SOLUTIONS MANUAL
Algebra and Trigonometry 5th Edition by James Stewart
CHAPTER P PREREQUISITES 1
P.1 Modeling the Real World with Algebra 1
P.2 Real Numbers 2
P.3 Integer Exponents and Scientific Notation 7
P.4 Rational Exponents and Radicals 12
P.5 Algebraic Expressions 16
P.6 Factoring 19
P.7 Rational Expressions 24
P.8 Solving Basic Equations 31
P.9 Modeling with Equations 36
Chapter P Review 42
Chapter P Test 48
¥ FOCUS ON MODELING: Making Optimal Decisions 51
,P PREREQUISITES
P.1 MODELING THE REAL WORLD WITH ALGEBRA
1. Using this model, we find that 15 cars have W 4 15 60 wheels. To find the number of cars that have a total of
W
W wheels, we write W 4X X .If the cars in a parking lot have a total of 124 wheels, we find that there are
4
X 124
4 31 cars in the lot.
2. If each gallon of gas costs $350, then x gallons of gas costs $35x. Thus, C 35x. We find that 12 gallons of gas would
cost C 35 12 $42.
3. If x $120 and T 006x, then T 006 120 72. The sales tax is $720.
4. If x 62,000 and T 0005x, then T 0005 62,000 310. The wage tax is $310.
5. If 70, t 35, and d t, then d 70 35 245. The car has traveled 245 miles.
6. V r 2 h 32 5 45 1414 in3
N 240
7. (a) M 30 miles/gallon 8. (a) T 70 0003h 70 0003 1500 655 F
G 8
175 175 (b) 64 70 0003h 0003h 6 h 2000 ft
(b) 25 G 7 gallons
G 25
9. (a) V 95S 95 4 km3 38 km3 10. (a) P 006s 3 006 123 1037 hp
(b) 19 km3 95S S 2 km3 (b) 75 006s 3 s 3 125 so s 5 knots
11. (a) (b) We know that P 30 and we want to find d, so we solve the
Depth (ft) Pressure (lb/in2 ) equation 30 147 045d 153 045d
0 045 0 147 147 153
d 340. Thus, if the pressure is 30 lb/in2 , the depth
10 045 10 147 192 045
20 045 20 147 237 is 34 ft.
30 045 30 147 282
40 045 40 147 327
50 045 50 147 372
60 045 60 147 417
12. (a) (b) We solve the equation 40x 120,000
Population Water use (gal) 120,000
x 3000. Thus, the population is about 3000.
0 0 40
1000 40 1000 40,000
2000 40 2000 80,000
3000 40 3000 120,000
4000 40 4000 160,000
5000 40 5000 200,000
13. The number N of cents in q quarters is N 25q.
ab
14. The average A of two numbers, a and b, is A .
2
1
,2 CHAPTER P Prerequisites
15. The cost C of purchasing x gallons of gas at $350 a gallon is C 35x.
16. The amount T of a 15% tip on a restaurant bill of x dollars is T 015x.
17. The distance d in miles that a car travels in t hours at 60 mi/h is d 60t.
d
18. The speed r of a boat that travels d miles in 3 hours is r .
3
19. (a) $12 3 $1 $12 $3 $15
(b) The cost C, in dollars, of a pizza with n toppings is C 12 n.
(c) Using the model C 12 n with C 16, we get 16 12 n n 4. So the pizza has four toppings.
20. (a) 3 30 280 010 90 28 $118
daily days cost miles
(b) The cost is , so C 30n 01m.
rental rented per mile driven
(c) We have C 140 and n 3. Substituting, we get 140 30 3 01m 140 90 01m 50 01m
m 500. So the rental was driven 500 miles.
21. (a) (i) For an allelectric car, the energy cost of driving x miles is Ce 004x.
(ii) For an average gasoline powered car, the energy cost of driving x miles is C g 012x.
(b) (i) The cost of driving 10,000 miles with an allelectric car is Ce 004 10,000 $400.
(ii) The cost of driving 10,000 miles with a gasoline powered car is C g 012 10,000 $1200.
22. (a) If the width is 20, then the length is 40, so the volume is 20 20 40 16,000 in3 .
(b) In terms of width, V x x 2x 2x 3 .
4a 3b 2c 1d 0 f 4a 3b 2c d
23. (a) The GPA is .
abcd f abcd f
(b) Using a 2 3 6, b 4, c 3 3 9, and d f 0 in the formula from part (a), we find the GPA to be
463429 54
284.
649 19
P.2 THE REAL NUMBERS
1. (a) The natural numbers are 1 2 3 .
(b) The numbers 3 2 1 0 are integers but not natural numbers.
p
(c) Any irreducible fraction with q 1 is rational but is not an integer. Examples: 32 , 12
5 , 1729 .
23
q
p
(d) Any number which cannot be expressed as a ratio of two integers is irrational. Examples are 2, 3, , and e.
q
2. (a) ab ba; Commutative Property of Multiplication
(b) a b c a b c; Associative Property of Addition
(c) a b c ab ac; Distributive Property
3. (a) In setbuilder notation: x 3 x 5 (c) As a graph:
_3 5
(b) In interval notation: 3 5
4. The symbol x stands for the absolute value of the number x. If x is not 0, then the sign of x is always positive.
5. The distance between a and b on the real line is d a b b a. So the distance between 5 and 2 is 2 5 7.
6. (a) If a b, then any interval between a and b (whether or not it contains either endpoint) contains infinitely many
ba
numbers—including, for example a n for every positive n. (If an interval extends to infinity in either or both
2
directions, then it obviously contains infinitely many numbers.)
, SECTION P.2 The Real Numbers 3
(b) No, because 5 6 does not include 5.
7. (a) No: a b b a b a in general.
(b) No; by the Distributive Property, 2 a 5 2a 2 5 2a 10 2a 10.
8. (a) Yes, absolute values (such as the distance between two different numbers) are always positive.
(b) Yes, b a a b.
9. (a) Natural number: 100 10. (a) Natural numbers: 2, 9 3, 10
(b) Integers: 0, 100, 8 (b) Integers: 2, 100
2 50, 9 3, 10
(c) Rational numbers: 15, 0, 52 , 271, 314, 100, 8 (c) Rational numbers: 45 92 , 13 , 16666 53 ,
(d) Irrational numbers: 7, 2, 100
2 , 9 3, 10
(d) Irrational numbers: 2, 314
11. Commutative Property of addition 12. Commutative Property of multiplication
13. Associative Property of addition 14. Distributive Property
15. Distributive Property 16. Distributive Property
17. Commutative Property of multiplication 18. Distributive Property
19. x 3 3 x 20. 7 3x 7 3 x
21. 4 A B 4A 4B 22. 5x 5y 5 x y
23. 2 x y 2x 2y 24. a b 5 5a 5b
25. 5 2x y 5 2 x y 10x y 26. 43 6y 43 6 y 8y
27. 52 2x 4y 52 2x 52 4y 5x 10y 28. 3a b c 2d 3ab 3ac 6ad
29. (a) 23 57 14 15 29
21 21 21 30. (a) 25 38 16 15 1
40 40 40
5 3 10 9 1
(b) 12 (b) 32 58 16 36 15 4 25
8 24 24 24 24 24 24 24
2
2
31. (a) 23 6 32 23 6 23 32 4 1 3 32. (a) 2 3 2 32 23 12 3 13 93 13 83
2
3
(b) 3 14 1 45 12 4 4
1 5 4 13 1 13
5 5 4 5 20 2 1 2 1 2 1
(b) 15 23 51 21 51 21 10 45 9
10 12 3 3
10 15 10 5 10 5
33. (a) 2 3 6 and 2 72 7, so 3 72 34. (a) 3 23 2 and 3 067 201, so 23 067
(b) 6 7 (b) 23 067
(c) 35 72 (c) 06 06
35. (a) False 36. (a) False: 3 173205 17325.
(b) True (b) False
37. (a) True (b) False 38. (a) True (b) True
