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178 Chapter 1 • Equations and Inequalities
147. What's wrong with this argument? Suppose x and y represent two real numbers, where x > y.
2> 1
2(y - x) > l(y - x)2y - 2x > Y - x
y-2x>-x
y>x
This is a true statement.
Multiply both sides byy - x.
Usethe distributive property.
Subtract y from both sides.
Add 2x to both sides.
The final inequality, y > x, is impossible because we wereinitially given x > y.
148. Write an absolute value inequality for which the intervalshown is the solution.
Solutions lie within3 units of 4.
a. x-2 -1 0 1 2 3 4 5 6 7 8
b. x-2 -1 0 1 2 3 4 5 6 7 8
Chapter 1 - ~_Summa ry,_Review,_a nd_TestSummary
DEFINITIONS AND CONCEPTS
1.1 Graphs and Graphing utilities
149. Here are two inequalities that describe the range ofmonthly average temperatures, T, in degrees Fahrenheit fortwo American cities:
Modell: IT - 571 < 7Model 2: IT - 501 < 22.
Which model describes Albany, New York, and whichmodel describes San Francisco, California?
B Group Exercise150. Each group member should research one situation that pro
vides two different pricing options. These can involve areassuch as public transportation options (with or withoutcoupon books), cell phone plans, long-distance telephoneplans, or anything of interest. Be sure to bring in all thedetails for each option. At a second group meeting, select thetwo pricing situations that are most interesting and relevant.Using each situation, write a word problem about selectingthe better of the two options. The word problem should beone that can be solved using a linear inequality. The groupshould turn in the two problems and their solutions.
a. The rectangular coordinate system consists of a horizontal number line, the x-axis, and a vertical numberline, the y-axis, intersecting at their zero points, the origin. Each point in the system corresponds to anordered pair of real numbers (x, y). The first number in the pair is the x-coordinate; the second number isthe y-coordinate. See Figure 1.1 on page 84.
b. An ordered pair is a solution of an equation in two variables if replacing the variables by the correspondingcoordinates results in a true statement. The ordered pair is said to satisfy the equation. The graph of theequation is the set of all points whose coordinates satisfy the equation. One method for graphing an equation is to plot ordered-pair solutions and connect them with a smooth curve or line.
c. An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis.The y-coordinate corresponding to an x-intercept is always zero.A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis.The x-coordinate corresponding to a y-intercept is always zero.
1.2 Linear Equations and Rational Equations3. A linear equation in one variable x can be written in the form ax + b = 0, a '" O.b. The procedure for solving a linear equation is given in the box on page 97.
c. If an equation contains fractions, begin by multiplying both sides by the least common denominator,thereby clearing fractions.
d. A rational equation is an equation containing one or more rational expressions. If an equation containsrational expressions with variable denominators, avoid in the solution set any values of the variable thatmake a denominator zero.
e. An identity is an equation that is true for all real numbers for which both sides are defined. A conditionalequation is not an identity and is true for at least one real number. An inconsistent equation is an equationthat is not true for even one real number.
EXAMPLES
Ex.1,p.85
Ex. 2, p. 86;Ex. 3,p. 86
Ex. 5,p. 89
Ex.1,p.96;Ex. 2,p. 97
Ex. 3,p. 99
Ex. 4, p. 100;Ex. 5, p. 101;Ex.6,p.101Ex.7,p.103
Summary, Review, and Test 179
1.3 Models and Applications3. A five-step procedure for solving word problems using equations that model verbal conditions is given in
the box on page 108.
b. Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side ofthe equation.
1.4 Complex Numbers3. The imaginary unit i is defined as
I = -v=l, where i2 = -1.The set of numbers in the form a + bi is called the set of complex numbers; a is the real part and b is theimaginary part. If b = 0, the complex number is a real number. If b '" 0, the complex number is an imaginary number. Complex numbers in the form bi are called pure imaginary numbers.
b. Rules for adding and subtracting complex numbers are given in the box on page 124.
c. To multiply complex numbers, multiply as if they are polynomials. After completing the multiplication,replace i2 with -1 and simplify.
d. The complex conjugate of a + bi is a - bi and vice versa. The multiplication of complex conjugates gives areal number:
(a + bi)(a - bi) = a2 + b2.
e. To divide complex numbers, multiply the numerator and the denominator by the complex conjugate of thedenominator.
f. When performing operations with square roots of negative numbers, begin by expressing all square roots interms of i. The principal square root of -b is defined by
yCb = iYb.1.5 Quadratic Equations
3. A quadratic equation in x can be written in the general form ax2 + bx + c = 0, a '" O.
b. The procedure for solving a quadratic equation by factoring and the zero-product principle is given in thebox on page 131.
c. The procedure for solving a quadratic equation by the square root property is given in the box on page 132.
d. All quadratic equations can be solved by completing the square. Isolate the binomial with the two variableterms on one side of the equation. If the coefficient of the x2-term is not one, divide each side of the equation by this coefficient. Then add the square of half the coefficient of x to both sides.
e. All quadratic equations can be solved by the quadratic formula
-b ± Vh2 - 4acx- 2a
The formula is derived by completing the square of the equation ax: + bx + c = O.
f. The discriminant, b2 - 4ac, indicates the number and type of solutions to the quadratic equationax2 + bx + c = 0, shown in Table 1.3 on page 140.
g. Table 1.4 on page 141 shows the most efficient technique to use when solving a quadratic equation.
1.6 other 'TYpes of Equations3. Some polynomial equations of degree 3 or greater can be solved by moving all terms to one side, obtaining
zero on the other side, factoring, and using the zero-product principle. Factoring by grouping is often used.
b. A radical equation is an equation in which the variable occurs in a square root, cube root, and so on. A radical equation can be solved by isolating the radical and raising both sides of the equation to a power equalto the radical's index. When raising both sides to an even power, check all proposed solutions in the originalequation. Eliminate extraneous solutions from the solution set.
c. A radical equation with rational exponents can be solved by isolating the expression with the rationalexponent and raising both sides of the equation to a power that is the reciprocal of the rational exponent.See the details in the box on page 156.
Ex. 1, p. 108;Ex. 2, p. 110;Ex. 3, p. 111;Ex. 4, p. 113;Ex. 5, p. 114;Ex.6,p.115
Ex. 7, p. 117;Ex. 8, p.117
Ex.1,p.124
Ex.2,p.125
Ex.3,p.126
Ex.4,p.127
Ex.1, p.131
Ex.2,p.133
Ex. 4, p. 135;Ex.5,p.136
Ex. 6, p. 138;Ex. 7, p.139
Ex.8,p.140
Ex. 9, p.142;Ex. 10, p.143
Ex. 1, p. 151;Ex.2,p.151
Ex. 3, p. 153;Ex.4,p.154
Ex.5,p.156
180 Chapter 1 • Equations and Inequalities
d. An equation is quadratic in form if it can be written in the form au2 + bu + c = 0, where u is an algebraicexpression and a '" O. Solve for u and use the substitution that resulted in this equation to find the valuesfor the variable in the given equation.
e. Absolute value equations in the form Ixi = c, c > 0, can be solved by rewriting the equation withoutabsolute value bars: X = c or X = -c.
1.7 Linear Inequalities and Absolute Value Inequalities
3. Solution sets of inequalities are expressed using set-builder notation and interval notation. In interval notation, parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpointsthat are included in an interval. See Table 1.5 on page 165.
b. A procedure for finding intersections and unions of intervals is given in the box on page 166.
c. A linear inequality in one variable x can be expressed as ax + b ::; c, ax + b < c, ax + b ~ c, orax + b > c, a '" O.
d. A linear inequality is solved using a procedure similar to solving a linear equation. However, when multiplying or dividing by a negative number, change the sense of the inequality.
e. A compound inequality with three parts can be solved by isolating the variable in the middle.
f. Inequalities involving absolute value can be solved by rewriting the inequalities without absolute valuebars. The ways to do this are shown in the box on page 171.
Review Exercises1.1Graph each equation in Exercises 1-4. Let x = -3, -2, -1,0,1,2, and 3.1. Y = 2x - 2 2. Y = x2 - 3
3. y = x 4. y = Ixl - 2
5. What does a [-20,40, lOJ by [-5,5,1 J viewing rectanglemean? Draw axes with tick marks and label the tick marks toillustrate this viewing rectangle.
In Exercises 6-8, use the graph and determine the x-intercepts, ~fany, and the y-intercepts, if any. For each graph, tick marks alongthe axes represen.t one unit each.6. 7. yy
8. y
Ex. 6, p. 157;Ex.7,p.158
Ex. 8, p. 159;Ex.9,p.159
Ex.1,p.166
Ex.2,p.166
Ex. 3, p. 168;Ex. 4, p. 169;Ex.5,p.170
Ex.6,p.171
Ex. 7, p. 172;Ex. 8, p. 172;Ex. 9, p.173
Afghanistan accounts for 76% of the world's illegal opium production. Opium-poppy cultivation nets big money in a countrywhere most people earn less than $1 per day. (Source: Newsweek)The line graph shows opium-poppy cultivation, in thousands ofacres, in Afghanistan from 1990 through 2004. Use the graph tosolve Exercises 9-14.
Afghanistan's Opium Crop
300275250225
:::0 200''= ~.~ ~ 175_ u
- '"ut; 150>,,,, 1250."00.:::o '" 1000... '", :lS 0 75:l£.~_.. 500
25
90 91 92 93 94 95 96 97 98 99 00 01 02 03 04ThM
Source: U.N. Office on Drugs and Crime
9. What are the coordinates of point A? What does this mean interms of the information given by the graph?
10. In which year were 150 thousand acres used for opiumpoppy cultivation?
11. For the period shown, when did opium cultivation reach aminimum? How many thousands of acres were used to cultivate the illegal crop?
12. For the period shown, when did opium cultivation reach amaximum? How many thousands of acres were used to cultivate the illegal crop?
13. Between which two years did opium cultivation not change?14. Between which two years did opium cultivation increase at
the greatest rate? What is a reasonable estimate of theincrease, in thousands of acres, used to cultivate the illegalcrop during this period?
1.2In Exercises 15-35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation.15. 2x - 5 = 7 16. 5x + 20 = 3x17. 7(x - 4) = x + 2 18. 1 - 2(6 - x) = 3x + 219. 2(x - 4) + 3(x + 5) = 2x - 220. 2x - 4(5x + 1) = 3x + 1721. 7x + 5 = 5(x + 3) + 2x22. 7x + 13 = 2(2x - 5) + 3x + 23
23 2x = ~ + 1·36
x 1 x 124 ---=-+-
• 2 10 5 2
2x x25 -=6--. 3 4
3x + 1 13 1 - x27. ----=--3 2 4
x x-326-=2---. 4 3
91428.---=-
4 2x x
7 x+2 1 1 229. -- + 2 = -- 30. -- - -- = -,-x-5 x-5 x-1 x+1 r-151831. -- + -- = ----
x + 3 x - 2 x2 + X - 6132. -- = 0
x+54 3 10
33. -- + - =x + 2 X xZ + 2x
34. 3 - 5(2x + 1) - 2(x - 4) = 0
x + 2 135. -- + -1 = 0x + 3 x2 + 2x - 3
1.3In Exercises 36-43, use the five-step strategy for solving wordproblems.36. The fast-food chains may be touting their "new and improved"
salads, but how do they measure up in terms of calories?
Burger KingChicken Caesar
Taco BellExpress Taco
Salad
Wendy'sMandarin Chicken
Salad
Number of caloriesexceeds the ChickenCaesar by 125.
Number of caloriesexceeds the Chicken
Caesar by 95.
Source: Newsweek
Combined, the three salads contain 1705 calories. Determinethe number of calories in each salad.
ReviewExercises 181
37. The bar graph shows that in 1970, 37.4% of US. adultssmoked cigarettes. For the period from 1970 through 2002,the percentage of smokers among US. adults decreased at anaverage rate of 0.5% per year. If this trend continues, whenwill only 18.4% of US. adults smoke cigarettes?
Butt Out: Percentage ofCigarette Smokers Among U.S. Adults
50%00E 40%oE~ 30%o0,)
~ 20%t:~'"c,
7 4%371% 41% 1°13. . 3. 32. 1°288% 255%)41%225%
. 25.5%. . . .
10%
1970 1974 1978 1982 1986 1990 1994 1998 2002~~
Source: Centers for Disease Control and Prevention
38. You are choosing between two long-distance telephoneplans. One plan has a monthly fee of $15 with a charge of$0.05 per minute. The other plan has a monthly fee of $5 witha charge of $0.07 per minute. For how many minutes of longdistance calls will the costs for the two plans be the same?
39. After a 20% price reduction, a cordless phone sold for $48.What was the phone's price before the reduction?
40. A salesperson earns $300 per week plus 5% commission ofsales. How much must be sold to earn $800 in a week?
41. You invested $9000 in two funds paying 4% and 7% annualinterest, respectively. At the end of the year, the total interestfrom these investments was $555. How much was invested ateach rate?
42. You invested $8000 in two funds paying 2% and 5% annualinterest, respectively. At the end of the year, the interest fromthe 5% investment exceeded the interest from the 2% investment by $85. How much money was invested at each rate?
43. The length of a rectangular field is 6 yards less than triple thewidth. If the perimeter of the field is 340 yards, what are itsdimensions?
44. In 2007, there were 14,100 students at college A, with a projected enrollment increase of 1500 students per year. In thesame year, there were 41,700 students at college B, with aprojected enrollment decline of 800 students per year.a. Let x represent the number of years after 2007. Write, but
do not solve, an equation that can be used to find how manyyears after 2007 the colleges will have the same enrollment.
b. The following table is based on your equation in part (a).Y1 represents one side of the equation and Yz representsthe other side of the equation. Use the table to answer thefollowing questions: In which year will the colleges havethe same enrollment? What will be the enrollment in eachcollege at that time?
v '...'1 Y:::,.,~ :::'161)1) 3611)1)B :::61(11) 3~31)(19 :::761)1) 3't~(1I)11) :::911)1) 3371)1)11 3(161)1) 3:::91)1)1::: :~;::11)1) 3:::11)1)13 :~361)1) 3131)1)
>~=7
182 Chapter 1 • Equations and Inequalities
In Exercises 45-47, solve each formula for the specified variable.45. vt + gt2 = s for g 46. T = gr + gvt for g
A-P47. T = --- for PPr
1.4In Exercises 48-57, perform the indicated operations and writethe result in standard form.48. (8 - 3i) - (17 - 7i) 49. 4i(3i - 2)
50. (7 - i)(2 + 3i) 51. (3 - 4i)2
53. _6_5 + i
55. v=32 - v=I84 + v=s
57. 2
52. (7 + 8i) (7 - 8i)
3 + 4i54·4_2i
1.5Solve each equation in Exercises 58-59 by factoring.58. 2X2 + 15x = 8 59. 5x2 + 20x = 0
Solve each equation in Exercises 60-63 by the square rootproperty.60. 2x2 - 3 = 125
62. (x + 3)2 = -10
x261 - + 5 = -3. 2
63. (3x - 4)2 = 18
In Exercises 64-65, determine the constant that should be addedto the binomial so that it becomes a perfect square trinomial.Then write and factor the trinomial.64. x2 + 20x 65. x2 - 3x
Solve each equation in Exercises 66-67 by completing-the square.66. x2 - 12x + 27 = 0 67. 3x2 - 12x + 11 = 0
Solve each equation in Exercises 68-70 using the quadraticformula.68. x2 = 2x + 4
70. 2X2 = 3 - 4x
69. x2 - 2x + 19 = 0
In Exercises 71-72, without solving the given quadratic equation,determine the number and type of solutions.71. x2 - 4x + 13 = 0 72. 9x2 = 2 - 3x
Solve each equation in Exercises 73-81 by the method of yourchoice.73. 2x2 - 1lx + 5 = 0
75. 3x2 - 7x + 1 = 0
77. (x - 3)2 - 25 = 0
79. 3x2 - lOx = 8
5 x - 181. --+--=2
x + 1 4
82. The formula W = 3t2 madels the weight of a human fetus, W,in grams, after t weeks, where 0 ::; t ::; 39. After how manyweeks does the fetus weigh 588 grams?
74. (3x + 5)(x - 3) = 5
76. x2 - 9 = 0
78. 3x2 - x + 2 = 0
80. (x + 2? + 4 = 0
83. In 1945,35.4% of taxes collected by the U.S.Treasury camefrom corporate income taxes. Since then, corporations haveworked hard to convince lawmakers that they shouldn't paytaxes. The bar graph shows the percentage of federal taxesfrom corporate income taxes for selected years from 1985through 2003.The data can be modeled by the formula
P = -0.035x2 + 0.65x + 7.6,where P represents the percentage of federal taxes fromcorporations x years after 1985. If these trends continue, bywhich year (to the nearest year) will corporations pay notaxes?
Percentage of Federal Taxesfrom Corporate Income Taxes
12%
en 10%<l.)
<l.) K 8%OIlFlB~::; '" 6%<l.) <l.)
8""2~~ 4%.....
02%
11%10%
9%
7.4%
1985 1990 1995 2000 2003Year
Source: White House Office of Management and Budget
84. An architect is allowed 15 square yards of floor space to adda small bedroom to a house. Because of the room's design inrelationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length.Find the length and width of the rectangular floor that thearchitect is permitted.
85. A building casts a shadow that is double the length of itsheight. If the distance from the end of the shadow to the topof the building is 300 meters, how high is the building?Round to the nearest meter.
1.6Solve each polynomial equation in Exercises 86-87.86. 2X4 = 50x2 87. 2x3 - x2 - 18x + 9 = 0
Solve each radical equation in Exercises 88-89.88. v'2X'=3 + x = 3 89. v:x=-4 + v'.X+l = 5
Solve the equations with rational exponents in Exercises 90-91.3 2
90. 3x4 - 24 = 0 91. (x - 7)3 = 25
Solve each equation in Exercises 92-93 by making an appropriatesubstitution.92. X4 - 5x2 + 4 = 0
1 193. Xl + 3x4 - 10 = 0
Solve the equations containing absolute value in Exercises 94-95.94. 12x + 11 = 7 95. 21x - 31 - 6 = 10
Solve each equation in Exercises 96-102 by the method of yourchoice.
4 296. 3x3 - 5x3 + 2 = 0 97. 2vx=l = x98. 12x - 51 - 3 = 0 99. x3 + 2x2 = 9x + 18
100. vs=2X - x = 0 101. x3 + 3x2 - 2x - 6 = 0
102. -41x + 11 + 12 = 0
103. By 2010, India could become the world's most HIV-afflictedcountry. The bar graph shows the increase in the country'sHIV infections from 1998 through 2001. The formulaN = 0.3VX + 3.4 models the number of HIV infections inIndia, N, in millions, x years after 1998.
AIDS in India
4.0~ 3.88 3.903.8 >- 3.70-3.6[3
3.40.4 -
3.2 f-
3.0 f-
1998 1999 2000 2001Year
Source: UNAIDS
If trends indicated by the data continue, use the model todetermine when the number of HIV infections in India willreach 4.3 million.
1.7in Exercises 104-106, express each interval in set-builder notationand graph the interval on a number line.104. [-3,5) 105. (-2,00) 106. (-00,0]
Chapter 1 Test 183
in Exercises 107-110, use graphs to find each set.107. (-2,lJn[-1,3)109. [1,3)n(0,4)
108. (-2,1] U [-1,3)110. [1,3) U (0,4)
in Exercises 111-121, solve each inequality. Other than 0, useinterval notation to express solution sets and graph each solutionset on a number line.111. -6x + 3 ,,; 15 112. 6x - 9 ~ -4x - 3
x 3 x113 - - - - 1 > - 114 6x + 5 > -2(x - 3) - 25• 3 4 2 •
115. 3(2x - 1) - 2(x - 4) ~ 7 + 2(3 + 4x)
116. 5 (x - 2) - 3 (x + 4) ~ 2x - 20
117. 7 < 2x + 3 ::; 9 118. 12x + 31 ,,; 15
12x + 61119. -3- > 2 120. 12x + 51 - 7 ~ -6
121. -41x + 21 + 5 ::; -7
in Exercises 122-123, use interval notation to represent all valuesof x satisfying the given conditions.122. Yl = -10 - 3(2x + l),yz = 8x + 1,andYl > yz.123. Y = 3 - 12x - 51 and Y is at least -6.124. A car rental agency rents a certain car for $40 per day with
unlimited mileage or $24 per day plus $0.20 per mile. Howfar can a customer drive this car per day for the $24 optionto cost no more than the unlimited mileage option?
125. To receive a B in a course, you must have an average of atleast 80% but less than 90% on five exams. Your grades onthe first four exams were 95%,79%,91 %, and 86%. Whatrange of grades on the fifth exam will result in a B for thecourse?
126. A retiree requires an annual income of at least $9000 froman investment paying 7.5% annual interest. How muchshould the retiree invest to achieve the desired return?
in Exercises 1-23, solve each equation or inequality. Other than0, use interval notation to express solution sets of inequalities andgraph these solution sets on a number line.1. 7(x - 2) = 4(x + 1) - 21
2. -10 - 3(2x + 1) - 8x - 1 = 0
2x - 3 x - 4 x + 13 --=-----• 4 2 4
2 4 84 -----=--. x - 3 x + 3 XZ - 9
5. 2xz - 3x - 2 = 0 6. (3x - l)z = 75
7. (x + 3? + 25 = 0 8. x(x - 2) = 4
9. 4x2 = 8x - 511. -v.x=-3 + 5 = x
10. x3 - 4xz - x + 4 = 012. V8 - 2x - x = 0
313. v:x+4 + '\.I'X'-=-I = 5 14. 5x2 - 10 = 0
Z 115. x3 - 9x3 + 8 = 0 16. I ~x - 61 = 2
1 418. - - - + 1 = 0XZ x
17. -314x - 71 + 15 = 0
19 2x 2 _ x. +-----XZ + 6x + 8 x + 2 x + 4
20. 3(x + 4) ~ 5x - 12
2x + 522. -3 ::; -- < 63
x 1 x 321 - + - ::; - - -• 6 8 2 4
23. 13x + 21 ~ 3
In Exercises 24-25, use interval notation to represent all values ofx satisfying the given conditions.24. Y = 2x - 5, and Y is at least -3 and no more than 7.
12 - xl25. Y = -4- and Y is at least 1.
184 Chapter 1 • Equations and Inequalities
In Exercises 26-27, use graphs to find each set.26. [-1,2) U (0, 5J 27. [-1,2) n (0,5]In Exercises 28-29, solve each formula for the specified variable.
128. V = '3lwh for h 29. y - Yl = m(x - Xl) for X
In Exercises 30-31, graph each equation in a rectangular coordinate system.30. Y = 2 - Ixl 31. Y = x2 - 4
In Exercises 32-34, perform the indicated operations and writethe result in standard form.
32. (6 - 7i)(2 + 5i)
34. 2v=49 + 3v=64
33. __22 - i
Without changes, the graphs show projections for the amountbeing paid in Social Security benefits and the amount going intothe system. All data are expressed in billions of dollars.
Social Insecurity: Projected Income andOutflow of the Social Security System
$2000
~? $1600o OJ~=;:;.g $12000 ....~ 00)§ g $800u:.:.5 e $400
2004 2008 2012 2016 2020 2024Year
Source: 2004 Social Security Trustees Report
Exercises 35-37 are based on the data shown by the graphs.35. In 2004, the system's income was $575 billion, projected to
increase at an average rate of $43 billion per year. In whichyear will the system's income be $1177 billion?
36. The data for the system's outflow can be modeled by theformula
B = 0.07X2 + 47.4x + 500,
where B represents the amount paid in benefits, in billions ofdollars, x years after 2004.According to this model, when willthe amount paid in benefits be $1177 billion? Round to thenearest year.
37. How well do your answers to Exercises 35 and 36 model thedata shown by the graphs?
38. From 2002 through 2004, there were 2598 books categorizedas U.S.politics and government. The number of books in 2003exceeded the number in 2002 by 62 and the number in 2004exceeded the number in 2002 by 190. How many books onU.S. politics were there for each of the three years?(Source: AndrewGrabois,R. R. Bowker)
39. The costs for two different kinds of heating systems for athree-bedroom home are given in the following table. Afterhow many years will total costs for solar heating and electricheating be the same? What will be the cost at that time?
System Cost to Install Operating CostIYearSolar $29,700 $150Electric $5000 $1100
40. You invested $10,000 in two accounts paying 8% and 10%annual interest, respectively. At the end of the year, the totalinterest from these investments was $940. How much wasinvested at each rate?
41. The length of a rectangular carpet is 4 feet greater than twiceits width. If the area is 48 square feet, find the carpet's lengthand width.
42. A vertical pole is to be supported by a wire that is 26 feetlong and anchored 24 feet from the base of the pole. How farup the pole should the wire be attached?
43. After a 60% reduction, a jacket sold for $20. What was thejacket's price before the reduction?
44. You are choosing between two telephone plans for localcalls. Plan A charges $25 per month for unlimited calls. PlanB has a monthly fee of $13 with a charge of $0.06 per localcall. How many local telephone calls in a month make plan Athe better deal?
