The mean (expected) time y served in prison for a serious crime can be approximated by a function of the mean sentence length x, with y = 0.55x - 2.886, where x and y are measured in months. According to this model, to how many months should a judge sentence a convicted criminal so that the criminal will serve between 37 and 78 months? (Source: National Center for Policy Analysis)
SOLUTION
We seek values of x that give y-values between 37 and 78, so we solve the inequality 37 … 0.55x - 2.886 … 78 for x:
37 … 0.55x - 2.886 … 78
37 + 2.886 … 0.55x … 78 + 2.886
39.886 … 0.55x … 80.886
72.52 … x … 147.07
Thus, the judge could impose a sentence of 73 to 147 months if she wants the criminal to actually serve between 37 and 78 months.
The x-values of the points of intersection are 44 and 94, so the solution to 80 … 356 + x
a. Solve the equation f (x) = g(x). b. Solve the inequality f (x) 6 g(x).
18. The graphs of three linear functions f, g, and h are shown in the following figure.
a. Solve the equation f (x) = g(x). b. Solve the inequality h(x) … g(x). c. Solve the inequality f (x) … g(x) … h(x).
Skills CHECK 2.4
In Exercises 1–12, solve the inequalities both algebra- ically and graphically. Draw a number line graph of each solution. 1. 3x - 7 … 5 - x 2. 2x + 6 6 4x + 5 3. 4(3x - 2) … 5x - 9 4. 5(2x - 3) 7 4x + 6 5. 4x + 1 6 -3 5 x + 5 6. 4x - 1 2 … -2 + x 3 7. x - 5 2 6 18 5 8. x - 3 4 6 16 3 9. 3(x - 6) 2 Ú 2x 5 - 12 10. 2(x - 4) 3 Ú 3x 5 - 8 11. 2.2x - 2.6 Ú 6 - 0.8x 12. 3.5x - 6.2 … 8 - 0.5x
In Exercises 13 and 14, solve graphically by the intersec- tion method. Give the solution in interval notation.
13. 7x + 3 6 2x - 7 14. 3x + 4 … 6x - 5
In Exercises 15 and 16, solve graphically by the x-intercept method. Give the solution in interval notation.
15. 5(2x + 4) Ú 6(x - 2) 16. -3(x - 4) 6 2(3x - 1)
17. The graphs of two linear functions f and g are shown in the following figure. (Domains are all real numbers.)
4 2 2 4 10 5 5 0 10 (1, 3) g(x) f(x) y x g(x) (10, 400) (30, 200) f(x) h(x) 10 20 30 40 50 100 0 200 300 400 500 x y
In Exercises 19–28, solve the double inequalities.
19. 17 … 3x - 5 6 31 20. 120 6 20x - 40 … 160 21. 2x + 1 Ú 6 and 2x + 1 … 21 22. 16x - 8 7 12 and 16x - 8 6 32 23. 3x + 1 6 -7 and 2x - 5 7 6 24. 6x - 2 … -5 or 3x + 4 7 9 25. 3 4x - 2 Ú 6 - 2x or 2 3x - 1 Ú 2x - 2 26. 1 2x - 3 6 5x or 2 5x - 5 7 6x 27. 37.002 … 0.554x - 2.886 … 77.998 28. 70 … 60 + 88 + 73 + 65 + x 5 6 80
DUI % Male % 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1 2 3 4 5 6 7
Number of 12-oz beers
Blood alcohol percent
(Source: Pennsylvania Liquor Control Board)
31. Freezing The equation F = 9
5 C + 32 gives the rela-
EXERCISES
2.4
29. Depreciation Suppose a business purchases equip- ment for $12,000 and depreciates it over 5 years with the straight-line method until it reaches its salvage value of $2000 (see the figure below). Assuming that the depreciation can be for any part of a year, do the following:
a. Write an equation that represents the depreciated value V as a function of the years t.
b. Write an inequality that indicates that the depreci- ated value V of the equipment is less than $8000. c. Write an inequality that describes the time t during
which the depreciated value is at least half of the original value. 2 4 Years Depreciated Value 6 5000 0 10,000 Dollars 15,000 x y 12,000 10,000 8000 6000 4000 2000
30. Blood Alcohol Percent The blood alcohol percent
p of a 220-pound male is a function of the number
of 12-oz beers consumed, and the percent at which a person is considered legally intoxicated (and guilty of DUI if driving) is 0.1% or higher (see the following figure).
a. Use an inequality to indicate the percent of alcohol in the blood when a person is considered legally intoxicated.
b. If x is the number of beers consumed by a 220- pound male, write an inequality that gives the number of beers that will cause him to be legally intoxicated.
32. Boiling The equation C = 5
9 (F - 32) gives the rela- tionship between temperatures measured in degrees Celsius and degrees Fahrenheit. We know that a temperature at or below 32°F is “freezing.” Use an inequality to represent the corresponding “freezing” Celsius temperature.
tionship between temperatures measured in degrees Celsius and degrees Fahrenheit. We know that a temperature at or above 100°C is “boiling.” Use an inequality to represent the corresponding “boiling” Fahrenheit temperature.
33. Job Selection Deb Cook is given the choice of two positions, one paying $3100 per month and the other paying $2000 per month plus a 5% commission on all sales made during the month. What amount must she sell in a month for the second position to be more profitable?
34. Stock Market Susan Mason purchased 1000 shares of stock for $22 per share, and 3 months later the value had dropped by 20%. What is the minimum percent increase required for her to make a profit?
35. Grades If Stan Cook has a course average score between 80 and 89, he will earn a grade of B in his algebra course. Suppose that he has four exam scores of 78, 69, 92, and 81 and that his teacher said the final exam score has twice the weight of each of the
other exams. What range of scores on the final exam will result in Stan earning a grade of B?
36. Grades If John Deal has a course average score between 70 and 79, he will earn a grade of C in his algebra course. Suppose that he has three exam scores of 78, 62, and 82 and that his teacher said the final exam score has twice the weight of the other exams. What range of scores on the final exam will result in John earning a grade of C?
1995. In what years does this model call for the percent to be greater than 88?
(Source: U.S. Census Bureau)
41. HID Headlights The new high-intensity discharge (HID) headlights containing xenon gas have an expected life of 1500 hours. Because a complete system costs $1000, it is hoped that these lights will last for the life of the car. Suppose that the actual life of the lights could be 10% longer or shorter than the advertised expected life. Write an inequality that gives the range of life of these new lights.
(Source: Automobile, July 2000)
42. Prison Sentences The mean time y spent in prison for a crime can be found from the mean sentence length x, using the equation y = 0.554x - 2.886, where x and y are measured in months. To how many months should a judge sentence a convicted criminal if she wants the criminal to actually serve between 4 and 6 years?
(Source: Index of Leading Cultural Indicators)
43. Marriage Rate According to data from the
National Vital Statistics Report 2010, the mar-
riage rate (marriages per 1000) can be described by
y = -0.146x + 11.074, where x is the number of
years after 1980. For what years does this model indicate that the marriage rate was above 9 marriages per 1000? Was below 8 marriages per 1000?
44. Earnings and Minorities The relation between the median annual salaries of blacks and whites can be modeled by the function B = 1.05W - 18.691, where B and W represent the median annual sala- ries (in thousands of dollars) for blacks and whites, respectively. What is the median salary range for whites that corresponds to a salary range of at least $100,000 for blacks?
(Source: Statistical Abstract of the United States)
45. Home Appraisal A home purchased in 1996
for $190,000 was appraised at $270,000 in 2000. Assuming the rate of increase in the value of the home is constant, do the following:
a. Write an equation for the value of the home as a function of the number of years, x, after 1996. b. Assuming that the equation in part (a) remained
accurate, write an inequality that gives the range of years (until the end of 2010) when the value of the home was greater than $400,000.
c. Does it seem reasonable that this model remained accurate until the end of 2010?
37. Cigarette Use For the period 1997–2009, the percent
y of students in grade 12 who used cigarettes can be
modeled by 2.1x + y = 82.1, where x is the number of years after 1990.
a. Solve the equation for x to represent the number of years after 1990 as a function of the percent. b. Use the equation from part (a) to determine the
range of percent of cigarette use for the years 2000 to 2009.
(Source: MonitoringtheFuture.org)
38. SAT Scores The College Board began reporting SAT scores with a new scale in 1996, with the new scale score y defined as a function of the old scale score
x by the equation y = 0.97x + 128.3829. Suppose
a college requires a new scale score greater than or equal to 1000 to admit a student. To determine what old score values would be equivalent to the new scores that would result in admission to this college, do the following:
a. Write an inequality to represent the problem, and solve it algebraically.
b. Solve the inequality from part (a) graphically to verify your result.
39. Doctorates For the period 2005–2009, the number of new doctorates in mathematics employed in academic positions can be modeled by y = 28.5x + 50.5, where x is the number of years after 2000.
a. If the model is accurate, algebraically determine the year in which the number of doctorates employed was 250.
b. Use a graph to verify your answer to part (a). c. Use your graph to find when the number of doctor-
ates employed was below 250.
(Source: www.ams.org)
40. Internet Access The percent of households in the United States with Internet access is given by
46. Car Sales Profit A car dealer purchases 12 new cars for $32,500 each and sells 11 of them at a profit of 5.5%. For how much must he sell the remaining car to average a profit of at least 6% on the 12 cars?
47. Electrical Components Profit A company’s daily profit from the production and sale of electrical com- ponents can be described by the equation P(x) = 6.45x - 2000 dollars, where x is the number of units produced and sold. What level of produc- tion and sales will give a daily profit of more than $10,900?
48. Profit The yearly profit from the production and sale of Plumber’s Helpers is P(x) = -40,255 + 9.80x dollars, where x is the number of Plumber’s Helpers produced and sold. What level of production and sales gives a yearly profit of more than $84,355? 49. Break-Even A large hardware store’s monthly profit
from the sale of PVC pipe can be described by the equation P(x) = 6.45x - 9675 dollars, where x is the number of feet of PVC pipe sold. What level of monthly sales is necessary to avoid a loss?
50. Break-Even The yearly profit from the production and sale of Plumber’s Helpers is P(x) = -40,255 + 9.80x dollars, where x is the number of Plumber’s Helpers produced and sold. What level of production and sales will result in a loss?
51. Break-Even A company produces a logic board for computers. The annual fixed cost for the board is $345,000, and the variable cost is $125 per board. If the logic board sells for $489, write an inequality that gives the number of logic boards that will give a profit for the product.
52. Temperature The temperature T (in degrees Fahrenheit) inside a concert hall m minutes after a 40-minute power outage during a summer rock concert is given by
T = 0.43m + 76.8. Write and solve an inequality that
describes when the temperature in the hall is not more than 85°F.
53. Hispanic Population Using data and projections from 1990 through 2050, the percent of Hispanics in the U.S. population is given by H(x) = 0.224x + 9.0, where x is the number of years after 1990. Find the years when the Hispanic population is projected to be at least 14.6% of the U.S. population.
(Source: U.S. Census Bureau)
54. Reading Tests The average reading score of 17-year- olds on the National Assessment of Progress tests is given by y = 0.155x + 244.37 points, where x is the number of years after 1970. Assuming that this model was valid, write and solve an inequality that describes when the average 17-year-old’s reading score on this test was between but not including 245 and 248. (Your answer should be interpreted discretely.)
(Source: U.S. Department of Education)
55. Cigarette Use The percent p of high school seniors who used cigarettes can be modeled by
p = 82.074 - 2.088x
where x is the number of years after 1990.
a. What percent does this model estimate for the year 2008?
b. Test integer values of x with the TABLE feature of your graphing utility to find the values of x for which p Ú 57.018.
c. For what years does this model say the percent is at least 57.018%?
56. Black Population Using data and projections from 1990 through 2050, the percent of the U.S. population that is black can be modeled by B(x) = 0.057x + 12.3, where x is the number of years after 1990. When does this model call for the percent of blacks to be at least 13.44%?
chapter 2
SUMMARY
In this chapter, we studied the solution of linear equations and systems of linear equa- tions. We used graphing utilities to solve linear equations. We solved business and eco- nomics problems involving linear functions, solved application problems, solved linear inequalities, and used graphing utilities to model linear functions.