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

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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

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

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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)

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?

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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?

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.

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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.

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 = 2×2 + 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

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. 2×2 − 6x + 7 = 0 33. x 2 + 4x + 7 = 0 34. 3x 2 + 5x − 8 = 0 35. 9x 2 − 30x + 25 = 0 36. 2×2 − 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. 2×2 − 8x − 5 = 0 41. 3×2 − 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

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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 0.3×2 + 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 − 0.5×2. 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.

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. √ —

3x − 1 − 2 = 0 21. √ —

x − 7 = 5 22. √ —

x − 1 = x − 7

23. √ —

3t + 5 = 7 24. √ —

t + 1 + 9 = 7 25. √ —

12 − x = x

26. √ —

2x + 3 − √ —

x + 2 = 2 27. √ —

3x + 7 + √ —

x + 2 = 1 28. √ —

2x + 3 − √ —

x + 1 = 1

For the following exercises, solve the equation involving absolute value. 29. |3x − 4| = 8 30. |2x − 3| = −2 31. |1 − 4x| − 1 = 5 32. |4x + 1| − 3 = 6 33. |2x − 1| − 7 = −2 34. |2x + 1| − 2 = −3 35. |x + 5| = 0 36. −|2x + 1| = −3

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

37. x 4 − 10x 2 + 9 = 0 38. 4(t − 1)2 − 9(t − 1) = −2 39. (x 2 − 1)2 + (x 2 − 1) − 12 = 0 40. (x + 1)2 − 8(x + 1) − 9 = 0 41. (x − 3)2 − 4 = 0

exTenSIOnS For the following exercises, solve for the unknown variable.

42. x−2 − x−1 − 12 = 0 43. √ —

|x|2 = x 44. t 25 − t 5 + 1 = 0 45. |x 2 + 2x − 36| = 12

ReAl-WORld APPlICATIOnS For the following exercises, use the model for the period of a pendulum, T, such that T = 2π √

___

L _ g , where the length of the pendulum is L and the acceleration due to gravity is g.

46. If the acceleration due to gravity is 9.8 m/s2 and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).

47. If the gravity is 32 ft/s2 and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.

For the following exercises, use a model for body surface area, BSA, such that BSA = √ _____

wh _ 3600

, where w = weight in kg and h = height in cm.

48. Find the height of a 72-kg female to the nearest cm whose BSA = 1.8.

49. Find the weight of a 177-cm male to the nearest kg whose BSA = 2.1.

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SECTION 2.7 sectioN exercises 149

2.7 SeCTIOn exeRCISeS

veRbAl

1. When solving an inequality, explain what happened from Step 1 to Step 2: Step 1 −2x > 6

Step 2 x < −3

2. When solving an inequality, we arrive at: x + 2 < x + 3 2 < 3 Explain what our solution set is.

3. When writing our solution in interval notation, how do we represent all the real numbers?

4. When solving an inequality, we arrive at: x + 2 > x + 3 2 > 3 Explain what our solution set is.

5. Describe how to graph y = |x − 3|

AlgebRAIC

For the following exercises, solve the inequality. Write your final answer in interval notation 6. 4x − 7 ≤ 9 7. 3x + 2 ≥ 7x − 1 8. −2x + 3 > x − 5

9. 4(x + 3) ≥ 2x − 1 10. − 1 _ 2 x ≤ − 5 _ 4 +

2 _ 5 x 11. −5(x − 1) + 3 > 3x − 4 − 4x

12. −3(2x + 1) > −2(x + 4) 13. x + 3 _ 8 − x + 5 _ 5 ≥

3 _ 10 14. x − 1 _ 3 +

x + 2 _ 5 ≤ 3 _ 5

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. 15. |x + 9| ≥ −6 16. |2x + 3| < 7 17. |3x − 1| > 11

18. |2x + 1| + 1 ≤ 6 19. |x − 2| + 4 ≥ 10 20. |−2x + 7| ≤ 13

21. |x − 7| < −4 22. |x − 20| > −1 23. | x − 3 _____ 4 | < 2

For the following exercises, describe all the x-values within or including a distance of the given values. 24. Distance of 5 units from the number 7 25. Distance of 3 units from the number 9

26. Distance of 10 units from the number 4 27. Distance of 11 units from the number 1

For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

28. −4 < 3x + 2 ≤ 18 29. 3x + 1 > 2x − 5 > x − 7

30. 3y < 5 − 2y < 7 + y 31. 2x − 5 < −11 or 5x + 1 ≥ 6

32. x + 7 < x + 2

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150 CHAPTER 2 equatioNs aNd iNequalities

gRAPhICAl

For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

33. |x − 1| > 2 34. |x + 3| ≥ 5 35. |x + 7| ≤ 4 36. |x − 2| < 7 37. |x − 2| < 0

For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.

38. x + 3 < 3x − 4 39. x − 2 > 2x + 1 40. x + 1 > x + 4 41. 1 __ 2

x + 1 > 1 __ 2

x − 5

42. 4x + 1 < 1 _ 2 x + 3

nUmeRIC

For the following exercises, write the set in interval notation.

43. {x|−1 < x < 3} 44. {x|x ≥ 7} 45. {x|x < 4} 46. { x| x is all real numbers}

For the following exercises, write the interval in set-builder notation.

47. (−∞, 6) 48. (4, ∞) 49. [−3, 5) 50. [−4, 1] ∪ [9, ∞)

For the following exercises, write the set of numbers represented on the number line in interval notation.

51. −2 −1

52. −1 −2

53. 4

TeChnOlOgy

For the following exercises, input the left-hand side of the inequality as a Y 1 graph in your graphing utility. Enter Y2= the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation.

54. |x + 2| − 5 < 2 55. − 1 _ 2 |x + 2| < 4 56. |4x + 1| − 3 > 2 57. |x − 4| < 3

58. |x + 2| ≥ 5

exTenSIOnS

59. Solve |3x + 1| = |2x + 3| 60. Solve x2 − x > 12

61. x − 5 _____ x + 7

≤ 0, x ≠ −7 62. p = −x 2 + 130x − 3,000 is a profit formula for a small business. Find the set of x-values that will keep this profit positive.

ReAl-WORld APPlICATIOnS

63. In chemistry the volume for a certain gas is given by V = 20T, where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, find the set of volume values.

64. A basic cellular package costs $20/mo. for 60 min of calling, with an additional charge of $0.30/min beyond that time. The cost formula would be C = $20 + .30(x − 60). If you have to keep your bill lower than $50, what is the maximum calling minutes you can use?

CHAPTER 2 review 151

ChAPTeR 2 RevIeW

Key Terms absolute value equation an equation in which the variable appears in absolute value bars, typically with two solutions, one

accounting for the positive expression and one for the negative expression

area in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: A = LW

Cartesian coordinate system a grid system designed with perpendicular axes invented by René Descartes

completing the square a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square

complex conjugate a complex number containing the same terms as another complex number, but with the opposite operator. Multiplying a complex number by its conjugate yields a real number.

complex number the sum of a real number and an imaginary number; the standard form is a + bi, where a is the real part and b is the complex part.

complex plane the coordinate plane in which the horizontal axis represents the real component of a complex number, and the vertical axis represents the imaginary component, labeled i.

compound inequality a problem or a statement that includes two inequalities

conditional equation an equation that is true for some values of the variable

discriminant the expression under the radical in the quadratic formula that indicates the nature of the solutions, real or complex, rational or irrational, single or double roots.

distance formula a formula that can be used to find the length of a line segment if the endpoints are known

equation in two variables a mathematical statement, typically written in x and y, in which two expressions are equal

equations in quadratic form equations with a power other than 2 but with a middle term with an exponent that is one- half the exponent of the leading term

extraneous solutions any solutions obtained that are not valid in the original equation

graph in two variables the graph of an equation in two variables, which is always shown in two variables in the two- dimensional plane

identity equation an equation that is true for all values of the variable

imaginary number the square root of −1: i = √ —

−1 .

inconsistent equation an equation producing a false result

intercepts the points at which the graph of an equation crosses the x-axis and the y-axis

interval an interval describes a set of numbers within which a solution falls

interval notation a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends

linear equation an algebraic equation in which each term is either a constant or the product of a constant and the first power of a variable

linear inequality similar to a linear equation except that the solutions will include sets of numbers

midpoint formula a formula to find the point that divides a line segment into two parts of equal length

ordered pair a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, (x, y)

origin the point where the two axes cross in the center of the plane, described by the ordered pair (0, 0)

perimeter in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: P = 2L + 2W

polynomial equation an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents

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152 CHAPTER 2 equatioNs aNd iNequalities

Pythagorean Theorem a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems

quadrant one quarter of the coordinate plane, created when the axes divide the plane into four sections

quadratic equation an equation containing a second-degree polynomial; can be solved using multiple methods

quadratic formula a formula that will solve all quadratic equations

radical equation an equation containing at least one radical term where the variable is part of the radicand

rational equation an equation consisting of a fraction of polynomials

slope the change in y-values over the change in x-values

solution set the set of all solutions to an equation

square root property one of the methods used to solve a quadratic equation, in which the x 2 term is isolated so that the square root of both sides of the equation can be taken to solve for x

volume in cubic units, the volume measurement includes length, width, and depth: V = LWH

x-axis the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right

x-coordinate the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin

x-intercept the point where a graph intersects the x-axis; an ordered pair with a y-coordinate of zero

y-axis the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top

y-coordinate the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin

y-intercept a point where a graph intercepts the y-axis; an ordered pair with an x-coordinate of zero

zero-product property the property that formally states that multiplication by zero is zero, so that each factor of a quadratic equation can be set equal to zero to solve equations

Key equations quadratic formula x = −b ± √

— b2 − 4ac ______________

2a

Key Concepts

2.1 The Rectangular Coordinate Systems and Graphs • We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as

displacement from the x-axis and displacement from the y-axis. See Example 1. • An equation can be graphed in the plane by creating a table of values and plotting points. See Example 2. • Using a graphing calculator or a computer program makes graphing equations faster and more accurate.

Equations usually have to be entered in the form y = . See Example 3. • Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the

axes. See Example 4. • The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment.

See Example 5 and Example 6. • The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the

x-coordinates and the sum of the y-coordinates of the endpoints by 2. See Example 7 and Example 8.

2.2 Linear Equations in One Variable • We can solve linear equations in one variable in the form ax + b = 0 using standard algebraic properties.

See Example 1 and Example 2. • A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation.

See Example 3 and Example 4. • All solutions to a rational equation should be verified within the original equation to avoid an undefined term,

or zero in the denominator. See Example 5, Example 6, and Example 7. • Given two points, we can find the slope of a line using the slope formula. See Example 8.

CHAPTER 2 review 153

• We can identify the slope and y-intercept of an equation in slope-intercept form. See Example 9. • We can find the equation of a line given the slope and a point. See Example 10. • We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See

Example 11. • The standard form of a line has no fractions. See Example 12. • Horizontal lines have a slope of zero and are defined as y = c, where c is a constant. • Vertical lines have an undefined slope (zero in the denominator), and are defined as x = c, where c is a constant.

See Example 13. • Parallel lines have the same slope and different y-intercepts. See Example 14 and Example 15. • Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other

is vertical. See Example 16.

2.3 Models and Applications • A linear equation can be used to solve for an unknown in a number problem. See Example 1. • Applications can be written as mathematical problems by identifying known quantities and assigning a variable

to unknown quantities. See Example 2. • There are many known formulas that can be used to solve applications. Distance problems, for example, are

solved using the d = rt formula. See Example 3. • Many geometry problems are solved using the perimeter formula P = 2L + 2W, the area formula A = LW, or

the volume formula V = LWH. See Example 4, Example 5, and Example 6.

2.4 Complex Numbers • The square root of any negative number can be written as a multiple of i. See Example 1. • To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is

the real axis, and the vertical axis is the imaginary axis. See Example 2. • Complex numbers can be added and subtracted by combining the real parts and combining the imaginary

parts. See Example 3. • Complex numbers can be multiplied and divided.

◦ To multiply complex numbers, distribute just as with polynomials. See Example 4 and Example 5. ◦ To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See Example 6 and Example 7.

• The powers of i are cyclic, repeating every fourth one. See Example 8.

2.5 Quadratic Equations • Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the

equation is a difference of squares. The zero-product property is then used to find solutions. See Example 1, Example 2, and Example 3.

• Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See Example 4 and Example 5.

• Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See Example 6 and Example 7.

• Completing the square is a method of solving quadratic equations when the equation cannot be factored. See Example 8.

• A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See Example 9 and Example 10.

• The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See Example 11.

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154 CHAPTER 2 equatioNs aNd iNequalities

• The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See Example 12.

2.6 Other Types of Equations • Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To

solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See Example 1, Example 2, and Example 3.

• Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See Example 4 and Example 5.

• We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See Example 6 and Example 7.

• To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See Example 8.

• Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See Example 9 and Example 10.

• Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See Example 11.

2.7 Linear Inequalities and Absolute Value Inequalities • Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a

system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See Table 1 and Example 1 and Example 2.

• Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See Example 3, Example 4, Example 5, and Example 6.

• Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See Example 7 and Example 8.

• Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See Example 9 and Example 10.

• Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See Example 11.

CHAPTER 2 review 155

ChAPTeR 2 RevIeW exeRCISeS

The ReCTAngUlAR COORdInATe SySTemS And gRAPhS

For the following exercises, find the x-intercept and the y-intercept without graphing.

1. 4x − 3y = 12 2. 2y − 4 = 3x

For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.

3. 5x = 3y − 12 4. 2x − 5y = 7

For the following exercises, find the distance between the two points.

5. (−2, 5)(4, −1) 6. (−12, −3)(−1, 5)

7. Find the distance between the two points (−71,432) and (511,218) using your calculator, and round your answer to the nearest thousandth.

For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

8. (−1, 5) and (4, 6) 9. (−13, 5) and (17, 18)

For the following exercises, construct a table and graph the equation by plotting at least three points.

10. y = 1 _ 2 x + 4 11. 4x − 3y = 6

lIneAR eQUATIOnS In One vARIAble

For the following exercises, solve for x.

12. 5x + 2 = 7x − 8 13. 3(x + 2) − 10 = x + 4 14. 7x − 3 = 5

15. 12 − 5(x + 1) = 2x − 5 16. 2x _ 3 − 3 _ 4 =

x _ 6

+ 21 _ 4

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

17. x _ x2 − 9

+ 4 _ x + 3 = 3 _

x2 − 9 x ≠ 3, −3 18. 1 _ 2 +

2 _ x

= 3 _ 4

For the following exercises, find the equation of the line using the point-slope formula.

19. Passes through these two points: (−2, 1),(4, 2). 20. Passes through the point (−3, 4) and has a slope of − 1 _ 3 .

21. Passes through the point (−3, 4) and is parallel to

the graph y = 2 _ 3 x + 5.

22. Passes through these two points: (5, 1),(5, 7).

mOdelS And APPlICATIOnS

For the following exercises, write and solve an equation to answer each question.

23. The number of males in the classroom is five more than three times the number of females. If the total number of students is 73, how many of each gender are in the class?

24. A man has 72 ft of fencing to put around a rectangular garden. If the length is 3 times the width, find the dimensions of his garden.

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156 CHAPTER 2 equatioNs aNd iNequalities

25. A truck rental is $25 plus $.30/mi. Find out how

many miles Ken traveled if his bill was $50.20.

COmPlex nUmbeRS

For the following exercises, use the quadratic equation to solve.

26. x2 − 5x + 9 = 0 27. 2×2 + 3x + 7 = 0

For the following exercises, name the horizontal component and the vertical component.

28. 4 − 3i 29. −2 − i

For the following exercises, perform the operations indicated.

30. (9 − i) − (4 − 7i) 31. (2 + 3i) − (−5 − 8i) 32. 2 √ —

−75 + 3 √ —

25

33. √ —

−16 + 4 √ —

−9 34. −6i(i − 5) 35. (3 − 5i)2

36. √ —

−4 · √ —

−12 37. √ —

−2  √ —

−8 − √ — 5  38.

2 _ 5 − 3i

39. 3 + 7i ______ i

QUAdRATIC eQUATIOnS

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

40. 2×2 − 7x − 4 = 0 41. 3×2 + 18x + 15 = 0 42. 25×2 − 9 = 0

43. 7×2 − 9x = 0

For the following exercises, solve the quadratic equation by using the square-root property.

44. x2 = 49 45. (x − 4)2 = 36

For the following exercises, solve the quadratic equation by completing the square.

46. x2 + 8x − 5 = 0 47. 4×2 + 2x − 1 = 0

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

48. 2×2 − 5x + 1 = 0 49. 15×2 − x − 2 = 0

For the following exercises, solve the quadratic equation by the method of your choice.

50. (x − 2)2 = 16 51. x2 = 10x + 3

CHAPTER 2 review 157

OTheR TyPeS OF eQUATIOnS

For the following exercises, solve the equations.

52. x 3 _ 2 = 27 53. x

1 _ 2 − 4x 1 _ 4 = 0 54. 4×3 + 8×2 − 9x − 18 = 0

55. 3×5 − 6×3 = 0 56. √ —

x + 9 = x − 3 57. √ —

3x + 7 + √ —

x + 2 = 1

58. |3x − 7| = 5 59. |2x + 3| − 5 = 9

lIneAR IneQUAlITIeS And AbSOlUTe vAlUe IneQUAlITIeS

For the following exercises, solve the inequality. Write your final answer in interval notation.

60. 5x − 8 ≤ 12 61. −2x + 5 > x − 7 62. x − 1 _____ 3

+ x + 2 _____ 5

≤ 3 __ 5

63. |3x + 2| + 1 ≤ 9 64. |5x − 1| > 14 65. |x − 3| < −4

For the following exercises, solve the compound inequality. Write your answer in interval notation.

66. −4 < 3x + 2 ≤ 18 67. 3y < 1 − 2y < 5 + y

For the following exercises, graph as described.

68. Graph the absolute value function and graph the constant function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. |x + 3| ≥ 5

69. Graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines. See the interval where the inequality is true. x + 3 < 3x − 4

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158 CHAPTER 2 equatioNs aNd iNequalities

ChAPTeR 2 PRACTICe TeST

1. Graph the following: 2y = 3x + 4. 2. Find the x- and y-intercepts for the following: 2x − 5y = 6.

3. Find the x- and y-intercepts of this equation, and sketch the graph of the line using just the intercepts plotted. 3x − 4y = 12

4. Find the exact distance between (5, −3) and (−2, 8). Find the coordinates of the midpoint of the line segment joining the two points.

5. Write the interval notation for the set of numbers represented by {x|x ≤ 9}.

6. Solve for x: 5x + 8 = 3x − 10.

7. Solve for x: 3(2x − 5) − 3(x − 7) = 2x − 9. 8. Solve for x: x _ 2 + 1 = 4 _ x

9. Solve for x: 5 _____ x + 4 = 4 + 3 _____

x − 2 . 10. The perimeter of a triangle is 30 in. The longest side

is 2 less than 3 times the shortest side and the other side is 2 more than twice the shortest side. Find the length of each side.

11. Solve for x. Write the answer in simplest radical form.

x 2 _ 3 − x = −

1 _ 2

12. Solve: 3x − 8 ≤ 4.

13. Solve: |2x + 3| < 5. 14. Solve: |3x − 2| ≥ 4.

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

15. Passes through the points (−4, 2) and (5, −3). 16. Has an undefined slope and passes through the point (4, 3).

17. Passes through the point (2, 1) and is perpendicular

to y = − 2 _ 5 x + 3.

18. Add these complex numbers: (3 − 2i) + (4 − i).

19. Simplify: √ —

−4 + 3 √ —

−16 . 20. Multiply: 5i(5 − 3i).

21. Divide: 4 − i ______ 2 + 3i

. 22. Solve this quadratic equation and write the two complex roots in a + bi form: x 2 − 4x + 7 = 0.

23. Solve: (3x − 1)2 − 1 = 24. 24. Solve: x 2 − 6x = 13.

25. Solve: 4x 2 − 4x − 1 = 0 26. Solve: √ —

x − 7 = x − 7

27. Solve: 2 + √ —

12 − 2x = x 28. Solve: (x − 1) 2 _ 3 = 9

For the following exercises, find the real solutions of each equation by factoring.

29. 2x 3 − x 2 − 8x + 4 = 0 30. (x + 5)2 − 3(x + 5) − 4 = 0

ODD ANSWERSC-2

5. y + 5 _____ y + 6

7. 3b + 3 9. x + 4 ______ 2x + 2

11. a + 3 _____ a − 3

13. 3n − 8 ______ 7n − 3

15. c − 6 _____ c + 6

17. 1 19. d 2 − 25 ________

25d 2 − 1

21. t + 5 _____ t + 3

23. 6x − 5 ______ 6x + 5

25. p + 6 ______ 4p + 3

27. 2d + 9 ______ d + 11

29. 12b + 5 _______ 3b−1

31. 4y − 1 ______ y + 4

33. 10x + 4y ________ xy

35. 9a − 7 __________ a2 − 2a − 3

37. 2y 2 − y + 9 __________

y 2 − y − 2 39. 5z

2 + z + 5 __________ z 2 − z − 2

41. x + 2xy + y _____________ x + xy + y + 1

43. 2b + 7a _______ ab2

45. 18 + ab _______ 4b

47. a − b 49. 3c 2 + 3c − 2 ___________

2c 2 + 5c + 2

51. 15x + 7 _______ x − 1

53. x + 9 _____ x − 9

55. 1 _____ y + 2

57. 4

Chapter 1 Review exercises 1. −5 3. 53 5. y = 24 7. 32m 9. Whole

11. Irrational 13. 16 15. a6 17. x 3

____ 32y 3

19. a

21. 1.634 × 107 23. 14 25. 5 √ — 3 27. 4 √

— 2 _____

5

29. 7 √ — 2 _ 50 31. 10 √

— 3 33. −3 35. 3x 3 + 4x 2 + 6

37. 5x 2 − x + 3 39. k2 − 3k − 18 41. x 3 + x 2 + x + 1 43. 3a2 + 5ab − 2b2 45. 9p 47. 4a2

49. (4a − 3)(2a + 9) 51. (x + 5)2 53. (2h − 3k)2

55. (p + 6)(p2 − 6p + 36) 57. (4q − 3p)(16q2 + 12pq + 9p2)

59. (p + 3) 1 _ 3 (−5p − 24) 61. x + 3 _ x − 4 63.

1 _ 2 65. m + 2 _ m − 3

67. 6x + 10y ________ xy

69. 1 _ 6

Chapter 1 practice test 1. Rational 3. x = 12 5. 3,141,500 7. 16

9. 9 11. 2x 13. 21 15. 3 √ — x _ 4 17. 21 √

— 6

19. 13q 3 − 4q 2 − 5q 21. n3 − 6n 2 + 12n − 8 23. (4x + 9)(4x − 9) 25. (3c − 11)(9c 2 + 33c + 121)

27. 4z − 3 ______ 2z − 1

29. 3a + 2b _______ 3b

ChapteR 2

Section 2.1 1. Answers may vary. Yes. It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants. 3. The y-intercept is the point where the graph crosses the y-axis. 5. The x-intercept is (2, 0) and the y-intercept is (0, 6). 7. The x-intercept is (2, 0) and the y-intercept is (0, −3). 9. The x-intercept is (3, 0) and the

y-intercept is  0, 9 _ 8  . 11. y = 4 − 2x 13. y = 5 − 2x _________ 3

15. y = 2x − 4 _ 5 17. d = √ —

74 19. d = √ —

36 = 6

21. d ≈ 62.97 23.  3, − 3 _ 2  25. (2, −1) 27. (0, 0) 29. y = 0

31. Not collinear

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5

(−1, 2) (2, 1)

(0, 4)

33. (−3, 2), (1, 3), (4, 0)

35. x −3 0 3 6 y 1 2 3 4

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(6, 4) (3, 3)

(0, 2)(−3, 1)

37. x −3 0 3 y 0 1.5 3

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(3, 3)

(0, 1.5)(−3, 0)

39.

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

(8, 0)

(0, −4)

41.

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6

(3, 0) (0, 2)

43. d = 8.246 45. d = 5 47. (−3, 4) 49. x = 0, y = −2 51. x = 0.75, y = 0 53. x = −1.667, y = 0 55. 15 − 11.2 = 3.8 mi shorter 57. 6.042 59. Midpoint of each diagonal is the same point (2, 2). Note this is a characteristic of rectangles, but not other quadrilaterals. 61. 37 mi 63. 54 ft

Section 2.2 1. It means they have the same slope. 3. The exponent of the x variable is 1. It is called a first-degree equation. 5. If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator). 7. x = 2

9. x = 2 _ 7 11. x = 6 13. x = 3 15. x = −14

17. x ≠ −4; x = −3 19. x ≠ 1; when we solve this we get x = 1, which is excluded, therefore NO solution 21. x ≠ 0; x = − 5 _ 2

23. y = − 4 ___ 5 x + 14 ___ 5

25. y = − 3 ___ 4 x + 2 27. y = 1 __ 2

x + 5 __ 2

29. y = −3x − 5 31. y = 7 33. y = −4 35. 8x + 5y = 7 37.

x

y

4

–2–4–6–8–10 –2 –4 –6 –8

–10

2

6

6

42 8

8

10

10

Parallel 39. Perpendicular

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

ODD ANSWERS C-3

41. m = − 9 _ 7 43. m = 3 _ 2 45. m1 = −

1 _ 3 , m2 = 3; perpendicular

47. y = 0.245x − 45.662. Answers may vary. ymin = −50, ymax = −40 49. y = −2.333x + 6.667. Answers may vary. ymin = −10, ymax = 10

51. y = − A _ B

x + C _ B

53. The slope for (−1, 1) to (0, 4) is 3.

The slope for (−1, 1) to (2, 0) is − 1 _ 3 . The slope for (2, 0) to (3, 3) is

3. The slope for (0, 4) to (3, 3) is − 1 _ 3 . Yes they are perpendicular. 55. 30 ft 57. $57.50 59. 220 mi

Section 2.3 1. Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading. 3. 2,000 − x 5. v + 10 7. Ann: 23; Beth: 46 9. 20 + 0.05m 11. 300 min 13. 90 + 40P 15. 6 devices 17. 50,000 − x 19. 4 hr 21. She traveled for 2 hr at 20 mi/hr, or 40 miles. 23. $5,000 at 8% and $15,000 at 12% 25. B = 100 + 0.05x 27. Plan A 29. R = 9 31. r = 4 _ 5 or 0.8

33. W = P − 2L __ 2 = 58 − 2(15)

_ 2 = 14

35. f = pq

_ p + q = 8(13)

_ 8 + 13 = 104 _ 21 37. m = −

5 _ 4

39. h = 2A _ b1 + b2

41. Length = 360 ft; width = 160 ft

43. 405 mi 45. A = 88 in.2 47. 28.7 49. h = V _ πr 2

51. r = √ ___

V _ πh

53. C = 12π

Section 2.4 1. Add the real parts together and the imaginary parts together. 3. Possible answer: i times i equals −1, which is not imaginary.

5. −8 + 2i 7. 14 + 7i 9. − 23 _ 29 + 15 _ 29 i

11.

r

i

−1−1 −2 −3

−5 −4

−2−3−4−5 321 4 5

3 2 1

4 5

13.

r

i

−1−1 −2 −3

−5 −4

−2−3−4−5 321 4 5

3 2 1

4 5

15. 8 − i 17. −11 + 4i 19. 2 − 5i 21. 6 + 15i

23. −16 + 32i 25. −4 − 7i 27. 25 29. 2 − 2 __ 3

i

31. 4 − 6i 33. 2 _ 5 + 11 _ 5 i 35. 15i 37. 1 + i √

— 3

39. 1 41. −1 43. 128i 45.  √ — 3 ____

2 + 1 __

2 i 

6

= −1

47. 3i 49. 0 51. 5 − 5i 53. −2i 55. 9 __ 2

− 9 __ 2

i

Section 2.5 1. It is a second-degree equation (the highest variable exponent is 2). 3. We want to take advantage of the zero property of multiplication in the fact that if a ∙ b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0 or b = 0.

5. One, when no linear term is present (no x term), such as x 2 = 16. Two, when the equation is already in the form (ax + b)2 = d.

7. x = 6, x = 3 9. x = − 5 _ 2 , x = − 1 _ 3 11. x = 5, x = −5

13. x = − 3 _ 2 , x = 3 _ 2 15. x = −2, 3 17. x = 0, x = −

3 ___ 7

19. x = −6, x = 6 21. x = 6, x = −4 23. x = 1, x = −2

25. x = −2, x = 11 27. x = 3 ± √ —

22 29. z = 2 _ 3 , z = − 1 _ 2

31. x = 3 ± √ —

17 ________ 4

33. Not real 35. One rational

37. Two real; rational 39. x = −1 ± √ —

17 __________ 2

41. x = 5 ± √ —

13 ________ 6

43. x = −1 ± √ —

17 __________ 8

45. x ≈ 0.131 and x ≈ 2.535 47. x ≈ − 6.7 and x ≈ 1.7 49. ax 2 + bx + c = 0

x 2 + b _ a x = − c _ a

x2 + b _ a x + b2 _

4a2 = − c _ a +

b _ 4a2

 x + b _ 2a  2 = b

2 − 4ac _ 4a2

x + b ___ 2a

= ± √ ________

b 2 − 4ac _

4a2

x = −b ± √ —

b2 − 4ac __ 2a

51. x(x + 10) = 119; 7 ft. and 17 ft. 53. Maximum at x = 70 55. The quadratic equation would be (100x − 0.5×2) − (60x + 300) = 300. The two values of x are 20 and 60. 57. 3 feet

Section 2.6 1. This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation. 3. He or she is probably trying to enter negative 9, but taking the square root of −9 is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in −27. 5. A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised. 7. x = 81 9. x = 17 11. x = 8, x = 27

13. x = −2, 1, −1 15. y = 0, 3 _ 2 , − 3 _ 2 17. m = 1, −1

19. x = 2 _ 5 , ± 3i 21. x = 32 23. t = 44 _ 3 25. x = 3

27. x = −2 29. x = 4, − 4 _ 3 31. x = − 5 _ 4 ,

7 _ 4

33. x = 3, −2 35. x = −5 37. x = 1, −1, 3, −3 39. x = 2, −2 41. x = 1, 5 43. All real numbers 45. x = 4, 6, −6, −8 47. 10 in. 49. 90 kg

Section 2.7 1. When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes. 3. (−∞, ∞)

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ODD ANSWERSC-4

5. We start by finding the x-intercept, or where the function = 0. Once we have that point, which is (3, 0), we graph to the right the straight line graph y = x − 3, and then when we draw it to the left we plot positive y values, taking the absolute value of them.

7.  −∞, 3 __ 4  9.  − 13 ____ 2 , ∞  11. (−∞, 3) 13.  −∞, −

37 ___ 3  15. All real numbers (−∞, ∞) 17.  −∞, − 10 _ 3  ∪ (4, ∞) 19. (−∞, −4] ∪ [8, +∞) 21. No solution 23. (−5, 11) 25. [6, 12] 27. [−10, 12] 29. x > −6 and x > −2 Take the intersection of two sets. x > −2, (−2, +∞) 31. x < − 3 or x ≥ 1 Take the union of the two sets. (−∞, −3) ∪ [1, ∞) 33. (−∞, −1) ∪ (3, ∞)

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

35. [−11, −3]

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

37. It is never less than zero. No solution.

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

39. Where the blue line is above the red line; point of intersection is x = −3. (−∞, −3)

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

41. Where the blue line is above the red line; always. All real numbers. ( −∞, −∞)

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

43. (−1, 3) 45. (−∞, 4) 47. {x|x < 6} 49. {x|−3 ≤ x < 5} 51. (−2, 1] 53. (−∞, 4]

55. Where the blue is below the red; always. All real numbers. ( −∞, +∞).

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

57. Where the blue is below the red; (1, 7).

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

59. x = 2, − 4 _ 5 61. (−7, 5]

63. 80 ≤ T ≤ 120; 1, 600 ≤ 20T ≤ 2, 400; [1, 600, 2, 400]

Chapter 2 Review exercises 1. x-intercept: (3, 0); y-intercept: (0, −4) 3. y = 5 _ 3 x + 4

5. √ —

72 = 6 √ — 2 7. 620.097 9. Midpoint is  2, 23 _ 2 

11. x 0 3 6 y −2 2 6

2

x

y

–1–2–3–4–5–6 –1 –2 –3 –4 –5 –6

1

3

3

21 4

4

5 6

5 6 (6, 6)

(3, 2)

(0, −2)

13. x = 4

15. x = 12 _ 7

17. No solution

19. y = 1 _ 6

x + 4 _ 3

21. y = 2 __ 3

x + 6

23. Females 17, males 56 25. 84 mi

27. x = − 3 _ 4 ± i √

— 47 _ 4

29. Horizontal component −2; vertical component −1

31. 7 + 11i 33. 16i 35. −16 − 30i 37. −4 − i √ —

10

39. x = 7 − 3i 41. x = −1, −5 43. x = 0, 9 __ 7

45. x = 10, −2 47. x = −1 ± √ — 5 _________

4 49. x = 2 __

5 , − 1 _ 3

51. x = 5 ± 2 √ — 7 53. x = 0, 256 55. x = 0, ± √

— 2

57. x = −2 59. x = 11 ___ 2

, − 17 ____ 2 61. (−∞, 4)

63.  − 10 ____ 3 , 2  65. No solution 67.  − 4 _ 3 ,

1 _ 5  69. Where the blue is below the red line; point of intersection is x = 3.5. (3.5, ∞)

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4

5

5

ODD ANSWERS C-5

Chapter 2 practice test 1. y = 3 _ 2 x + 2

x 0 2 4 y 2 5 8

4

x

y

–2–4–6–8–10 –2 –4 –6 –8

–10 –12

2

6

6

42 8

8

10

10 12

3. (0, −3) (4, 0)

x

y

2

–1–2–3–4–5 –1 –2 –3 –4 –5

1

3

3

21 4

4 5

5

(−1, 2) (2, 1)

(0, 4)

5. (−∞, 9] 7. x = −15 9. x ≠ −4, 2; x = − 5 _ 2 , 1

11. x = 3 ± √ — 3 _______

2 13. (−4, 1) 15. y = − 5 _ 9 x −

2 _ 9

17. y = 5 _ 2 x − 4 19. 14i 21. 5 ___

13 − 14 ___

13 i 23. x = 2, − 4 _ 3

25. x = 1 _ 2 ± √

— 2 _ 2 27. 4 29. x =

1 _ 2 , 2, −2

ChapteR 3

Section 3.1 1. A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate. 3. When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function. 5. When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input. 7. Function 9. Function 11. Function 13. Function 15. Function 17. Function 19. Function 21. Function 23. Function 25. Not a function 27. f (−3) = −11, f (2) = −1, f (−a) = −2a − 5, −f (a) = −2a + 5, f (a + h) = 2a + 2h − 5 29. f (−3) = √

— 5 + 5, f (2) = 5,

f (−a) = √ —

2 + a + 5, −f (a) = − √ —

2 − a − 5, f (a + h) = √ —

2 − a − h + 5 31. f (−3) = 2, f (2) = −2, f (−a) = ∣ −a − 1 ∣ − ∣ −a + 1 ∣ , −f (a) = − ∣ a − 1 ∣ + ∣ a + 1 ∣ , f (a + h) = ∣ a + h − 1 ∣ − ∣ a + h + 1 ∣ 33.

g(x) − g(a) _ x − a = x + a + 2, x ≠ a 35. a. f (−2) = 14 b. x = 3

37. a. f (5) = 10 b. x = 4 or −1 39. a. r = 6 − 2 __ 3 t

b. f (−3) = 8 c. t = 6 41. Not a function 43. Function 45. Function 47. Function 49. Function 51. Function 53. a. f (0) = 1 b. f (x) = −3, x = −2 or 2 55. Not a function, not one-to-one 57. One-to-one function 59. Function, not one-to-one 61. Function 63. Function 65. Not a function 67. f (x) = 1, x = 2 69. f (−2) = 14; f (−1) = 11; f (0) = 8; f (1) = 5; f (2) = 2

71. f (−2) = 4; f (−1) = 4.414; f (0) = 4.732; f (1) = 5; f (2) = 5.236

73. f (−2) = 1 __ 9 ; f (−1) = 1 __ 3 ; f (0) = 1; f (1) = 3; f (2) = 9 75. 20

77. The range for this viewing window is [0, 100].

x

y

20

–20 –40 –60 –80

–100

40 60 80

100

–5 5–10 10

79. The range for this viewing window is [−0.001, 0.001].

x

y

0.0002

–0.0002 –0.0004 –0.0006 –0.0008 –0.001

0.0004 0.0006 0.0008

0.001

–0.05–0.1 0.05 0.1

81. The range for this viewing window is [−1,000,000, 1,000,000].

x

y

2.105

–2.105 –4.105 –6.105 –8.105

–10.105

4.105 6.105 8.105

10.105

–50–100 50 100

83. The range for this viewing window is [0, 10].

10 8 6 4 2

–20 x

y

20 40 60 80 100

85. The range for this viewing window is [−0.1, 0.1].

x

y

0.02 0.04 0.06 0.08

0.1

–0.02 –0.0005–0.0001 0.0001

–0.04 –0.06 –0.08 –0.1

0.0005

87. The range for this viewing window is [−100, 100].

x

y

20 40 60 80

100

–20–5 .105–10.105 5.105 10.105

–40 –60 –80

–100

89. a. g(5000) = 50 b. The number of cubic yards of dirt required for a garden of 100 square feet is 1. 91. a. The height of the rocket above ground after 1 second is 200 ft. b. The height of the rocket above ground after 2 seconds is 350 ft.

Section 3.2 1. The domain of a function depends upon what values of the independent variable make the function undefined or imaginary. 3. There is no restriction on x for f (x) = 3 √

— x because you can

take the cube root of any real number. So the domain is all real numbers, (−∞, ∞). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x-values are restricted for f (x) = √

— x to nonnegative numbers and the

domain is [0, ∞). 5. Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate included endpoints with a solid circle and excluded endpoints with an open circle. Use an arrow to indicate −∞ or ∞. Combine the graphs to find the graph of the piecewise function. 7. (−∞, ∞) 9. (−∞, 3]

11. (−∞, ∞) 13. (−∞, ∞) 15.  −∞, − 1 _ 2  ∪  − 1 _ 2 , ∞ 

17. (−∞, −11)∪(−11, 2)∪(2, ∞) 19. (−∞, −3)∪(−3, 5)∪(5, ∞)

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