East Greenwich Mathematics Summer Review Material for Students Entering Algebra I, Part II 2016-2017 Directions: In order to earn full credit, show all work in the spaces provided. Do all work without a calculator. Remember to do EVERY problem to the best of your ability. If you cannot complete a problem, do as much as you can, and then explain why you cannot go any further. Be specific! When finished, remember to write your thoughtful reflection on the packet as a whole. Discuss the questions that were easy for you and those that were difficult for you and why you felt that way. Page 1 of 16
Evaluating Algebraic Expressions Helpful links: http://www.math.com/school/subject2/lessons/s2u2l3dp.html Evaluate each of the following when a = 2, b = -3, x = 4, and y = 0. 1. b a xy 2. -b 2 3. b a 4. a b x b(y a) Perform the indicated operation and simplify. Helpful links: https://www.youtube.com/watch?v=t5imqdkdhyy http://www.mathguide.com/lessons/polynomials.html 5. 3x! + x! 4x 10 + (5x! 4x! + 5) 6. 7x! 2x! + 6 (2x! 6x! 4) 7. x + 5! 8.!"!!!!!!!!! 9. Consider the polynomial 7x! 2x 4x! + 1. a. Write in standard form. b. State the degree. c. State the leading coefficient. d. Classify the polynomial by degree. e. Classify the polynomial by number of terms. f. State the constant term. Page 2 of 16
Helpful Links: http://www.studygs.net/equations.htm http://www.sosmath.com/algebra/solve/solve0/solve0.html Solve each equation. Linear Equations and Inequalities in One Variable 10. 4 x = 11 11. 9(3 2x) = 2(10 8x) 12. 5 (2x + 10) = 2( x + 5) + 12x + 60 13. 3x 2 =!!!!! 14. 6 = b 1 9 7 15. 2 3 x 1= 1 4 x + 1 2 16. 2! x + 5 =! x + 1 17. y + 8 5 = 2!! Page 3 of 16
Helpful Links: http://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-linear-inequalities Solve each inequality and graph its solution. Also express your answers using Interval Notation. 18. 6x + 4 <10 19. 2(x 4) 7 20. 1 < 9 + n < 17 21. r + 5 12 or!! < 0 22. 3 2 n 2 > 5 23. 8 + 4v 7 17 Page 4 of 16
Linear Equations in Two Variables Helpful Links: http://www.mathsisfun.com/data/cartesian-coordinates.html http://www.mathsisfun.com/algebra/line-equation-point-slope.html http://www.mathsisfun.com/equation_of_line.html http://www.purplemath.com/modules/systlin1.htm 24. Graph and LABEL the following points: A (0, 4) B (7,0) C (5,6) D (3, -1) E (9, -4) F (-4, -2) G (-7, 8) H (-1, 3) a) Using the points plotted, list points in quadrant I? b) Using the points plotted, list points in quadrant II? c) Using the points plotted, list points in quadrant III? d) Using the points plotted, list points in quadrant IV? 25. Graph the line y = 2 x + 3. 26. Graph the line y 4 =! (x + 3)! Label at least two points on the line. Label at least two points on the line. Page 5 of 16
27. Graph the line 5x + 2y =10. Label at least two points on the line. 28. Find the slope: a) of a line containing the two points, (4, 7) and (-2, 1). b) of a linear function described by the table. x 1 5 7 8 y 7 15 19 21 c) of a graph shown. Page 6 of 16
29. Two different linear relationships are shown below. x y 2 4 7 14 12 24 17 34 Linear relationship A Linear relationship B Which linear relationship has the lower rate of change? Justify your answer. Marie left her home and headed to work. The graph below shows Marie s distance from home to work over time. 30. During which segment did Marie stop? Explain. 31. During which segment did Marie travel the fastest? Explain how you can tell by looking at the graph. 32. What was Marie s speed during segment A of her trip? Page 7 of 16
33. Write an equation of the line that passes through the points (-5, 4) and (2, 3). 34. Write the equation of a line that includes the point (5, 9) and is perpendicular to the line y =! x + 9.! 35. Write the equation of a line that includes the point (5, 9) and is parallel to the line y =! x + 9.! 36. What is the slope of a vertical line? 37. Write the equation of a vertical line that passes through the point (4, -2). 38. What is the slope of a horizontal line? 39. Write the equation of a horizontal line that passes through the point (4. -2). 40. Solve the system by graphing. y = x + 3 4x 6y = 12 Page 8 of 16
y = 5x 41. Solve the system using substitution. 10x + 2y = 0 42. Solve the system algebraically using elimination. 2x + 4y = 10 x 2y = 7 Graph the following linear inequalities. Helpful Links: https://www.youtube.com/watch?v=0vvqyomrlkq 43. y > 4 44. 5x 3y 15 Graph the system of linear inequalities. 45. # y + 5 < 4(x + 2) $ % y x 2 Page 9 of 16
Reasoning with Numbers Helpful links: http://www.charleston.k12.il.us/cms/teachers/math/prealgebra/paunit7/l7-4.pdf http://www.glencoe.com/sec/math/prealg/prealg04/add_lesson/using_sampling_pa1.pdf 46. Use your estimation skills to place each of the following radicals onto the number line below. A) 3 B) 20 C)!"!! D) 5 E) 14 F) π G) -3.2 H) (.7) 2 I)!! J)!! Using Mathematics Helpful Links: http://www.mathplanet.com/education/pre-algebra/ratios-and-percent/solving-problems-with-percent http://www.mathsisfun.com/geometry/perimeter.html http://www.regentsprep.org/regents/math/algebra/ao2/indexao2.htm Use problem solving skills. 1) Understand the problem (Use sketches, diagrams or tables to help with the understanding.) 2) Develop a plan 3) Execute the plan 4) Check. Does your answer make sense? 47. The manager of a sporting goods store uses a markup rate of 60%. What is the selling price of a set of golf clubs that cost him $300? Page 10 of 16
48. The perimeter of a rectangle is 144 feet. The length of the rectangle is three times the width. Find the length, width, and area of the rectangle. 49. The table shows the results of a random survey of students in grade 7 and grade 8. Every student surveyed gave a response. Each student was asked if he or she exercised less than 5 hours last week or 5 or more hours last week. Based on the results of the survey, which statements are true? Select each correct statement. a. More grade 8 students were surveyed than grade 7 students. b. A total of 221 students were surveyed. c. Less than 50% of the grade 8 students surveyed exercised 5 or more hours last week. d. More than 50% of the students surveyed exercised less than 5 hours last week. e. A total of 107 grade 7 students were surveyed. 50. Which equation has both 4 and -4 as possible values of y? a. y! = 8 b. y! = 8 c. y! = 16 d. y! = 64 51. Eric planted a seedling in his garden and recorded its height each week. The equation, h =! w +!, can be used to estimate the height, h, in inches, of the seedling after w, weeks since!! Eric planted the seedling. a. What does the slope of the graph of the equation h =!! w +!! represent? i. the height, in inches, of the seedling after w weeks ii. the height, in inches, of the seedling when Eric first planted it iii. the increase in the height, in inches, of the seedling each week iv. the total increase in the height, in inches, of the seedling after w weeks b. The equation h =! w +!!! many weeks? estimates the height of the seedling t be 8.25 inches after how Page 11 of 16
Interpreting Rates of Change Helpful Links: http://www.regentsprep.org/regents/math/algebra/ac1/rate.htm http://hotmath.com/hotmath_help/topics/rate-of-change.html http://www.purplemath.com/modules/slope.htm http://www.rcstn.net/technology/files/ch_3-4_equations_of_lines.ppt 52. Line t and ΔECA and ΔFDB are shown on the coordinate grid. Which statements are true? Select ALL that apply. A. The slope of AC is equal to the slope of BC. B. The slope of AC is equal to the slope of BD. C. The slope of AC is equal to the slope of line t. D. The slope of line t is equal to EC AC. E. The slope of line t is equal to FB FD. F. The slope of line t is equal to AE FD. 53. The functions A, B, and C are linear functions. Some values of function A are shown in the table. The graph of function B has a y-intercept of (0,3) and an x-intercept of (-5,0). Function C is defined by the equation y = 3x+1. Order the linear functions based on the rate of change, from least to greatest. Page 12 of 16
Systems of linear equations Helpful Links: http://www.purplemath.com/modules/systlin1.htm http://www.regentsprep.org/regents/math/algebra/ae3/indexae3.htm 54. a) Solve the system algebraically: y = 3x + 1 y = 4 b) What does your solution(s) represent? c) How many solutions does the system have (one solution, no solution, or infinitely many solutions)? How could you have known this answer just by looking at the equations in the system? 55. a) Solve the system algebraically: y = x y = x 4 b) What does your solution(s) represent? c) How many solutions does the system have (one solution, no solution, or infinitely many solutions)? How could you have known this answer just by looking at the equations in the system? 56. a) Solve the system algebraically: y = x 8y = 8x b) What does your solution(s) represent? c) How many solutions does the system have (one solution, no solution, or infinitely many solutions)? How could you have known this answer just by looking at the equations in the system? Page 13 of 16
57. A school is selling t-shirts and sweatshirts for a fundraiser. The table shows the number of t- shirts and the number of sweatshirts in each of three recent orders. The total cost of orders A and B are given. Each t-shirt has the same cost and each sweatshirt has the same cost. The system of equations shown can be used to represent this situation. 2x + 2y = 38 3x + y = 35 a) What is the total cost of one t- shirt and one sweatshirt? b) In the system above, x represents and y represents c) If the system of equations is graphed in a coordinate plane, what are the coordinates (x, y) of the intersection of the two lines? d) What is the total cost in dollars of order C? Page 14 of 16
Helpful Links: https://www.youtube.com/watch?v=0vvqyomrlkq 58. Skateboarder Lance Stunning is about to enter a free-style competition and is busy preparing his routine. His two key moves are the pool drop-in and the rail slide. Lance has to decide how many of each move he should include in the routine to maximize his score. Lance figures he has the stamina to make at most 25 of these moves in his performance. He also knows that his performance can be at most three minutes long. Each pool drop-in takes four seconds to perform, and each rail slide takes nine seconds. Lance also knows that if he includes more than 20 of either move, the judges will get bored, and he will get a poor score. a. Define your variables. b. Write a system of inequalities to model the situation. I. II. III. IV. V. x 0 VI. y 0 c. Graph your inequalities. d. Suppose the judges give each pool drop-in 5 points and each rail slide 10 points. What combination of moves will give Lance the best possible score? Page 15 of 16
Helpful Links: http://www.mathplanet.com/education/algebra-1/formulating-linear-equations/scatterplots-and-linear-models 59. 60. Page 16 of 16