Math 205 Elementary Algebra Fall 2010

Final Exam Study Guide The exam is on Tuesday, December 14th from 3:30pm–5:30pm. You are allowed a scientific calculator and a 4" by 6" index card for notes. On your index card be sure to write any formulas you needed for any of the problems listed. I will not provide you with any formulas on the exam. The Final Exam is comprehensive; however, not all of the problems that appeared in the previous exams will appear in the final. Use this study guide to know which math concepts you need to review. Don’t forget there is new material from chapter 9.3 and 6.6 on the final exam. For the Final Exam, you will need to be able to: 1. Solve linear equations, including equations with fractions. The solution to these equations can be a

unique solution, no solution, or all real numbers. Remember that the correct final answer for these types of problems is either x = #, all real numbers or no solution. Remember to clear your fractions multiply each term by the LCD.

2. Solve an application problem. These include but are not limited to problems involving money and percents.

3. Solve an application problem involving geometry—measurement of the sides of a triangle and the perimeter of a rectangle.

4. Solve an application problem by setting up a proportion and solving that proportion. 5. Solve linear inequalities and graph the solution. The solution to these inequalities can be a unique

interval, no solution, or all real numbers. 6. Graph linear equations in two variables. * graph using a point and the slope. * graph using the x- and y-intercepts * graph by plotting points 7. Find the slope of the line given two points or given an equation. This includes horizontal and

vertical lines. 8. Given a graph, find the slope of the line or state that the slope is undefined. 9. Write an equation of the line in slope-intercept form. * write an equation given a point and the slope. * write an equation given two points. 10. Graph linear inequalities. Shade the regions that make the statement true. 11. Solve a system of linear equations graphically. This includes systems that have one unique

solution, no solution, or an infinite number of solutions. 12. Solve a system of linear equations by substitution or addition. You may choose either method to

solve the system. The system might have one unique solution, no solution or an infinite number of solutions.

13. Solve an application problem involving interest, mixtures, or distance. You saw these types of problems twice. Once you had to solve using 1 variable (linear equations) and again using 2 variables(systems of equations). You can solve these problems either way.

14. Simplify expressions using the rules of exponents, including expressions with both positive, negative and zero exponents. The final answer must have only positive exponents.

15. Convert numbers to and from scientific notation and perform calculations. 16. Add or subtract two polynomials. Write the answer in descending order. 17. Multiply two polynomials. Write the answer in descending order. 18. Divide a polynomial by a binomial. If there is a remainder, use appropriate notation. 19. Factor completely any given polynomial using the methods learned.

a) Factor a polynomial by factoring out the GCF of all the terms. b) Factor a polynomial by grouping.

c) Factor a trinomial with a leading coefficient that is one. d) Factor a trinomial with a leading coefficient that is not one. e) Factor a difference of squares. Recognize that a sum of squares is prime. f) Factor using a mixture of these tools.

20. Solve quadratic equations by factoring. 21. Simplify a rational expression. 22. Multiply or divide two rational expressions. Simplify the result. 23. Add or subtract two rational expressions that have common denominators or different

denominators. Simplify the result, if possible. 24. Simplify complex fractions. * get it into one fraction over one fraction, then flip and multiply the bottom fraction

* multiply each term by the LCD of all the denominators. 25. Solve a rational equation. Remember to eliminate as solutions any values that make an

expression in the original equation undefined (make one of the denominators = 0). 26. Simplify a square root expression by using the multiplication property of square roots. You might

first need to multiply or divide two square root expressions and then simplify. 27. Add or subtract radical expressions. You might need to simplify terms before they can be

combined. 28. Multiply square root expressions by using the distributive property or the ―FOIL‖ method. 29. Simplify a square root expression by using the division property of square roots. You might first

need to divide two square root expressions and then simplify. 30. Simply expressions with a square root in the denominator by rationalizing the denominator. You

might need to simplify first. 31. Solve a radical equation. If the equation has no solution, state so. Remember to check for

extraneous solutions. (In the process of solving the radical equation you might need to solve a quadratic equation.)

32. Solve application problems involving square roots—the only problems I will put on the exam will involve the Pythagorean Theorem.

33. Solve a quadratic equation by using the quadratic formula. If possible, simplify radicals and rationalize denominators. No decimal approximations. *You must solve the quadratic equations using the indicated method in order to receive full credit.*

34. Solve application problems involving a quadratic equation. You will either be given the quadratic equation and you will need to know how to use it to answer the question OR you will need to come up the equation yourself—these involve the area of a rectangle, or a right triangle so you will use the Pythagorean Theorem. You can solve the equation by either factoring or using the quadratic formula (if you don’t like to factor)

. Repeating from above! On your index card be sure to write any formulas you needed for any of the problems listed above. I will not provide you with any formulas on the exam. Practice Problems for the Final To study for the final do the following problems AND look at the problems that were on the exams (but only those similar to those in this handout). The answer to the problems listed below, (For those of you who have the Chapter Test Prep Video cd that came with the book, you can view it to see someone working out each of the problems that are in the Chapter Tests.)

Math 205 Name____________________ Final Exam Review Directions: Solve each equation.

1. )3(5)32(2 rr

2. zz7

52

7

2

3. 5)23()6(4 kk

4. 3)32(3

1)18(

9

1ppp

5. Solve for d:

pd

155

11

6. Given the area formula for a trapezoid, find h ,

when 5.32A , 61b , and 72b

)(2

121 bbhA

7. A “Lucky-duck” found two-thirds as much money on Saturday as he did on Friday. Throughout both days he found a total of $105. How much money did he find on Friday?

8. Given the area formula for finding the perimeter of a rectangle, solve forW .

WLP 22

9. How many liters of 10% alcohol should you mix with 40 liters of 50% alcohol to obtain a blend of 40% alcohol?

10. Say you and your friend are 230 miles apart and want to meet up with each other. Both of you left at the same time, but your friend drives 15 mph slower than you. You both finally meet up after 2 hours. How fast were you each going?

Rate Time Distance

You

Your

friend

11. Solve the following inequality. Graph it on the number line and state it in INTERVAL NOTATION.

12336 xx

12. Solve the following inequality. Graph it on the number line and state it in INTERVAL NOTATION.

8437 x

13. A car rental agency charges $180 per week plus $0.25 per mile to rent a car. How many miles can you travel in one

week if you want to spend $395?

14. The perimeter of a triangle is 28 feet. The medium side is 4 feet longer than the shorter side, while the longest side

is twice as long as the shortest side. What are the measures of each of the three sides?

15. Deter mine whether the ordered pair 3

1,2 is a solution to the equation yx 332

16. Write the equation, in slope-intercept form, of the line that has a slope of 6 and passes through the point (2, -4).

17. Find an equation of a line through the points (-2, 5) and (-3, 4)

18. Graph:

3056 yx

x

y

19. Graph

1243 yx

20. Solve the following linear system by graphing:

8

1923

yx

yx

21. Solve the following linear system by elimination or substitution:

3963

852

yx

yx

22. A coin purse contains a mixture of 39 coins in quarters and dimes. The coins have a total value of

$7.50. Determine the number of quarters and number of dimes in the coin purse.

x

y

x

y

23. With the current, you can row 16 miles in 2 hours. Against the same current, you can only row 12 miles in 3 hours. Find your rowing rate in still water and the rate of the current.

24. Say you invested twice as much money at a 5% simple interest rate than you did in an account paying 4% simple interest. At the end of the year, the two accounts earn $350 in simple interest. How much was invested at each rate?

25. The other day I went to Staples to buy some office supplies. I bought 2 pens and 3 binders, which together cost me $9.75. My friend bought 3 of the same pens and 2 of the same binders, and it cost her $9.00. What is the cost for a single binder and the cost for a single pen?

26. Evaluate.

233

27. Evaluate.

323

28. Simplify: 543423 )()4( yzzyx

29. Simplify: 3

2

32

p

nm

30. Evaluate:

42

1

31. Convert the following to scientific notation. 562000000000

32. Convert the following to standard form. 51079.3

33. Multiply. 27 108109

34. Subtract.

xxxxx 534736 232

35. Multiply.

1512 2 xxx

36. Multiply.

5

1

5

1nn

37. Multiply. 2

2xx

38. Divide.

yx

yxxyyx2

3234

7

286349

39. Divide using long division.

365124 2234 xxxxx

40. Factor

34

5

4

5

1xx

41. Factor

3652 xx

42. Factor

94 2x

43. Factor 22 62525 yxyx

44. Solve for w by factoring. 6)5(ww

45. Multiply:

1

1

34

232

2

w

w

ww

ww

46. Add:

96

2

9

322 aaa

47. Divide:

9

62

3

22k

k

k

48. Subtract:

1

4

1

22pp

p

49. Solve for y .

yyyy 2

64

2

32

50. Solve for a .

1

16

44

162

22 aa

a

a

a

51. Find each indicated root. If the root is not a real number, say so.

a. 400 b. 20 c. 3 729 d. 4 16

52. Find the missing length.

Simplify each expression completely. Assume all variables are real, non-negative numbers.

53. Simplify:

7481 yx

54. Rationalize the denominator:

2

142

55. Rationalize the denominator:

x32

24

56. Simplify:

2)25( yx

10

8

x

57. Solve by using the square root property:

25)2( 2k

58. Solve by using the square root property:

2536 2x

59. Solve by using the quadratic formula:

xx 252

60. Solve by using the quadratic formula:

023 2 xx

61. Find the length and width of a rectangle if the width is 3 inches less than the length and the area of the rectangle is

180 square inches.

62. The lengths of the two legs of a right triangle are consecutive integers. The hypotenuse is one less than twice the

length of the shorter leg. Find the length of each of the sides.

Answers:

1. r=-11 2. z=2 3. k=21

4. p=0 5. 33

5

11pd

6. h=5

7. Found $63 Friday and $42 Saturday 8. l

pw

2

9. 13.33 L of 40%

10. You went 65 mph and your friend went 50 mph

11. ]3,( 12. [-1,4)

13. 860 miles 14. Shortest 6 ft Medium 10 ft Longest 12 ft

15. Not a solution 16. y=6x-16 17. y=x+7

18.

19.

20.

21. (-9, -2) 22. 24 quarters, 15 dimes 23. Your rate in still water 6

mph current’s rate 2 mph

24. $2500 at 4%, $5000 at 5% 25. Pens cost $1.50 Binders cost $2.25

26. 729

27. -729 28. 3211964 zyx

29. 6

398

p

nm

30. 16 31. 111062.5 32. .0000379

33. 6102.7 34. 7834 23 xxx 35. 1710 23 xxx

36. 25

12n 37. xxx 44 23

38. xx

yyx 4

97 22

39. 242 xx 40. )4(

5

1 3 xx 41. )4)(9( xx

42. )32)(32( xx 43. )25)(35( yxyx 44. w=2, w=3

45. 3

2

w

w 46.

)3()3(

352 aa

a

47. -1

48. 1

)2(2

p

p

49. No solution 50. a=20

51. 20, 52 , -9, 2 52. 412x 53. yyx329

54. 72 55.

x

x

56. yx 25

57. k=3, k=-7 58.

6

5 59. 61x

60. 3

2,1 xx

61. 12 inches by 15 inches 62. Shortest 3 ft Medium 4 ft Longest 5 ft

(7, 1)