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1.1 Homework Solve these linear equations, check your solutions: 1. 2x÷5=9 2. 5x÷8=3 3. 5—3x=—4 18. 3x+3x—3=7 19. 2(x—7)=5(x+3)—x 20. (5x—1)5(x--1) 4. 3—4x=7 5. 5x—18=7 6. 5—7x=—9 21. 22. x—4=1O—x 7. 5x—7=9 8. 4x—2=3x÷4 9. 5x+13=3x÷7 23. 24 5 7 2 10. 3x+7=12—2 11. 7—x=21—8 12. 5—9=2x+17 25. 26. 13. 4x—7=8x+2 14. 3÷2=7x+2 15. 8(x—2)6(3x—4) 16 27 6 4 28. 5x ( 4 2x—7) 6 — 17. =24 7 CHAPTER 1 8 Linear Equations and Inequalities Solve these rational equations, check the domain to ensure that the solution Is valid. 1 3 4 2x÷1_ 1 30. —2z 31. lOx÷25(x—3)=—5 32. —26=—19 x—5 x—5 ‘‘. 44• 45. x÷7 x÷7 x x÷4 46. x—3 +2 = 3x 4 33. x+= 6 (x6) (x—6) — 34. 35 5_7 36. 4x-12=60 37. X+X 38. O.9.x—O.7x=42 48. 2x (x—3) 49. 5 (3x+8) 50. 51. 39. 3x—7=—7—5x 52. 6 (x—3) + 2 (x-1) 3121 (33—x) 2 5x 111 40. 2(x+6)=8x (÷3)8 (x—2) 3 41. 80=1O(3x÷2) 42. ——--1=3—---— x—3 x—3 I I Which of the following are polynomials and which are not? 7 +2x—7 2 54. 3x 58. 55. 59. 56. —7 60. 3\/-i-5x—1 57. 5x+2 Vx2+5x_1 Section 1.1 Linear Equations in One Variable 9 61, Five more than two times a number is 39. What is the number? 62. Three less than 4 times a number is 25. What is the number? 63. Ten more than three times a number is the same as five times that same number. What is the number? 64. Dan opens a savings account with $500. He plans to add $10 a week. How many weeks will it take for him to have a total of $810 in this savings account? 65. While vacationing at a beach, Mike rents a bicycle. The rental fee for the bicycle includes $5.00 to rent the bicycle and 20 cents per mile that he rides. Write a linear equation which shows how much Mike will owe for renting the bicycle and riding m miles. Then use this equation to determine how much he will owe after riding 12 miles. 66. A car rental company rents cars for $80.00 plus 15 cents per mile. Write a linear equation which shows how much renting a car will cost if the car is driven x miles. Then use this equation to determine the total rental cost if the car is driven 220 miles. 67. Sam has been building a fence to enclose his property. He now has 25 sections of fencing completed. He will be able to build another 8 sections of fencing every week. Write a linear equation which shows how many sections of fencing he will have built after w weeks. Use this equation to find out how many sec tions of fencing he will have built after 15 weeks. 68. The perimeter of a rectangle is 700 feet and the length of the rectangle is four times as long as the width. Find the dimensions of the rectangle. 69. The perimeter of a rectangle is 24 inches and the length of the rectangle is five times as long as the width. Find the dimensions of the rectangle. 70. A collector bought two rare coins for $10,000. A year later, the collector sold these coins for $10,400 or a $400 profit. One of the coins was sold for a 3% profit and the other was sold for 7% profit. What was the purchase price of each coin? 71. A car dealership has two vehicles which were bought for $14,000 (total). The dealership was able to sell the first vehicle for an 11% profit and the second for a 6% profit. The total profit for the dealership was $1,100. What did the dealership originally pay for each vehicle? 72. A real estate investor bought two parcels of land for $1,000,000. Several years later, the investor sold these parcels for a $10,000 profit. One of the parcels was sold for a 1% loss and the other was sold for a 4% profit. What was the purchase price of each parcel of land? Temperatures. The relationship between Fahrenheit and Celsius temperature readings is: F where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius. 73. Find the temperature in degrees Fahrenheit when the temperature is 74. Find the temperature in degrees Celsius when the temperature is 500 350 Celsius. Fahrenheit. = C + 32, 10 CHAPTER 1 Linear Equations and Inequalities 75. Determine the temperature where both of these temperature scales have the same numerical reading. Hint: Change the equation F = C + 32 by replacing the F with C since both F and C will have the same numerical value for this particular temperature. Then solve the resulting equation to find C. * Future Value of an Investment. When money is invested and will receive simple interest, the future value of the investment is given by S = P + Prt, where S is the future value of the investment, P is the principal amount invested, r is the interest rate per time interval, and t is the time interval. (Usually the time interval is in years so the interest rate will be the annual interest rate.) 76. At what annual interest rate must $1,000 be invested so that its Future Value is $2,300 after 5 years? 77. For how many years would $5,000 need to be invested at a 5% annual interest rate to have an invest ment with a Future Value of $7,000? 78. How much of a principal amount would need to be invested at a 2% annual interest rate for 10 years to have an investment with a Future Value of $9,000? 79. For how many years would $1,000 need to be invested at a 1% annual interest rate to have an invest ment with a Future Value of $2,000? 80. Sierra has 2 liters of a mixture containing 45% of boric acid. How much water must be added to make the mixture 36% boric acid? 81. There are three consecutive even integers. Half the sum of the second and third numbers is 55. What are the integers? 82. A hardware store employee is paid a base salary of $2,000 per month plus an 8% commission on sales over $7,000. How much must the employee sell to earn $4,000 for the month? 83. If the low temperature for a 24-hour period at an arctic outpost was —49F°, vhat was the temperature in degrees Celsius? 84. A 30-foot piece of cable is cut into two pieces. One piece is 2 feet longer than the other. How long are the pieces of cable? 85. Five plus twice a number is seven times the number. What is the number? 86. The perimeter of a basketball court is 104 ft. and the length is 15 ft. more than the width. What are the dimensions of the court? 87. A rectangular field has a perimeter of 170 m. The length is Sm more than the width. Find the dimen sions of the field. 88. A car rents for $45 per day with unlimited mileage, or for $25 per day plus 18 per mile. How many miles need to be traveled for the rates to be equal? 89. Find three consecutive integers whose sum is 36. Section 1.1 Linear Equations in One Variable ii 90. A mixture of silver and copper alloy weighs 128 pounds and contains 12 pounds of silver. How many pounds of silver must be added in order that every 8 pound s of the resulting alloy will contain 1 pound of silver. (Hint: find the constant quantity.) 91. IQ (Intelligence Quotient) is measured by dividing Menta l Age (MA) by chronological age (CA) and multiplying by 100. A person with a mental age of 12 and a chronological age of 8 has an IQ of 150. If a 9-year-old girl has an IQ of 140, what is her mental age? 92. If a group of 14-year-old children have IQs that range from 85 to 137, what is their range of mental ages? 93. A 30-foot piece of rope is cut into two pieces. One piece is 5 feet longer than the other. How long are the pieces of rope? 94. A student has test scores of 93, 89, 72, 80, and 96. What must the sixth test score be to have an average of 88? 95. A loan is obtained at 7% simple interest. After one year $856.00 pays off the loan. How much was the loan? 96. A car sells for $22,461.00 and this price includes 6.75% sales tax. How much is the sales tax? 97. A small business reported profits during 2008 of $170,000. If this was 147% of the 2007 year profits, what was the profit in 2007? 98. Morgan received her paycheck this week. From her gross pay, 37% was deducted for federal taxes, 8% for FICA, 8% for state taxes, and 3% for her 401K. She receive d $660 after the deductions. What is her gross pay? - ,? ‘ ‘;:- :.- 1.2 Homework For problems 1-45, solve the linear inequalities, and graph the solu tions on a real number line: 1. 2+5<9 16. 5-x>-5 2. x+28 3. 1+x 17. 5—x<6 4. 5x+6>21 18. 5. 4x+7>—17 6. 5—3x--1O 7. 9—2x3 19. 20. 8. 4—6x<2 9. 5x+3—3 3x > 3 3 + x 2 21. 1 10. 22. 2+4<+5+i 11. 23. 12. 24. 13. 25. 14. x—25 26. 2x+35 +1<3 27. —4<2x—3<9 15. 1—x—4 28. 17 18 CHAPTER 1 Linear Equations and Inequalities 29. 2—x4(x—2) 38. (2w—1)>10 3 30. —1+5i0 S 39. .(3x+4)<25 31. 11<x—631 40. ——--5-4 —12 32. 2 6 ) 3 41. —5(y+ 33. 5(x+3)+9<3(x—2)+6 42. 20>2(q+i1) 34. .(6x+24)—20—(12—72) 43. —22>—7+15 35. y+(y—3)(y+2)—(y+i) 44. -+3<12 36. (5÷4x)+1738 I — 45. v+24<3v+29 37. 3x+7<11 I For probiems 46—50, translate to an inequality: 46. The price of a movie ticket is less than $8. 47. A number is greater than —4.3. 48. A number is less than 12 and greater than 7. 49. A number is less than —4 and greater than—il. 50. Three times a number is less than 13 and greater than —3. 4 For problems 51—70, answer the given question. 51. Four less than three times a number is no more than eleven. What range of values can this number have? 52. Five more than four times a number is at least 21. What range of values can this number have? 53. wo times a less than 10 1 What range of values can this number have? eas of 54. Six more than two times a number is at least as much as four times that same number. What range values can this number have? 55. Maria has gotten scores of 75, 77, 81 and 84 (out of a possible 100) on the four midterm examinations for a math class. If each of these midterms counts as 15% of the total grade in the class and the final exam counts as 40% of the total grade, what must she get as a score (out of 100 possible) on the final examination to get at least 85% as her total grade in the class? Section 1.2 Linear Inequalities in One Variable 19 56. An investor wants to invest a total of $10,000 in two different accounts. The riskier investment yields an annual average of 9.5% profit and the safer investment has an annual yield of 4.5%. How much money should be invested in each account in order to earn at least $600 profit in a year? 57. A company invests a total of $100,000 in two different accounts. The riskier investment yields an annual average of 7.5% profit and the safer investment has an annual yield of 4.0%. How much money should be invested in each account in order to earn at least $6,000 profit in a year? 58. During a 24-hotir period at an Arctic station, the temperature range was —SIF° to hF 0 (—51 What is this range in degrees Celsius? F 11). 59. An employee is given a base salary of $2,000 a month plus an 8% commission on sales over $7,000 dur ing the month. How much in total sales must the employee have to make at least $4,000 for the month? 60. Your new job allows you to choose between two ways of being paid: Monthly Salary Sales Commission (1) $800 (2) $1000 4% of all sales 6% of sales over $10,000 When is plan (1) better than plan (2)? 61. If the revenue function is R(x) be sold to produce a profit? = 40x and the cost function is C() = 20x + 1600, how many items must 62. The wind chill factor is a temperature in still air that is the equivalent of the ambient temperature plus the depressing effect a wind has on temperature due to evaporation of water on skin. For a wind speed of 25 mph the wind chill can be expressed by: WC= 1.479t—43.821, where t= temperature. What range of temperatures does the wind chill factor of 30°F represent? (Hint: Find t such that WCt—30.) 63. A video camera sells for $695 and tapes for $5.75 each. How many tapes can be purchased along with the camera for $900? 64. A line of t-shirts sells for $3.50 each. The fixed costs to manufacture them is $6500 and it costs $1.30 to make each t-shirt. How many t-shirts must be sold to realize a profit? 65. A magazine costs $0.60 to produce and is sold to dealers for $0.55. The publisher also gets advertising fees of 10% of the amount received for all magazines sold to dealers over 10,000. How many magazines must be published to make a profit? 66. The relationship between height, H, in inches and age, A, in boys 4 to 16 years of age is related by: H=2.31A +31.26 Normal variation at these ages is ± 5% of H. Express the variability, as an inequality, in height of boys 10.5 years old. F 1.3 Homework For problems 1—15, find the slope of the line through each set of two points: 1. (1, 2) and (3, 4) 9. (—3, 3) and (3, 3) (i-, -) and (2, 3) 2. (—8, 8) and (5, —5) 10. 3. (3, 7) and (8, —8) 11. (—16.3, 12.4) and (—5.2, 8.7) 4. (3, —5) and (2, 5) 12. 5. (—1, —5) and (5, —8) 13. (14, 3) and (2, 12) 6. (1, —1) and (—2, —3) 14. (3, 2) and (—7, —5) 7. (8, —8) and (8, 7) 15. (7, 1) and (10, —3) (, -i-) and (—* -) 8. (—2, —10) and (6, —6) Determine whether lines with these slopes are parallel, perpendicular or neither: =1, 2 1 16. m m = 1 17. 1 m = 2, 18. 1 , 19. 1 = m 20. m = 1 —, 2 m = —3 2 m -2 1 ni = 2 —— 3 3 2 1 m = ,m — 21. 1 2 m = 3 Find the slope and y-intercept for these lines, fthe’ exist: 22. x+y2 25. 4y—5x=12 23. 6x+y=—1 26. 8x+3y+9=0 24. 3x—2y=6 27. 7x—lOy=4 27 28 CHAPTER 1 Linear Equations and Inequalities 34. 3p—2x=5+9p—3 28. p—8x=12 35. 4x=9y+36 29. 3x=12+y 36. —6x=4p+3 30. 5x—6=15 37. 5x+4y=16 31. 2x—17=--p 38. p=3.8x 32. 4—5p÷7x=—10 39. —8x--7p=24 33. 17p+4x+3=7+4x tch the line. the slope and p-intercept, and ske Write the equation of the line given = p-intercept is (0, —5) 40. m = 1, p-intercept is (0, —2) 46. m 41. m =—1, p-intercept is (0, 9) 47. rn = 5, p-intercept is (0, 8) —, 3, p-intercept is (0, —8) 42. m 48.m 43. m =0, p-intercept is (0, 2) Y-intercePt is (0, 28) = 49. m = 3, p-intercept is (0, —3) 44. m = 45. m = , p-intercept is (0, —, ) p-intercept is (0, 4) 50. m =0, p-intercept is (0, 5) (2, 0) 51. m is undefined, x-intercept is passes through, and sketch the line. en the slope and a point that the line Write the equation of the line giv 5 point is (1, 2) 52. m = 7, point is (0, 6) 58. m = 53. m=2, pointis(1,1) 59. m = 54. m =—1, point is (1, 9) 60. m=5, pointis (5, 4) 55. m=—3, point is (—2, 4) 56. m = 57. m = , 4, point is (3, 2) 61. m=, pointis(6,7) 1, point is (—2, —1) —i, point is (10, —4) 62. m = —4 point is (4, 5) 0 0 0 (1 Section 1.3 Equations of Lines 29 Write the equation of the line that passes through the two given points, and sketch the line. 63. (1, 3) and (—1, 5) 69. (—4, 3) and (8, 10) 64. (1, —6) and (—1, 3) 70. (1, 4) and (5, 6) 65. (—9, —1) and (6, —6) 71. (—1, —1) and (2, 2) 66. (—7, —2) and (—5, 1) 72. (0, —3) and (7, 0) 67. (—7, —2) and (5, 1) 73. (0, 0) and (—3, 4) 68. (—8, 1) and (4, 5) 74. (, _) (., and 6) Determine whether the lines are parallel or perpendicular to each other: 75. y=x÷6, y—x=—2 76. y+7x=—9, y=—7x—1 77. 2x—3yr7, 3xzr—2y--6 78. y4x—5, 4y=8—x 79. 2x—7=y, y—2x=8 Write the equation of the line that passes through the given point and is parallel to the given line: 80. The point (2, —1) and the line 3 + = 10 83. The point (3, 7) and the line x + 81. The point (—3, 8) and the line 2x + = 3 84. The point (0, 3) and the line 2x 82. The point (2, —8) and the line 4x — y=4 6 = — y=7 85. The point (—7, 0) and the line 2y + 5x = 6 Write the equation of the line that passes through the given point and is perpendicular to the given line: 86. The point (2, —1) and the line 3x 87. The point (7, 8) and the line x + 4y =10 89. The point (—3, 4) and the line 2x 7 90. The point (5, 2) and the line 3x y=5 91. The point (—2, 2) and the line x + 88. The point (—3, —1) and the line 8x — — — + y=5 = 6 5y = 6