Finding a rental 2.6 7 a2 answers

2

73

Equations and Inequalities 2

Figure 1

Introduction For most people, the term territorial possession indicates restrictions, usually dealing with trespassing or rite of passage and takes place in some foreign location. What most Americans do not realize is that from September through December, territorial possession dominates our lifestyles while watching the NFL. In this area, territorial possession is governed by the referees who make their decisions based on what the chains reveal. If the ball is at point A (x1, y1), then it is up to the quarterback to decide which route to point B (x2, y2), the end zone, is most feasible.

ChAPTeR OUTlIne

2.1 The Rectangular Coordinate Systems and graphs 2.2 linear equations in One variable 2.3 models and Applications 2.4 Complex numbers 2.5 Quadratic equations 2.6 Other Types of equations 2.7 linear Inequalities and Absolute value Inequalities

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

84 CHAPTER 2 equatioNs aNd iNequalities

2.1 SeCTIOn exeRCISeS

veRbAl 1. Is it possible for a point plotted in the Cartesian

coordinate system to not lie in one of the four quadrants? Explain.

2. Describe the process for finding the x-intercept and the y-intercept of a graph algebraically.

3. Describe in your own words what the y-intercept of a graph is.

4. When using the distance formula d = √ ——

(x2 − x1) 2 + (y2 − y1)

2 , explain the correct order of operations that are to be performed to obtain the correct answer.

AlgebRAIC For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

5. y = −3x + 6 6. 4y = 2x − 1 7. 3x − 2y = 6 8. 4x − 3 = 2y

9. 3x + 8y = 9 10. 2x − 2 _ 3 = 3 _ 4 y + 3

For each of the following exercises, solve the equation for y in terms of x. 11. 4x + 2y = 8 12. 3x − 2y = 6 13. 2x = 5 − 3y 14. x − 2y = 7 15. 5y + 4 = 10x 16. 5x + 2y = 0

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

17. (−4, 1) and (3, −4) 18. (2, −5) and (7, 4) 19. (5, 0) and (5, 6) 20. (−4, 3) and (10, 3)

21. Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth. (19, 12) and (41, 71)

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points. 22. (−5, −6) and (4, 2) 23. (−1, 1) and (7, −4) 24. (−5, −3) and (−2, −8) 25. (0, 7) and (4, −9)

26. (−43, 17) and (23, −34)

gRAPhICAl

For each of the following exercises, identify the information requested.

27. What are the coordinates of the origin? 28. If a point is located on the y-axis, what is the x-coordinate?

29. If a point is located on the x-axis, what is the y-coordinate?

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).

30. (4, 1)(−2, −3)(5, 0) 31. (−1, 2)(0, 4)(2, 1)

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
SECTION 2.1 sectioN exercises 85

32. (−3, 0)(−3, 4)(−3, −3) 33. Name the coordinates of the points graphed.

34. Name the quadrant in which the following points would be located. If the point is on an axis, name the axis. a. (−3, −4) b. (−5, 0) c. (1, −4) d. (−2, 7) e. (0, −3)

For each of the following exercises, construct a table and graph the equation by plotting at least three points. 35. y = 1 _ 3 x + 2 36. y = −3x + 1 37. 2y = x + 3

nUmeRIC

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.

38. 4x − 3y = 12 39. x − 2y = 8 40. y − 5 = 5x 41. 3y = −2x + 6 42. y = x − 3 _____ 2

For each of the following exercises, use the graph in the figure below.

43. Find the distance between the two endpoints using the distance formula. Round to three decimal places.

44. Find the coordinates of the midpoint of the line segment connecting the two points.

45. Find the distance that (−3, 4) is from the origin.

46. Find the distance that (5, 2) is from the origin. Round to three decimal places.

47. Which point is closer to the origin?

TeChnOlOgy

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the 2nd CALC button and 1:value button, hit ENTER. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

48. Y1 = −2x + 5 49. Y1 = 3x − 8 ______

4 50. Y1 =

x + 5 _____ 2

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu. After graphing it, use the 2nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

A B

C

2

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

Nicol
Highlight
86 CHAPTER 2 equatioNs aNd iNequalities

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

51. Y1 = −8x + 6 52. Y1 = 4x − 7 53. Y1 = 3x + 5 ______

4 Round your answer to the nearest thousandth.

exTenSIOnS 54. A man drove 10 mi directly east from his home,

made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

55. If the road was made in the previous exercise, how much shorter would the man’s one-way trip be every day?

56. Given these four points: A(1, 3), B(−3, 5), C(4, 7), and D(5, −4), find the coordinates of the midpoint of line segments

_  AB and

_  CD .

57. After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

58. Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.

59. In the previous exercise, find the coordinates of the midpoint for each diagonal.

ReAl-WORld APPlICATIOnS 60. The coordinates on a map for San Francisco are

(53, 17) and those for Sacramento are (123, 78). Note that coordinates represent miles. Find the distance between the cities to the nearest mile.

61. If San Jose’s coordinates are (76, −12), where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.

62. A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were (49, 64). One rescue boat is at the coordinates (60, 82) and a second Coast Guard craft is at coordinates (58, 47). Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

63. A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?

(20, 50)

20

50

(0, 0) 64. If we rent a truck and pay a $75/day fee plus $.20

for every mile we travel, write a linear equation that would express the total cost y, using x to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.

2

x

y

–1–2–3–4–5–7 –1 –2 –3

1

3

321 4

4

5 6 7 8 9 11

5 6

(10, –1)

(10, 5) (–6, 5)

(–6, –1)

100 CHAPTER 2 equatioNs aNd iNequalities

2.2 SeCTIOn exeRCISeS

veRbAl

1. What does it mean when we say that two lines are parallel?

2. What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?

3. How do we recognize when an equation, for example y = 4x + 3, will be a straight line (linear) when graphed?

4. What does it mean when we say that a linear equation is inconsistent?

5. When solving the following equation: 2 _ x − 5 = 4 _ x + 1

explain why we must exclude x = 5 and x = −1 as possible solutions from the solution set.

AlgebRAIC

For the following exercises, solve the equation for x. 6. 7x + 2 = 3x − 9 7. 4x − 3 = 5 8. 3(x + 2) − 12 = 5(x + 1)

9. 12 − 5(x + 3) = 2x − 5 10. 1 _ 2 − 1 _ 3 x =

4 _ 3 11. x _ 3 −

3 _ 4 = 2x + 3 _ 12

12. 2 _ 3 x + 1 _ 2 =

31 _ 6

13. 3(2x − 1) + x = 5x + 3 14. 2x _ 3 − 3 _ 4 =

x _ 6

+ 21 _ 4

15. x + 2 _ 4 − x − 1 _ 3 = 2

For the following exercises, solve each rational equation for x. State all x-values that are excluded from the solution set.

16. 3 _ x − 1 _ 3 =

1 _ 6

17. 2 − 3 _ x + 4 = x + 2 _ x + 4 18.

3 _ x − 2 = 1 _ x − 1 +

7 __ (x − 1)(x − 2)

19. 3x _ x − 1 + 2 = 3 _ x − 1 20.

5 _ x + 1 + 1 _____ x − 3 =

−6 __________ x2 − 2x − 3

21. 1 _ x = 1 _ 5 +

3 _ 2x

For the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form.

22. (0, 3) with a slope of 2 _ 3 23. (1, 2) with a slope of − 4 _ 5

24. x-intercept is 1, and (−2, 6) 25. y-intercept is 2, and (4, −1)

26. (−3, 10) and (5, −6) 27. (1, 3) and (5, 5)

28. parallel to y = 2x + 5 and passes through the point (4, 3)

29. perpendicular to 3y = x − 4 and passes through the point (−2, 1).

For the following exercises, find the equation of the line using the given information.

30. (−2, 0) and (−2, 5) 31. (1, 7) and (3, 7)

32. The slope is undefined and it passes through the point (2, 3).

33. The slope equals zero and it passes through the point (1, −4).

34. The slope is 3 _ 4 and it passes through the point (1,4). 35. (−1, 3) and (4, −5)

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
SECTION 2.2 sectioN exercises 101

gRAPhICAl

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

36. y = 2x + 7 y = − 1 _ 2 x − 4

37. 3x − 2y = 5 6y − 9x = 6

38. y = 3x + 1 _ 4 y = 3x + 2

39. x = 4 y = −3

nUmeRIC

For the following exercises, find the slope of the line that passes through the given points. 40. (5, 4) and (7, 9) 41. (−3, 2) and (4, −7) 42. (−5, 4) and (2, 4) 43. (−1, −2) and (3, 4) 44. (3, −2) and (3, −2)

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

45. (−1, 3) and (5, 1) (−2, 3) and (0, 9)

46. (2, 5) and (5, 9) (−1, −1) and (2, 3)

TeChnOlOgy

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

47. 0.537x − 2.19y = 100 48. 4,500x − 200y = 9,528 49. 200 − 30y

_ x = 70

exTenSIOnS 50. Starting with the point-slope formula

y − y1 = m(x − x1), solve this expression for x in terms of x1, y, y1, and m.

51. Starting with the standard form of an equation Ax + By = C, solve this expression for y in terms of A, B, C, and x. Then put the expression in slope-intercept form.

52. Use the above derived formula to put the following standard equation in slope intercept form: 7x − 5y = 25.

53. Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular. (−1, 1), (2, 0), (3, 3), and (0, 4)

54. Find the slopes of the diagonals in the previous exercise. Are they perpendicular?

ReAl-WORld APPlICATIOnS 55. The slope for a wheelchair ramp for a home has to

be 1 _ 12 . If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the home in order to comply with the needed slope.

x feet 2.5 feet

56. If the profit equation for a small business selling x number of item one and y number of item two is p = 3x + 4y, find the y value when p = $453 and x = 75.

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be y = 45 + 0.25x, where x is the number of miles traveled.

57. What is your cost if you travel 50 mi? 58. If your cost were $63.75, how many miles were you charged for traveling?

59. Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
108 CHAPTER 2 equatioNs aNd iNequalities

2.3 SeCTIOn exeRCISeS

veRbAl

1. To set up a model linear equation to fit real-world applications, what should always be the first step?

2. Use your own words to describe this equation where n is a number: 5(n + 3) = 2n

3. If the total amount of money you had to invest was $2,000 and you deposit x amount in one investment, how can you represent the remaining amount?

4. If a man sawed a 10-ft board into two sections and one section was n ft long, how long would the other section be in terms of n ?

5. If Bill was traveling v mi/h, how would you represent Daemon’s speed if he was traveling 10 mi/h faster?

ReAl-WORld APPlICATIOnS

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.

6. Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

7. Beth and Ann are joking that their combined ages equal Sam’s age. If Beth is twice Ann’s age and Sam is 69 yr old, what are Beth and Ann’s ages?

8. Ben originally filled out 8 more applications than Henry. Then each boy filled out 3 additional applications, bringing the total to 28. How many applications did each boy originally fill out?

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.

9. Find the model of the total cost of Company A’s plan, using m for the minutes.

10. Find the model of the total cost of Company B’s plan, using m for the minutes.

11. Find out how many minutes of calling would make the two plans equal.

12. If the person makes a monthly average of 200 min of calls, which plan should for the person choose?

For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use P for the number of devices that need data plans as part of their cost.

13. Find the model of the total cost of the Family Plan. 14. Find the model of the total cost of the Mobile Share Plan.

15. Assuming they stay under their data limit, find the number of devices that would make the two plans equal in cost.

16. If a family has 3 smart phones, which plan should they choose?

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
SECTION 2.3 sectioN exercises 109

For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.

17. If we let x be the amount the woman invests in the 15% bond, how much will she be able to invest in the CD?

18. Set up and solve the equation for how much the woman should invest in each option to sustain a $6,000 annual return.

19. Two planes fly in opposite directions. One travels 450 mi/h and the other 550 mi/h. How long will it take before they are 4,000 mi apart?

20. Ben starts walking along a path at 4 mi/h. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at 6 mi/h. How long will it be before Amanda catches up to Ben?

21. Fiora starts riding her bike at 20 mi/h. After a while, she slows down to 12 mi/h, and maintains that speed for the rest of the trip. The whole trip of 70 mi takes her 4.5 h. For what distance did she travel at 20 mi/h?

22. A chemistry teacher needs to mix a 30% salt solution with a 70% salt solution to make 20 qt of a 40% salt solution. How many quarts of each solution should the teacher mix to get the desired result?

23. Paul has $20,000 to invest. His intent is to earn 11% interest on his investment. He can invest part of his money at 8% interest and part at 12% interest. How much does Paul need to invest in each option to make get a total 11% return on his $20,000?

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.

24. Write the model equation for the cost of renting a truck with plan A.

25. Write the model equation for the cost of renting a truck with plan B.

26. Find the number of miles that would generate the same cost for both plans.

27. If Tim knows he has to travel 300 mi, which plan should he choose?

28. A = P(1 + rt) is used to find the principal amount P deposited, earning r% interest, for t years. Use this to find what principal amount P David invested at a 3% rate for 20 yr if A = $8,000.

29. The formula F = mv 2 _

R relates force (F), velocity (v),

mass (m), and resistance (R). Find R when m = 45, v = 7, and F = 245.

30. F = ma indicates that force (F) equals mass (m) times acceleration (a). Find the acceleration of a mass of 50 kg if a force of 12 N is exerted on it.

31. Sum = 1 _ 1 − r is the formula for an infinite series

sum. If the sum is 5, find r.

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

32. Solve for W: P = 2L + 2W 33. Use the formula from the previous question to find the width, W, of a rectangle whose length is 15 and whose perimeter is 58.

34. Solve for f: 1 _ p + 1 _ q =

1_ f 35. Use the formula from the previous question to find f

when p = 8 and q = 13.

36. Solve for m in the slope-intercept formula: y = mx + b

37. Use the formula from the previous question to find m when the coordinates of the point are (4, 7) and b = 12.

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
110 CHAPTER 2 equatioNs aNd iNequalities

38. The area of a trapezoid is given by A = 1 _ 2 h(b1 + b2).

Use the formula to find the area of a trapezoid with h = 6, b1 = 14, and b2 = 8.

39. Solve for h: A = 1 _ 2 h(b1 + b2)

40. Use the formula from the previous question to find the height of a trapezoid with A = 150, b1 = 19, and b2 = 11.

41. Find the dimensions of an American football field. The length is 200 ft more than the width, and the perimeter is 1,040 ft. Find the length and width. Use the perimeter formula P = 2L + 2W.

42. Distance equals rate times time, d = rt. Find the distance Tom travels if he is moving at a rate of 55 mi/h for 3.5 h.

43. Using the formula in the previous exercise, find the distance that Susan travels if she is moving at a rate of 60 mi/h for 6.75 h.

44. What is the total distance that two people travel in 3 h if one of them is riding a bike at 15 mi/h and the other is walking at 3 mi/h?

45. If the area model for a triangle is A = 1 _ 2 bh, find the area of a triangle with a height of 16 in. and a base of 11 in.

46. Solve for h: A = 1 _ 2 bh 47. Use the formula from the previous question to find the height to the nearest tenth of a triangle with a base of 15 and an area of 215.

48. The volume formula for a cylinder is V = πr2 h. Using the symbol π in your answer, find the volume of a cylinder with a radius, r, of 4 cm and a height of 14 cm.

49. Solve for h: V = πr2h

50. Use the formula from the previous question to find the height of a cylinder with a radius of 8 and a volume of 16π

51. Solve for r: V = πr2h

52. Use the formula from the previous question to find the radius of a cylinder with a height of 36 and a volume of 324π.

53. The formula for the circumference of a circle is C = 2πr. Find the circumference of a circle with a diameter of 12 in. (diameter = 2r). Use the symbol π in your final answer.

54. Solve the formula from the previous question for π. Notice why π is sometimes defined as the ratio of the circumference to its diameter.

Nicol
Highlight
118 CHAPTER 2 equatioNs aNd iNequalities

2.4 SeCTIOn exeRCISeS

veRbAl 1. Explain how to add complex numbers. 2. What is the basic principle in multiplication of

complex numbers? 3. Give an example to show that the product of two

imaginary numbers is not always imaginary. 4. What is a characteristic of the plot of a real number

in the complex plane?

AlgebRAIC For the following exercises, evaluate the algebraic expressions.

5. If y = x2 + x − 4, evaluate y given x = 2i. 6. If y = x3 − 2, evaluate y given x = i. 7. If y = x2 + 3x + 5, evaluate y given x = 2 + i. 8. If y = 2x2 + x − 3, evaluate y given x = 2 − 3i.

9. If y = x + 1 _ 2 − x , evaluate y given x = 5i. 10. If y = 1 + 2x _ x + 3 , evaluate y given x = 4i.

gRAPhICAl For the following exercises, plot the complex numbers on the complex plane.

11. 1 − 2i 12. −2 + 3i 13. i 14. −3 − 4i

nUmeRIC For the following exercises, perform the indicated operation and express the result as a simplified complex number.

15. (3 + 2i) + (5 − 3i) 16. (−2 − 4i) + (1 + 6i) 17. (−5 + 3i) − (6 − i) 18. (2 − 3i) − (3 + 2i)

19. (−4 + 4i) − (−6 + 9i) 20. (2 + 3i)(4i) 21. (5 − 2i)(3i) 22. (6 − 2i)(5)

23. (−2 + 4i)(8) 24. (2 + 3i)(4 − i) 25. (−1 + 2i)(−2 + 3i) 26. (4 − 2i)(4 + 2i)

27. (3 + 4i)(3 − 4i) 28. 3 + 4i _ 2 29. 6 − 2i _ 3 30.

−5 + 3i _ 2i

31. 6 + 4i _ i 32. 2 − 3i _

4 + 3i 33. 3 + 4i _

2 − i 34. 2 + 3i _

2 − 3i

35. √ —

−9 + 3 √ —

−16 36. − √ —

−4 − 4 √ —

−25 37. 2 + √ —

−12 __________ 2

38. 4 + √ —

−20 __________ 2

39. i8 40. i15 41. i22

TeChnOlOgy For the following exercises, use a calculator to help answer the questions.

42. Evaluate (1 + i)k for k = 4, 8, and 12. Predict the value if k = 16.

43. Evaluate (1 − i)k for k = 2, 6, and 10. Predict the value if k = 14.

44. Evaluate (l + i)k − (l − i)k for k = 4, 8, and 12. Predict the value for k = 16.

45. Show that a solution of x6 + 1 = 0 is √ — 3 ____

2 + 1 _ 2 i.

46. Show that a solution of x8 −1 = 0 is √ — 2 ____

2 + √

— 2 ____

2 i.

exTenSIOnS For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

47. 1 __ i + 4 __

i3 48. 1 __

i11 − 1 __

i21 49. i7(1 + i2) 50. i−3 + 5i7

51. (2 + i)(4 − 2i) ____________ (1 + i)

52. (1 + 3i)(2 − 4i) _____________ (1 + 2i)

53. (3 + i) 2

_______ (1 + 2i)2

54. 3 + 2i ______ 2 + i

+ (4 + 3i)

55. 4 + i _____ i + 3 − 4i ______

1 − i 56. 3 + 2i ______

1 + 2i − 2 − 3i ______

3 + i

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
SECTION 2.5 sectioN exercises 129

2.5 SeCTIOn exeRCISeS

veRbAl 1. How do we recognize when an equation is

quadratic? 2. When we solve a quadratic equation, how many

solutions should we always start out seeking? Explain why when solving a quadratic equation in the form ax2 + bx + c = 0 we may graph the equation y = ax2 + bx + c and have no zeroes (x-intercepts).

3. When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

4. In the quadratic formula, what is the name of the expression under the radical sign b2 − 4ac, and how does it determine the number of and nature of our solutions?

5. Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.

AlgebRAIC For the following exercises, solve the quadratic equation by factoring.

6. x 2 + 4x − 21 = 0 7. x 2 − 9x + 18 = 0 8. 2x 2 + 9x − 5 = 0 9. 6x 2 + 17x + 5 = 0 10. 4x 2 − 12x + 8 = 0 11. 3x 2 − 75 = 0 12. 8x 2 + 6x − 9 = 0 13. 4x 2 = 9 14. 2x 2 + 14x = 36 15. 5x 2 = 5x + 30 16. 4x 2 = 5x 17. 7x 2 + 3x = 0

18. x _ 3 − 9 _ x = 2

For the following exercises, solve the quadratic equation by using the square root property. 19. x2 = 36 20. x 2 = 49 21. (x − 1)2 = 25 22. (x − 3)2 = 7 23. (2x + 1)2 = 9 24. (x − 5)2 = 4

For the following exercises, solve the quadratic equation by completing the square. Show each step. 25. x2 − 9x − 22 = 0 26. 2x 2 − 8x − 5 = 0 27. x 2 − 6x = 13 28. x 2 + 2 _ 3 x −

1 _ 3 = 0 29. 2 + z = 6z2 30. 6p2 + 7p − 20 = 0 31. 2x 2 − 3x − 1 = 0

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

32. 2x2 − 6x + 7 = 0 33. x 2 + 4x + 7 = 0 34. 3x 2 + 5x − 8 = 0 35. 9x 2 − 30x + 25 = 0 36. 2x2 − 3x − 7 = 0 37. 6x 2 − x − 2 = 0

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

38. 2x 2 + 5x + 3 = 0 39. x 2 + x = 4 40. 2x2 − 8x − 5 = 0 41. 3x2 − 5x + 1 = 0

42. x 2 + 4x + 2 = 0 43. 4 + 1 _ x − 1 _ x2

= 0

TeChnOlOgy

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero, enter), then right bound (move your cursor to the right of the zero, enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

44. Y1 = 4x 2 + 3x − 2 45. Y1 = −3x

2 + 8x − 1 46. Y1 = 0.5x 2 + x − 7

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
Nicol
Highlight
130 CHAPTER 2 equatioNs aNd iNequalities

47. To solve the quadratic equation x2 + 5x − 7 = 4, we can graph these two equations Y1 = x

2 + 5x − 7 Y2 = 4 and find the points of intersection. Recall 2nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

48. To solve the quadratic equation 0.3x2 + 2x − 4 = 2, we can graph these two equations Y1 = 0.3x

2 + 2x − 4 Y2 = 2 and find the points of intersection. Recall 2nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

exTenSIOnS 49. Beginning with the general form of a quadratic

equation, ax2 + bx + c = 0, solve for x by using the completing the square method, thus deriving the quadratic formula.

50. Show that the sum of the two solutions to the

quadratic equation is − b _ a .

51. A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.2. Solve the quadratic equation to find the length and width.

52. Abercrombie and Fitch stock had a price given as P = 0.2t2 − 5.6t + 50.2, where t is the time in months from 1999 to 2001. ( t = 1 is January 1999). Find the two months in which the price of the stock was $30.

53. Suppose that an equation is given p = −2x 2 + 280x − 1000, where x represents the number of items sold at an auction and p is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2nd CALC maximum. To obtain a good window for the curve, set x [0,200] and y [0,10000].

ReAl-WORld APPlICATIOnS 54. A formula for the normal systolic blood pressure for

a man age A, measured in mmHg, is given as P = 0.006A2 − 0.02A + 120. Find the age to the nearest year of a man whose normal blood pressure measures 125 mmHg.

55. The cost function for a certain company is C = 60x + 300 and the revenue is given by R = 100x − 0.5x2. Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $300.

56. A falling object travels a distance given by the formula d = 5t + 16t 2 ft, where t is measured in seconds. How long will it take for the object to traveled 74 ft?

57. A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft 2. Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.

x

x

x x

12 feet

15 feet

58. An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, P, who contracted the flu t days after it broke out is given by the model P = − t 2 + 13t + 130, where 1 ≤ t ≤ 6. Find the day that 160 students had the flu. Recall that the restriction on t is at most 6.

Nicol
Highlight
Nicol
Highlight
SECTION 2.6 sectioN exercises 141

2.6 SeCTIOn exeRCISeS

veRbAl 1. In a radical equation, what does it mean if a number

is an extraneous solution? 2. Explain why possible solutions must be checked in

radical equations. 3. Your friend tries to calculate the value − 9

3 _ 2 and keeps getting an ERROR message. What mistake is he or she probably making?

4. Explain why |2x + 5| = −7 has no solutions.

5. Explain how to change a rational exponent into the correct radical expression.

AlgebRAIC For the following exercises, solve the rational exponent equation. Use factoring where necessary.

6. x 2 _ 3 = 16 7. x

3 _ 4 = 27 8. 2 x 1 _ 2 − x

1 _ 4 = 0 9. (x − 1) 3 _ 4 = 8

10. (x + 1) 2 _ 3 = 4 11. x

2 _ 3 − 5x 1 _ 3 + 6 = 0 12. x

7 _ 3 − 3x 4 _ 3 − 4x

1 _ 3 = 0

For the following exercises, solve the following polynomial equations by grouping and factoring. 13. x 3 + 2x 2 − x − 2 = 0 14. 3x 3 − 6x 2 − 27x + 54 = 0 15. 4y 3 − 9y = 0 16. x 3 + 3x 2 − 25x − 75 = 0 17. m3 + m2 − m − 1 = 0 18. 2x 5 −14x 3 = 0 19. 5x 3 + 45x = 2x 2 + 18

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

20. √ —