( ( ( It cubes ( Copright 00 Pearson Education, Inc. Publishing as Prentice Hall. 9

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