Level 2 Placement Test Study

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math teacher or someone else who understands this topic. TOPIC 1: ARITHMETIC OPERATIONS A. Fractions: Simplifying fractions: example: Reduce = = • = 1• = (Note that you must be able to find a common factor; in this case 9; in both the top and the bottom in order to reduce.) 27 : 36 27 36 9•3 9•4 9 9 3 4 3 4 3 4 Problems 1-3: Reduce: 1. 13 52 = 2. 26 65 = 3+6 3+9 3. = Equivalent fractions: example: 3 4 3 4 = 3 4 is equivalent to how many eighths? 8 = 1• 43 = 22 • 43 = 2• 3 2• 4 = 4 9 = 5. 72 6 8 = 3 5 5 6 and 8 . 15 5 6 = 20 2 3 and 2 9 7. 3 8 25 , 30 and 8 15 = 16 30 and 7 12 8. Which is larger, 57 or 43 ? (Hint: find the LCD fractions) Adding, subtracting fractions: if the denominators are the same combine the numerators: example: 7 10 − 101 = 7− 1 10 = 106 = 4 7 + 27 = 10. 5 6 + 16 = 13. 5 8 = −61 + 41 = Multiplying fractions: multiply the top numbers, multiply the bottom numbers, reduce if possible. example: 3 4 14. 2 3 • 3 8 = 15. 1 2 • 25 = 3• 2 4 •5 = 6 20 = 103 (43 ) = 2 17. (2 12 ) = 2 16. • 13 = 18. 3 2 ÷ 41 = 20. 3 4 3 4 21. = ÷2= 2 3 2 22. = 4 3 4 B. Decimals: Meaning of places: in 324.519, each digit position has a value ten times the place to its right. The part to the left of the point is the whole number part. Right of the point, the places have values: tenths, hundredths, etc., So, 324.519 = (3 ×100) + (2 ×10) + (4 ×1) +(5 × 110) + (1× 1100) + (9 × 11000). 23.Which is larger: .59 or .7? 3 5 Problems 9-11: Find the sum or difference (reduce if possible): 9. − 23 = 19. 11 83 ÷ Problems 6-7: Find equivalent fractions with the LCD: 6. 3 5 3− 4 6 3 2 3 4 3 4 •12 9 ÷ = = = 4 3 2 3 2 3 •12 8 7 7•6 42 42 example: 2 1 = 2 1 = = = 42 − 2 (3 − 2 )• 6 4 − 3 1 3 First find LCM of 6 and 15: 6 = 2• 3 15 = 3• 5 LCM = 2 • 3• 5 = 30 , so 12. − 23 = 63 − 46 = example: How to get the lowest common denominator (LCD) by finding the least common multiple (LCM) of all denominators: example: example: 22 + 23 = 12 + 10 = 15 = 1 157 15 15 Dividing fractions: make a compound fraction, then multiply the top and bottom (of the big fraction) by the LCD of both: Problems 4-5: Complete: 4. 4 5 1 2 example: 11. 78 − 58 = If the denominators are different, find equivalent fractions with common denominators, then proceed as before: To add or subtract decimals, like places must be combined (line up the points). example: 1.23 − .1 = 1.13 example: 4 + .3 = 4.3 example: 6.04 − (2 −1.4 ) = 6.04 − .6 = 5.44 24. 5.4 + .78 = 25. .36 − .63 = 2 26. 4 − .3+ .001− .01+ .1= 27. $3.54 − $1.68 = Multiplying decimals: Problems 52-55: Write each as a decimal. If the decimal repeats, show the repeating block of digits: 52. .3 × .5 = .15 .3 × .2 = .06 2 (.03) = .0009 example: example: example: 53. 2 Dividing decimals: change the problem to an equivalent whole number problem by multiplying both by the same power of ten. example: .3 ÷ .03 Multiply both by 100, to get 30 ÷ 3 = 10 .014 .07 example: 14 70 = 14 ÷ 70 = .2 32. .013 ÷100 = 33. .053 ÷ .2 = 34. = 3 100 = example: .4 = four tenths = 104 = 25 example: 3.76 = three and seventy six 76 hundredths = 3 100 = 3 19 25 Problems 56-58: Write as a fraction: 56. .01 = 57. 4.9 = 58. 1.25 = 8 100 example: .075 = 7.5 % example: 1 41 = 1.25 = 125% Problems 59-60: Write as a percent: 59. .3 = Problems 35-44: Find the value: 40. 100 = 60. 4 = 2 2 To change a percent to decimal form, move the point 2 places left and drop the % symbol. 37. −(3) = 41. (2.1) = 42. (−.1) = 38. −3 = 43. example: 8.76% = .0876 example: 67% = .67 39. (−2) = 44. 36. (−3) 2= 2 2 3 2 3 (23 ) = 3 (− 23 ) = 3 Problems 61-62: Write as a decimal: a is a non-negative real number if a ≥ 0 2 a = b means b = a , where b ≥ 0 . Thus 49 = 7 , because 72 = 49 . Also, − 49 = −7 . 45. 144 = 46. − 144 = 47. −144 = 48. 49. 50. 51. 1.44 = .09 = 4 = 9 D. Fraction-decimal conversion: Fraction to decimal: divide the top by the bottom. example: example: 3 = 3 ÷ 4 = .75 4 20 = 20 ÷ 3 = 6.66666... = 6.6 3 3 52 = 3 + 25 = 3 + 2 ÷ 5 = 3 + .4 ( ) 61. 10% = 62. .03% = To solve a percent problem which can be written in this form: a % of b is c First identify a, b, c : Problems 63-65: If each statement were written (with the same meaning) in the form of a % of b is c , identify a, b, and c : 8100 = example: = To change a decimal to percent form, multiply by 100: move the point 2 places right and write the % symbol. 34 = 3• 3• 3• 3 = 81 4 3 = 4 • 4 • 4 = 64 35. 3 = 55. example: 8% means 8 hundredths or .08 or C. Positive integer exponents and square roots of perfect squares: Meaning of exponents (powers): example: example: 54. 4 13 = E. Percents: Meaning of percent: translate ‘percent’ as ‘hundredths’: Multiply both by 1000, get 340 3.4 = = Non-repeating decimals to fractions: read the number as a fraction, write it as a fraction, reduce if possible: 30. (.51) = 31. 5 × .4 = 28. 3.24 × 10 = 29. .01× .2 = 5 8 3 7 = 3.4 63. 3% of 40 is 1.2 64. 600 is 150% of 400 65. 3 out of 12 is 25% Given a and b, change a % to decimal form and multiply (since ‘of’ can be translated ‘multiply’). Given c and one of the others, divide c by the other (first change percent to decimal, or if the answer is a , write it as a percent). 2 25 3 Problems 69-71: Round to one significant digit. example: What is 9.4% of $5000? ( a % of b is c : 9.4% of $5000 is ? ) 69. 45.01 9.4% = .094 .094 × $5000 = $470 (answer) example: 56 problems correct out of 80 is what percent? ( a % of b is c : ? % of 80 is 56) 56 ÷ 80 = .7 = 70% (answer) example: 5610 people vote in an election, which is 60% of the registered voters. How many are registered? ( a % of b is c : 60 % of ? is 5610); 60% = .6 ; 5610 ÷ .6 = 9350 (answer) 66. 4% of 9 is what? 67. What percent of 70 is 56? 68. 15% of what is 60? F. Estimation and approximation: Rounding to one significant digit: example: 3.67 rounds to 4 example: .0449 rounds to .04 example: 850 rounds to either 800 or 900 Answers: 1. 1 4 2. 2 5 3. 3 4 4. 32 5. 12 6. 6 9 , 2 9 7. 9 24 , 14 24 8. 3 4 (because 20 28 < 21 28 ) 9. 6 7 10. 1 11. 1 4 12. − 115 13. 7 8 14. 1 4 15. 1 6 16. 916 17. 25 4 18. 6 19. 15 1 6 20. 3 8 21. 8 3 22. 1 6 23. .7 24. 6.18 25. -.27 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. example: .0298 × .000513 Round and compute: .03 × .0005 = .000015 .000015 is the estimate Problems 72-75: Select the best approximation of the answer: 72. 1.2346825 × 367.0003246 = (4, 40, 400, 4000, 40000) 73. .0042210398 ÷ .0190498238 = (.02, .2, .5, 5, 20, 50) 74. 101.7283507 + 3.141592653 = (2, 4, 98, 105, 400) 3 75. (4.36285903) = (12, 64, 640, 5000, 12000) 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 27 − 8 27 12 –12 not a real number 90 1.2 .3 2 71. 00083 To estimate an answer, it is often sufficient to round each given number to one significant digit, then compute. 3.791 $1.86 32.4 .002 .2601 2 .00013 .265 100 9 9 –9 –9 –8 10000 4.41 -.001 8 70. 1.09 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. .625 .428571 4.3 .03 1 100 9 10 1 4 49 4 = 10 1 = 45 30% 400% .1 .0003 a b 3 40 150 400 25 12 .36 80% 400 50 1 .0008 400 .2 105 64 3 TOPIC 2: POLYNOMIALS A. Grouping to simplify polynomials: The distributive property says: a(b + c ) = ab + ac example: 3(x − y ) = 3x − 3y (a = 3, b = x, c = −y) c 1.2 600 3 4 x + 7x = (4 + 7)x = 11x (a = x, b = 4, c = 7) example: 4a + 6x − 2 = 2(2a + 3x − 1) example: Problems 1-3: Rewrite, using the distributive property: 3. −5(a − 1) = 1. 6(x − 3) = 2. 4x − x = Commutative and associative properties are also used in regrouping: example: 3x + 7 − x = 3x − x + 7 = 2x + 7 example: 5 − x + 5 = 5 + 5 − x = 10 − x example: 3x + 2y − 2x + 3y = 3x − 2x + 2y + 3y = x + 5y Problems 4-9: Simplify: 4. x + x = 7. 4x + 1+ x − 2 = 5. a + b − a + b = 8. 180 − x − 90 = 6. 9 x − y + 3y − 8x = 9. x − 2 y + y − 2 x = example: If x = 3 , then 7 − 4 x = 7 − 4(3) = 7 − 12 = −5 example: If a = −7 and b = −1, then 2 a 2 b = (−7) (−1) = 49(−1) = −49 2 example: If x = −2 , then 3x − x − 5 = 2 3(−2) − (−2) − 5 = 3• 4 + 2 − 5 = 12 + 2 − 5 = 9 Problems 10-19: Given x = −1, y = 3, and z = −3 . Find the value: 2x = −z = xz = y+z = y2 + z 2 = 15. 16. 17. 18. 19. 2x + 4y = 2 2x − x − 1 = (x + z )2 = 2 2 x +z = 2 −x z = C. Adding, subtracting polynomials: Combine like terms: ( ) example: 3x + x + 1 − (x − 1) = 2 3x + x + 1 − x + 1 = 3 x + 2 2 example: (x − 1) + (x + 2 x − 3)= 2 2 x − 1 + x + 2x − 3 = x + 3x − 4 2 2 example: (x + x − 1)− (6 x − 2 x + 1)= 2 2 x + x − 1 − 6 x + 2 x − 1 = −5 x + 3x − 2 2 2 2 ( ) 23. ) ( ) − 3y − 5)− (2 y − y + 5)= 2 2 2 2 24. (7 − x ) − (x − 7) = ( ) 25. x − x + x − 1 = 2 2 D. Monomial times polynomial: Use the distributive property: example: 3(x − 4 ) = 3• x + 3(−4 ) = 3x + (−12) = 3x − 12 example: (2 x + 3)a = 2ax + 3a example: −4 x x 2 −1 = −4 x 3 + 4 x ( ) 26. − ( x − 7) = 27. −2(3 − a ) = 28. x (x + 5) = 30. a(2x − 3) = 31. 32. 29. (3x − 1)7 = (x −1)(−1) = 8(3a + 2a − 7)= 2 2 example: (2x + 1) (x − 4 ) is a(b + c ) if: a = (2x + 1), b = x , and c = −4 So a(b + c ) = ab + ac = (2x + 1)x + (2x + 1)(−4) 2 2 = 2 x + x − 8 x − 4 = 2 x − 7x − 4 Short cut to multiply above two binomials (see example above): FOIL (do mentally and write the answer) 2 F: First times First: (2x )( x) = 2x O: multiply ‘Outers’: (2x )(−4) = −8x I: multiply ‘Inners’: (1)(x ) = x L: Last times Last: (1)(−4) = −4 2 Add, get 2 x − 7x − 4 example: (x − 3) + (5 − 2 x ) = (x + 2)(x + 3) = x 2 + 5x + 6 example: (2x −1)(x + 2) = 2x 2 + 3x − 2 example: (x − 5)(x + 5) = x − 25 example: −4( x − 3) = −4x + 12 2 2 example: (3x − 4) = 9x − 24x + 16 example: (x + 3)(a − 5) = ax − 5x + 3a −15 2 Problems 33-41: Multiply: 33. 34. (x + 3)2 = 2 (x − 3) = (x + 3) (x − 3) = 36. (2 x + 3)(2 x − 3) = 37. (x − 4 ) (x − 2) = 35. Problems 20-25: Simplify: 20. x 2 + x − (x +1) = 21. ( (y 22. 2 a − a + a + a − 1 = E. Multiplying polynomials: Use the distributive property: a(b + c ) = ab + ac B. Evaluation by substitution: 10. 11. 12. 13. 14. 4 38. −6x (3 − x ) = 39. 40. 41. (x − 12 ) 2 = (x −1) (x + 3) = (x 2 )( ) −1 x 2 + 3 = 5 F. Special products: These product patterns (examples of FOIL) should be remembered and recognized: 2 2 I. (a + b)(a − b) = a − b 2 2 2 II. (a + b) = a + 2ab + b 2 2 2 III. (a − b) = a − 2ab + b example: 4x y + 6xy = 2xy(2x + 3) Problems 42-44: Match each pattern with its example: 2 2 a. (3x − 1) = 9 x − 6x + 1 2 2 b. ( x + 5) = x + 10x + 25 a 2 + 2 ab + b 2 = (a + b) 2 a 2 − 2ab + b 2 = (a − b) c. 2 Difference of two squares: a 2 − b 2 = (a + b)(a − b) example: 9x 2 − 4 = (3x + 2)(3x − 2) Trinomial square: 2 example: x − 6x + 9 = (x − 3) 2 (x + 8) (x − 8) = x 2 − 64 42. I: 43. II: Trinomial: example: x − x − 2 = (x − 2)(x + 1) 2 example: 6 x − 7x − 3 = (3 x + 1)(2 x − 3) 2 44. III: Problems 45-52: Write the answer using the appropriate product pattern: Problems 53-67: Factor: 45. (3a + 1)(3a −1) = 49. (3a − 2)(3a − 2) = 46. (y −1) = 2 47. (3a + 2) = 48. (3a + 2) (3a − 2) = 2 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. (x − y)2 = (4x + 3y)2 = (3x + y ) (3x − y ) = G. Factoring: Monomial factors: ab + ac = a(b + c ) example: x − x = x (x − 1) 2 Answers: 1. 6x − 18 2. 3x 3. −5a + 5 4. 2x 5. 2b 6. x + 2y 7. 5x − 1 8. 90 − x 9. − x − y 10. –2 11. 3 12. 3 13. 0 14. 18 15. 10 16. 2 17. 16 18. 10 19. 3 2 20. x − 1 21. 2 − x 2 22. 3a − 1 2 23. − y − 2 y −10 24. 14 − 2x 2 a + ab = 3 2 2 a − a b + ab = 2 8x − 2 = 2 x − 10x + 25 = −4 xy + 10 x 2 = 2 2x − 3x − 5 = 2 x − x−6= x2 y − y2 x = 2 −x +1 −x + 7 −6 + 2a x 2 + 5x 21x − 7 2ax − 3a −x 2 + 1 24a2 + 16a − 56 x 2 + 6x + 9 2 x − 6x + 9 2 x −9 4x 2 − 9 x 2 − 6x + 8 −18x + 6x 2 x 2 − x + 14 2 40. x + 2x − 3 4 2 41. x + 2x − 3 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 42. 43. 44. 45. 46. 47. 48. c b a 61. 62. 63. 64. 65. 66. 67. 49. 50. 51. 52. 53. 2 2 9a 2 − 12a + 4 x 2 − 2xy + y 2 16x 2 + 24xy + 9y 2 9 x2 − y 2 a (a + b) ( 54. a a 2 − ab + b 2 ) 55. 2(2 x + 1)(2 x −1) 56. 57. 58. 59. 60. 61. 62. 63. 9a − 1 y 2 − 2y + 1 9a 2 + 12a + 4 9a 2 − 4 x − 3x − 10 = 2 2x − x = 3 2 8x + 8x + 2x = 2 9x + 12x + 4 = 6 x3y2 − 9 x 4y = 2 1 − x − 2x = 2 3x − 10x + 3 = 64. 65. 66. 67. (x − 5)2 −2x(2y − 5x ) (2 x − 5)(x + 1) (x − 3)(x + 2) xy(x − y ) (x − 5)(x + 2) x (2x −1) 2 2x (2x + 1) (3x + 2)2 3x 3 y (2 y − 3x ) (1− 2 x )(1+ x ) (3x −1)(x − 3) 6 TOPIC 3: LINEAR EQUATIONS and INEQUALITIES A. Solving one linear equation in one variable: Add or subtract the same value on each side of the equation, or multiply or divide each side by the same value, with the goal of placing the variable alone on one side. If there are one or more fractions, it may be desirable to eliminate them by multiplying both sides by the common denominator. If the equation is a proportion, you may wish to cross-multiply. (Be sure to check answer in the original equation.) x+ 3 example: 3x+ =5 1 Problems 1-11: Solve: Problems 20-25: Solve and check: 1. 2. 3. 4. 5. 6. 2x = 9 3 = 65x 3x + 7 = 6 x = 45 3 5− x =9 x = 25x + 1 7. 4x − 6 = x 8. x − 4 = x2 + 1 9. 6 − 4x = x 10. 7x − 5 = 2x + 10 11. 4x + 5 = 3 − 2x To solve a linear equation for one variable in terms of the other(s), do the same as above: example: Solve for F : C = 59 (F − 32) Multiply by 59 : 95 C = F − 32 Add 32: 59 C + 32 = F Thus, F = 95 C + 32 example: Solve for b : a + b = 90 Subtract a : b = 90 − a example: Solve for x : ax + b = c Subtract b : ax = c − b Divide by a : x = c−b a Problems 12-19: Solve for the indicated variable in terms of the other(s): 12. a + b = 180 16. y = 4 − x 13. 2a + 2b = 180 17. y = 23 x + 1 14. P = 2b + 2h 18. ax + by = 0 15. y = 3x − 2 19. by − x = 0 b= x= b= x= b= x= x= y= B. Solution of a one-variable equation reducible to a linear equation: Some equations which do not appear to be linear can be solved by using a related linear equation: 1 example: x+ = −1 3x Multiply by 3x : x + 1 = −3x Solve: 4x = −1 x = − 41 Think of 5 as 5 1 and cross-multiply: 5x + 5 = 3x + 3 2x = −2 x = −1 But x = −1 does not make the original equation true (thus it does not check), so there is no solution. 20. 21. 22. x−1 = 67 x+1 3x = 52 2 x+1 3x− 2 =4 2 x+1 23. 24. 25. x+ 3 =2 2x 1 = x+x 8 3 x− 2 =3 4 − 2x example: 3 − x = 2 Since the absolute value of both 2 and −2 is 2, 3 − x can be either 2 or −2 . Write these two equations and solve each: 3− x =2 or 3 − x = −2 − x = −1 x =1 or − x = −5 x =5 Problems 26-30: Solve: 26. x = 3 27. x = −1 28. x −1 = 3 29. 2 − 3x = 0 30. x + 2 = 1 C. Solution of linear inequalities: Rules for inequalities: If a > b, then: a +c > b+c a −c > b−c ac > bc (if c > 0 ) ac < bc (if c < 0 ) a > bc (if c > 0 ) c a < bc (if c < 0 ) c If a < b , then: a +c < b+c a −c < b−c ac < bc (if c > 0 ) ac > bc (if c < 0 ) a < bc (if c > 0 ) c a > bc (if c < 0 ) c example: One variable graph: solve and graph on a number line: 1 − 2x ≤ 7 (This is an abbreviation for {x :1 − 2x ≤ 7}) Subtract 1, get −2x ≤ 6 Divide by −2 , x ≥ −3 Graph: -4 -3 -2 -1 0 1 2 3 Problems 31-38: Solve and graph on a number line: 31. x − 3 > 4 32. 4x < 2 33. 2x + 1≤ 6 34. 3 < x − 3 35. 4 − 2x < 6 7 36. 5 − x > x − 3 37. x > 1 + 4 38. 6x + 5 ≥ 4x − 3 39. x + 2y = 7 3x − y = 28 40. x + y = 5 x − y = −3 41. 2 x − y = −9 x =8 42. 2 x − y = 1 y = x −5 D. Solving a pair of linear equations in two variables: The solution consists of an ordered pair, an infinite number of ordered pairs, or no solution. Problems 39-46: Solve for the common solution(s) by substitution or linear combinations: Answers: 1. 9 2 2. 5 2 3. − 1 3 4. 15 4 5. −4 6. 5 3 7. 2 8. 10 9. 6 5 10. 3 11. − 1 3 12. 180 − a 13. 90 − a (P − 2 h ) 14. 2 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 38. x ≥ −4 -4 x >7 7 17. (3y−3) 2 by 18. − 19. x b 0 = 3(y−1) 2 a 33. x ≤ 1 2 1 0 2 3 0 6 4 5 0 43. (2819, −1319) 44. (1 4 ,0) 45. no solution 46. any ordered pair of the form (a, 2a − 3) where a is any number. One example: (4, 5) . Infinitely many solutions. 5 2 34. x > 6 0 39. (9,−1) 40. (1, 4) 41. (8, 25) 42. (−4,−9) (y + 2) 15. 3 16. 4 − y 0 37. x > 5 –3, −1 0 -1 0 36. x < 4 3 32. x < 12 3x + 5y = 1 44. 4 x −1 = y 4x + y = 1 45. x + y = 3 x + y =1 46. 2 x − y = 3 6x − 9 = 3y 35. x > −1 13 − 54 − 65 1 4 no solution –3, 3 no solution –2, 4 2 43. 2x − 3y = 5 TOPIC 4: QUADRATIC EQUATIONS A. ax 2 + bx + c = 0 : A quadratic equation can always be written so it 2 looks like ax + bx + c = 0 where a, b , and c are real numbers and a is not zero. example: 5 − x = 3x 2 2 Add x: 5 = 3x + x 2 Subtract 5: 0 = 3x + x − 5 2 or 3x + x − 5 = 0 So a = 3, b = 1, c = −5 example: x = 3 2 Rewrite: x − 3 = 0 2 (Think of x + 0x − 3 = 0 ) So a = 1, b = 0, c = −3 2 Problems 1-4: Write each of the following in the 2 form ax + bx + c = 0 and identify a, b, c : 1. 3x + x − 4 = 0 2 2. 5 − x = 0 2 3. x = 3x − 1 2 4. x = 3x 2 5. 81x = 1 2 B. Factoring: Monomial factors: ab + ac = a(b + c ) example: x − x = x (x −1) 2 example: 4 x y + 6 xy = 2 xy (2 x + 3) 2 Difference of two squares: a − b = (a + b)(a − b) 2 2 example: 9 x − 4 = (3 x + 2)(3 x − 2) 2 8 Trinomial square: Note: there must be a zero on one side of the equation to solve by the factoring method. a + 2ab + b = (a + b) 2 2 2 a2 − 2ab + b 2 = (a − b) 2 Problems 25-31: Solve: example: x − 6x + 9 = (x − 3) 2 Trinomial: example: x − x − 2 = (x − 2)(x + 1) 2 example: 6 x − 7 x − 3 = (3 x + 1)(2 x − 3) 2 Problems 6-20: Factor: 6. a + ab = 3 2 2 7. a − a b + ab = 2 8. 8 x − 2 = 2 9. x − 10x + 25 = 2 10. −4xy + 10x = 2 11. 2x − 3x − 5 = 2 12. x − x − 6 = 2 2 13. x y − y x = 2 14. 15. 16. 17. 18. 19. 20. (x + 1)(x −1) = 0 26. 4x (x + 4 ) = 0 27. 0 = (2 x − 5)x 28. 0 = (2x + 3)(x − 1) 25. 2 x 2 − 3x − 10 = 2x 2 − x 2 x3 + 8 x2 + 8 x = 9 x 2 + 12 x + 4 = 6x 3 y 2 − 9x 4 y = 1− x − 2x 2 = 3x 2 −10x + 3 = 29. 30. 31. (x − 6)(x − 6) = 0 (2x − 3)2 = 0 x (x + 2)(x − 3) = 0 D. Solving quadratic equations by factoring: Arrange the equation so zero is on one side (in 2 the form ax + bx + c = 0 ), factor, set each factor equal to zero, and solve the resulting linear equations. example: Solve: 6x = 3x 2 Rewrite: 6x − 3x = 0 Factor: 3x (2x −1) = 0 So 3x = 0 or (2 x −1) = 0 2 Thus, x = 0 or x = 1 2 example: 0 = x − x − 12 2 C. Solving factored quadratic equations: The following statement is the central principle: If ab = 0 , then a = 0 or b = 0 . First, identify a and b in ab = 0 : example: (3 − x )(x + 2) = 0 Compare this with ab = 0 a = (3 − x ) ; b = (x + 2) Problems 32-43: Solve by factoring: 32. x (x − 3) = 0 Then, because ab = 0 means a = 0 or b = 0 , we can use the factors to make two linear equations to solve: example: If 2x (3x − 4) = 0 then (2x ) = 0 or (3 x − 4) = 0 and so x = 0 or 3x = 4 ; x = 43 . Thus, there are two solutions: 0 and 4 3 example: If (3 − x )(x + 2) = 0 then (3 − x ) = 0 or (x + 2) = 0 and thus x = 3 or x = −2 . example: If (2x + 7) = 0 then 2x + 7 = 0 39. (2 x + 1)(3 x − 2) = 0 34. 2 x = x 35. 3x (x + 4 ) = 0 40. 6 x = x + 2 2 41. 9 + x = 6x 36. x = 2 − x 2 37. x + x = 6 42. 1− x = 2x 2 43. x − x − 6 = 0 2 23. (2 x −1)(x − 5) = 0 24. 0 = (x − 1)(x + 1) 2 2 x = −7 x = − 72 (one solution) 38. 0 = (x + 2)(x − 3) 33. x − 2x = 0 2 Problems 21-24: Identify a and b in each of the following: 21. 3x (2 x − 5) = 0 22. (x − 3)x = 0 0 = (x − 4 )(x + 3) x − 4 = 0 or x + 3 = 0 x = 4 or x = −3 2 2 2 Another problem form: if a problem is stated in this form: ‘One of the solutions of ax 2 + bx + c = 0 is d’, solve the equation as above, then verify the statement. example: Problem: One of the solutions of 10x 2 − 5x = 0 is A. –2 B. −1 2 C. 1 2 D. 2 E. 5 2 Solve 10 x − 5 x = 0 by factoring: 5x (2x −1) = 0 so 5 x = 0 or (2x −1) = 0 thus x = 0 or x = 12 . Since x = 1 2 is one solution, answer C is correct. 9 44. One of the solutions of (x −1)(3x + 2) = 0 is A. −3 2 B. −2 3 C. 0 D. 2 3 E. 3 2 Answers: 1. 2. 3. 4. 5. a 1 -1 1 3 81 x + 3x − 4 = 0 −x 2 + 5 = 0 x 2 − 3x +1 = 0 3x 2 − x = 0 81x 2 −1 = 0 2 b 3 0 -3 -1 0 ( 7. a a 2 − ab + b 2 2 14. (x − 5) (x + 2) c -4 5 1 0 -1 29. 6 30. 3 2 31. –2, 0, 3 32. 0, 3 33. 0, 2 34. 0, 1 2 35. –4, 0 36. –2, 1 37. –3, 2 38. –2, 3 39. −1 2 , 2 3 40. −1 2 , 2 3 41. 3 42. –1, 1 2 43. –2, 3 44. B 45. B 15. x (2x −1) 16. 2x (x + 2) 2 17. (3x + 2) 2 18. 3x 3 y (2y − 3x ) Note: all signs could be the opposite. 6. a (a + b) 45. One solution of x − x − 2 = 0 is A. –2 B. –1 C. −1 2 D. 1 2 E. 1 19. (1− 2x ) (1+ x ) 20. (3x −1) (x − 3) a b 21. 3x 2x − 5 22. x − 3 x 23. 2 x −1 x − 5 24. x − 1 x +1 25. –1, 1 26. – 4, 0 27. 0, 5 2 28. −3 2 , 1 ) 8. 2(2 x +1) (2x -1) 9. (x − 5) 2 10. −2x (2y − 5x ) 11. (2x − 5) (x + 1) 12. (x − 3) (x + 2) 13. xy (x − y ) TOPIC 5: GRAPHING A. Graphing a point on the number line: Problems 1-7: Select the letter of the point on the number line with coordinate: C D E A B -2 -1 0 F 1 1. 0 2. 12 5. -1.5 6. 2.75 3. 7. 4. − 12 4 3 G 2 3 5 9 9. 3 4 − 23 10. 2 3 Problems 11-13: Solve each equation and graph the solution on the number line: example: x + 3 = 1 x = −2 13. 4 − x = 3 + x B. Graphing a linear inequality (in one variable) on the number line: Rules for inequalities: If a > b , then: Problems 8-10: Which letter best locates the given number: 8. 11. 2x − 6 = 0 12. x = 3x + 5 If a < b , then: a+c >b+c a−c >b−c ac > bc (if c > 0 ) ac < bc (if c < 0 ) a > bc (if c > 0 ) c a < bc (if c < 0 ) c a+c 0 ) ac > bc (if c < 0 ) a < bc (if c > 0 ) c a > bc (if c < 0 ) c example: One variable graph: solve and graph on a number line: 1 − 2x ≤ 7 − 2x ≤ 7} ) (This is an abbreviation for {x :1− Subtract 1, get −2x ≤ 6 Divide by -2, x ≥ −3 Graph: -4 -3 -2 -1 0 1 2 3 10 Problems 14-20: Solve and graph on a number line: 18. 4 − 2x < 6 x − 3> 4 4x < 2 19. 5 − x > x − 3 2x + 1 ≤ 6 20. x > 1+ 4 3< x −3 example: x > −3 and x < 1 14. 15. 16. 17. The two numbers -3 and 1 splits the number line into three parts: x > −3 , −3 < x < 1 , and x < 1. Check each part to see if both x > −3 and x < 1 are true: part x values x > −3 ? x < 1? both true? 1 no yes no x < −3 2 yes yes yes (solution) −3 < x < 1 3 yes no no x >1 Thus the solution is −3 < x < 1 and the graph is: example: x ≤ −3 or x < 1 (‘or’ means ‘and/or’) at least one part x values x ≤ −3 ? x < 1? true? 1 yes yes yes (solution) x ≤ −3 no yes yes (solution) 2 −3 ≤ x < 1 3 no no no x >1 So x ≤ −3 or −3 ≤ x < 1 ; these cases are both covered if x < 1. Thus the solution is x < 1 and the graph is: Problems 21-23: Solve and graph: To locate a point on the plane, an ordered pair of numbers is used, written in the form (x, y). The x-coordinate is always given first. Problems 24-27: Identify x and y in each ordered pair: 24. (3, 0) 25. (-2, 5) 26. (5, -2) 27. (0, 3) To plot a point, start at the origin and make the moves, first in the x-direction (horizontal) and the in the y-direction (vertical) indicated by the ordered pair. example: (-3, 4) Start at the origin, move left 3 (since x = −3 ), then (from there), up 4 (since y = 4 ) Put a dot there to indicate the point (-3, 4) 28. Join the following points in the given order: (-3, -2), (1, -4), (-3, 0), (2, 3), (-1, 2), (3, 0), (-3, -2), (-1, 2), (1, -4) 21. x < 1 or x > 3 22. x ≥ 0 and x > 2 23. x > 1 and x ≤ 4 29. Two of the lines you drew cross each other. What are the coordinates of this crossing point? C. Graphing a point in the coordinate plane: Problems 31-34: For each given 33 point, which of its coordinates, x or y, 32 is larger? If two number lines intersect at right angles so that: 1) one is horizontal with positive to the right and negative to the left, 2) the other is vertical with positive up and negative down, and 3) the zero points coincide Then they form a coordinate plane, and 1) the horizontal number line is called the x-axis, 2) the vertical line is the y-axis, 3) the common zero point is the origin, y 4) there are four quadrants, II I numbered x as shown: III IV 30. In what quadrant does the point (a, b) lie, if a > 0 and b < 0 ? 34 31 D. Graphing linear equations on the coordinate plane: The graph of a linear equation is a line, and one way to find the line is to join points of the line. Two points determine a line, but three are often plotted on a graph to be sure they are collinear (all in a line). Case I: If the equation looks like x = a , then there is no restriction on y, so y can be any number. Pick 3 numbers for values of y, and make 3 ordered pairs so each has x = a . Plot and join. example: x = −2 Select three y’s, say -3, 0, and 1 Ordered pairs: (-2, -3), (-2, 0), (-2, 1) Plot and join: Note the slope formula gives −2−−3(−−02) = −03 , 11 Problems 35-41: Graph each line on the number plane and find its slope (refer to section E below if necessary): which is not defined: a vertical line has no slope. 3 6 = 43. 5− 2 1−(−1) = 46. 0−1 −1−4 0 3 44. −6−(−1) 5−10 = 47. −2 0 42. example: y = −2 Select three x’s, say -1, 0, and 2 Since y must be -2, the pairs are: (-1, -2), (0, -2), (2, -2) 45. = = P2 (x 2 , y 2 ) has slope 0 −1 y 2 −y1 x 2 −x1 . example: A (3, -1), B (-2, 4) 4 −( −1) slope of AB = −2− 3 = −55 = −1 Problems 48-52: Find the slope of the line joining the given points: 48. (-3, 1) and (-1, -4) 49. (0, 2) and (-3, -5) 50. (3, -1) and (5, -1) 51. 52. = 13 = 3, And the line is neither horizontal nor vertical. Answers: 1. D 2. E 3. C 4. F 5. B 6. G 7. B 8. Q 9. T 10. S 11. 3 12. −5 17. x > 6 6 18. x > −1 -1 19. x < 4 0 4 20. x > 5 0 5 22. x > 2 2 -2 0 14. x > 7 1 2 16. x ≤ 5 2 23. 1 < x ≤ 4 x y 24. 3 0 25. -2 5 26. 5 -2 27. 0 3 28. 3 -3 13. ½ 15. x < 0 0 21. x < 1 or x > 3 0 0 1 0 7 0 0 ½ 1 2 3 = The line joining the points P1 (x1, y1 ) and The slope is = =0 And the line is horizontal. example: y = 3x −1 Select 3 x’s, say 0, 1, 2: If x = 0 , y = 3• 0 −1 = −1 If x = 1, y = 3•1−1 = 2 If x = 2 , y = 3• 2 −1 = 5 Ordered pairs: (0, -1), (1, 2), (2, 5) Note the slope is 2−(−1) 1− 0 39. x = −2 40. y = −2x 41. y = 12 x + 1 E. Slope of a line through two points: Problems 42-47: Find the value of each of the following: Case II: If the equation looks like y = mx + b , where either m or b (or both) can be zero, select any three numbers for values of x, and find the corresponding y values. Graph (plot) these ordered pairs and join. −2−(−2) −1− 0 y = 3x x−y=3 x = 1− y y =1 35. 36. 37. 38. 0 1 (0,-1) IV x y y x 3 3 0 2 0 29. 30. 31. 32. 33. 34. 35. 1 3 36. 1 4 -3 37. -1 1 -1 38. 0 1 39. none -2 12 40. -2 41. ½ 42. 43. 44. 45. 46. 47. 1 -2 48. −5 2 49. 7 3 50. 0 51. −3 5 52. 3 4 ½ 3 2 1 1 5 0 none (undefined) TOPIC 6: RATIONAL EXPRESSIONS A. Simplifying fractional expressions: 9• 3 9• 4 example: = = • = 1• = (note that you must be able to find a common factor—in this case 9 –in both the top and bottom in order to reduce a fraction.) example: 123aab = 33a•a•41b = 33aa • 41b = 1• 41b = 41b (common factor: 3a ) Problems 1-12: Reduce: 27 36 1. 2. 3. 4. 9 9 = = 13 52 26 65 8. 9. 10. = 5. 19 a 95 a 6. 14 x − 7 y 7y example: 3 5 • 3 5 3 4 7. 3+ 6 = 3+ 9 6 axy = 15 by 2 3 4 11. = 3 •y x 15 • 10y 2x = example: 3 4 5 a+b = 5 a+c x− 4 = 4 −x 2(x+ 4 )(x− 5) (x− 5)(x− 4 ) x − 9x x− 9 2 8(x −1) 6(x 2 −1) = = 2 12. 2 x2 − x −1 x − 2 x +1 3•10• x • y = 15• x • y 2 example: 3x + 2 x +1 4x 6 14. x 2 − 3x x− 4 • = 2 y x (x − 4 ) 2x− 6 −1+ 3 2( 2)−1 = 16. 17. 18. 6 = b x = a x = 3 a−y = b 20. 21. 22. 4 x−5y 3y − 2 x b = c b −z = c = z 4 9 24. 3x 7 25. x+ 3 x+ 2 = 8, 3 4 = 1• 43 = 22 • 43 = 2• 3 2• 4 2 72 = 7y 26. 30−15 a 15−15 b 27. x− 6 6−x = (1+b )(1−b ) = −2 = (x−1)(x+ 2) example: 5 6 and 8 15 . First find LCM of 6 and 15: 6 = 2•3 15 = 3• 5 LCM = 2 • 3• 5 = 30 , so 6 8 = 25 , and 15 = 16 5 30 30 3 4 and 1 6a : 4 = 2•2 6a = 2 • 3• a LCM = 2 • 2 • 3• a = 12a , so 3 = 129aa , and 61a = 122a 4 example: 3 x+ 2 and −1 x− 2 LCM = (x + 2)(x − 2), so = 3 x+ 2 3•(x− 2) = (x+ 2)(x− 2) , and −1 x− 2 −1•(x+ 2) = (x+ 2)(x− 2) Problems 28-33: Find equivalent fractions with the lowest common denominator: 28. C. Equivalent fractions: example: 3 4 is equivalent to how many eighths? 3 4 = 23. 2 3 19. (x− 2)(x−1) example: Problems 15-22: Find the value, given a = −1, b = 2 , c = 0 , x = −3 , y = 1, z = 2 : 15. = (x+1)(x− 2) x − 3x+ 2 = (x− 2)(x+1) = (x+ 1)(x− 2) x−1 x+1 = B. Evaluation of fractions: example: If a = −1 and b = 2 , find the value of 2ab+−31 Substitute: x−1 x+1 How to get the lowest common denominator (LCD) by finding the least common multiple (LCM) of all denominators: • 21 • xx • yy • 1y = 1 • 1 • 2 • 1 • 1 • 1y = • yxy2 • 32y = x+8 = 44 • 3xx++12 = 12 4 x+4 example: Problems 13-14: Simplify: 13. = Problems 23-27: Complete: = 2 6 5a 6 = 5 ab 5a b b • 6 = 56ab b 5a 3x+ 2 = 4 (x+1) x+1 = 6 8 29. 30. 2 3 3 x x 3 2 9 31. and 5 32. and and −4 x+1 33. 3 4 and 2−x x− 2 −4 and x+5 3 x− 3 1 3x and x+ x 1 13 D. Adding and subtracting fractions: If denominators are the same, combine the numbers: example: 3x y − xy = 3x−x y = 2x y Problems 34-38: Find the sum or difference as indicated (reduce if possible): 34. 35. 36. 4 7 + 27 = 52. 3y 2 x+ 2 − = 2 x + 2x xy 2 3a + b2 − ab = b 37. 3 − x−x 3 = x− 3 b−a − a−b = b+a b+a 38. If denominators are different, find equivalent fractions with common denominators, then proceed as before (combine numerators): − a4 = 24a − a4 = example: x−31 + x+1 2 example: = = a 2 3(x+ 2) (x−1)(x+ 2) 3x+ 6+x−1 (x−1)(x+ 2) + 2 a−a 4 = a 4 5 = (x−41x+ )(x+ 2) 39. 40. 41. 42. 43. − − − +2= −2= 44. a − 45. 1 a c b 1 b 46. a − 47. 48. 49. 50. = 51. + = 1 a = x x + 1−x = x−1 3x− 2 2 − x+ 2 = x− 2 2 x−1 − 2x−x−21 = x+1 x − x 2 −4 2 x = x− 2 x − x 24− 4 = x− 2 3 4 • 25 = 6 20 54. 2 7a 55. (43 ) 7. 55aa +b +c 8. -1 2(x + 4 ) 9. x− 4 10. x 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 2 = ( )= 2a 3 5b 3( x + 4 ) 5y 3 • 5y x 2 −16 58. = 2 x+3 • 2xx+ 6 3x 59. 5x 2y 3 4 2 3 ÷ 2x = = 3 4 = = = = a b c d 5x 2y 2x = 5x 2y a−4 = 3 − 2 a 67. x+7 x 2 −9 1 x−3 68. 63. a b ÷3= 69. 64. 3 a ÷ b3 = 70. 2 3 3 -1 -1 1 2 = 71. 22. 23. 24. 25. 26. 27. 28. 29. 9 undefined 20 -1 = 66. ÷2= 2a − b • 2y 2x • 2y 3 4 x −17 a b c d 62. 4 (x−1) 3(x +1) 2x +1 x−1 2 x2 ÷ dc = example: 61. 11 83 ÷ = 103 11. • ab = 12 2 3 57. a b 65. Answers: 1. 1 4 2. 2 5 3. 3 4 4. 25axb 5. a 5 2 x−y 6. y 56. 2 12 example: 60. E. Multiplying fractions: Multiply the tops, multiply the bottoms, reduce if possible: example: a b 3x+ 6 x−1 ( ) = • dc = 53. = • bd ad = • bd bc 7 7•6 42 example: 2 1 = 2 1 = 4−3 = 42 1 = 42 − − • 6 (3 2 ) 3 2 (x−1) (x−1)(x+ 2) 1 = 2a 2 = a 2 = x 2 3 • 3 8 3(x+1) x 2 − 4 • x 2 −1 x− 2 3( x+1)(x+ 2)( x− 2) = (x− 2)(x+1)(x−1) F. Dividing fractions: A nice way to do this is to make a compound fraction and then multiply the top and bottom (of the big fraction) by the LCD of both: Problems 39-51: Find the sum or difference: 3 a 3 x 4 5 2 5 a b example: 30. 31. 2 3 4 2 3 4 a b c a b c 5x 4 xy = 5 4y = = = = = 0 32 3xy x 2 + 2x − 3 or (x −1)(x + 3) 2 + 2b − a − ab or (1+ b)(2 − a ) 2 6 , 29 9 3 5x , x x x (x +1) 3(x +1) , −12 3(x +1) 3 −4 , x−2 x−2 32. 33. −4 (x +3) , 14 −5(x−3) 45. (x−3)(x +3) (x−3)(x +3) a +b ab a 2 −1 a 58. 59. 60. 61. 62. 63. 64. 46. 47. 0 2 +2x 48. 3xx 2 −4 2 x +1 , 3x x (x +1) x (x +1) 34. 6 7 35. -1 36. 2b−2a b +a 37. −2 x 38. 2ab+2 39. 2a5 40. 3a−2x ax 41. 4 x−10 5x 42. 12 5 43. a−2b b 44. ab−c b 49. 50. 51. 52. 53. 54. 55. 56. 57. −3(2x−1) (x +1)(x−2) x +2 x x 2 +2x−4 x 2 −4 1 4 ac bd b 42 9 16 25 4 8a 9 125b 3 3y 2 x −4 x 6 9 8 91 6 3 8 a 3b 9 ab 65. 4a − 2b 2 −4 a 66. a3−2 a x +7 67. x +3 68. 8 3 69. 1 6 70. bca 71. acb TOPIC 7: EXPONENTS and SQUARE ROOT A. Positive integer exponents: a b means use a as a factor b times. (b is the exponent or power of a.) example: 2 5 means 2 • 2 • 2 • 2 • 2 , and has value 32. example: c • c • c = c 3 Problems 1-14: Find the value: 8. (.2) = 2 1. 2 3 = 2. 32 = 9. 3. −4 2 = 10. 2 4. (−4 ) = = 13. (−1.1) = 6. 1 = () 2 = 3 14. 32 • 2 3 = example: Simplify: a • a • a • a • a = a5 Problems 15-18: Simplify: 15. 32 • x 4 = 17. 4 2 (−x )(−x )(−x ) = 4 16. 2 4 • b • b • b = 18. (−y ) = B. Integer exponents: I. II. III. IV. V. ab • ac = ab+c ab = ab− c ac (a ) = a c b (ab) c (ab ) c bc = a c • bc = ac bc 19. 2 3 • 2 4 = 2 x 3 20. 24 = 2 x 2 22. = ( ) 12. 2 23 4 2 4 3 = 9 5. 0 4 = Problems 19-28: Find x: 21. 3−4 = 11. (−2) = 2 7. ( ) 2 1 12 10 a 0 = 1 (if a ≠ 0 ) a− b = a1b VI. VII. 52 52 =5 1 3x x ( ) =2 23. 2 3 4 24. 8 = 2 x 25. a 3 • a = a x 26. 27. x 28. = bx b 10 b5 1 c −4 = cx a 3 y−2 a 2 y−3 = ax Problems 29-41: Find the value: 29. 7 x 0 = 36. 30. 3−4 = 37. 31. 2 3 • 2 4 = 38. 32. 0 5 = 39. 33. 5 0 = 40. 34. (−3) − 33 = 41. 3 35. x c + 3 • x c − 3 = x c+3 x c−3 2 x −3 6 x −4 = = (a ) = (x ) = (3x ) = (−2xy ) = x+3 x−3 3 2 3 2 2 3 C. Scientific notation: example: 32800 = 3.2800 ×10 4 if the zeros in the ten’s and one’s places are significant. If the one’s zero is not, write 3.280 ×10 4 ; if neither is significant: 3.28 ×10 4 . example: .004031 = 4.031 ×10−3 15 example: 2 ×10 = 200 example: 9.9 ×10−1 = .99 2 Note that scientific form always looks like a ×10 where 1 ≤ a < 10 , and n is an integer power of 10. n Problems 42-45: Write in scientific notation: 42. 93,000,000 = 43. .000042 = 65. 44. 5.07 = 45. −32 = 48. 4 ×10−6 = 5 example: 69. 5+2 5 =3 5 32 − 2 = 4 2 − 2 = 3 2 70. 5 + 5 = 72. 3 2 + 2 = 71. 2 3 + 27 − 75 = 73. 5 3 − 3 = F. Multiplying square roots: a • b = ab if a ≥ 0 and b ≥ 0 . (3.14 )(2) × 10 5 = 6.28 × 10 5 4.28×10 = 2.14×10 −2 6 4.28 10 × 10 −2 = 2.14 −3 2.01×10 = 8.04×10 −6 3 68. Problems 70-73: Simplify: (3.14 × 10 )(2) = example: 67. .09 = example: example: To compute with numbers written in scientific form, separate the parts, compute, then recombine. example: example: 6 • 24 = 6 • 24 = 144 = 12 example: 2 • 6 = 12 = 4 • 3 = 2 3 example: 5 2 3 2 = 15 4 = 15 • 2 = 30 6 2.00 ×10 8 ( )( ) Problems 74-79: Simplify: .250 ×10 = 2.50 ×10 2 Problems 49-56: Write answer in scientific notation: 74. 3• 3 = 77. 49. 10 40 ×10−2 = 75. 3• 4 = 78. 53. 50. 10 −40 10 −10 51. 1.86×10 4 3×10 −1 = 55. 52. 3.6×10 −5 1.8×10 −8 = 56. = 54. 1.8×10 −8 3.6×10 −5 = (4 × 10 ) = (2.5 × 10 ) = ( )(3 2 )= −3 2 76. 2 3 2 −1 (−2.92×10 )(4.1×10 ) 3 −8.2×10 7 −3 = ab = a • b if a and b are both non-negative ( a ≥ 0 and b ≥ 0 ). 32 = 16 • 2 = 4 2 75 = 3 • 25 = 3 • 5 = 5 3 example: If x ≥ 0 , x 6 = x 3 If x < 0 , x 6 = x 3 example: example: a = b means (by definition) that 1) b 2 = a , and 2) b ≥ 0 Problems 57-69: Simplify (assume all square roots are real numbers): 57. 81 = 58. − 81 = 59. 2 9 = 60. 4 9 = 79. ( 9) = ( 5) = ( 3) = 2 2 4 Problems 80-81: Find the value of x: D. Simplification of square roots: Note: x5 = 4x6 = a2 = a3 = 66. E. Adding and subtracting square roots: Problems 46-48: Write in standard notation: 46. 1.4030 ×10 3 = 47. −9.11×10−2 40 = 62. 3 12 = 63. 52 = 9 64. 16 = 61. 80. 4• 9= x 81. 3 2 • 5 = 3 x G. Dividing square roots: a÷ b= example: = a b a b , if a ≥ 0 and b > 0 . 2 ÷ 64 = 2 64 = 2 8 (or 1 8 2) Problems 82-86: Simplify: 82. 83. 84. 3÷ 4 = 9 = 25 49 2 85. 86. 36 ÷ 4 = = −8 16 = If a fraction has a square root on the bottom, it is sometimes desirable to find an equivalent fraction with no root on the bottom. This is called rationalizing the denominator. example: example: 5 8 1 3 = = 5 8 1 3 = • 5 8 3 3 • = 2 = 10 2 16 3 3 = 3 9 = 10 4 16 Problems 87-94: Simplify: 87. 88. 9 4 18 9 = = 89. 91. 4 9 3 2 90. Answers: 1. 8 2. 9 3. -16 4. 16 5. 0 6. 1 7. 16 81 8. .04 9. 9 4 10. 1024 11. -512 12. 64 9 13. -1.331 14. 72 15. 9x 4 16. 16 b 3 17. −16 x 3 18. y 4 19. 7 20. -1 21. 4 22. 0 23. 12 24. 3 25. 4 26. 5 27. 4 28. y + 1 29. 7 30. 1 81 31. 128 32. 0 33. 1 = 92. = 93. a b = 94. 2+ 1 2 = 68. a x 2 −9 a x6 9x 6 −8x 3 y 6 9.3×10 7 4.2 ×10−5 5.07 −3.2 ×10 1403.0 -.0911 .000004 1×10 38 1×10−30 6.2 ×10 4 2.0 ×10 3 5.0 ×10−4 1.6 ×10−5 4.0 ×10−3 1.46 ×1013 9 -9 6 12 2 10 6 3 2 13 3 = = 66. x 2 x 67. 2 x 3 34. -54 35. x 2 c 36. x 6 37. x 3 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 1 5 3 3 4 .3 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. a a 2 5 0 4 2 4 3 3 2 3 6 6 9 5 9 36 10 94. 3 2 2 3 3 7 3 2 5 2 2 -2 3 2 2 2 9 6 5 2 5 3 ab b TOPIC 8: GEOMETRIC MEASUREMENT A. Intersecting lines and parallels: If two lines intersect as shown, adjacent angles add to 180° . For example, a b c a + d = 180° . Non-adjacent angles d are equal: for example, a = c . If two lines, a and b, are parallel and are cut by a third line c, c forming angles w, x, y, z x w y as shown, then z x = z , w + y = 180°, a so z + y = 180° . b example: If a = 3x and c = x , find the measure of c. b = c , so b = x . a b a + b = 180 , so 3 x + x = 180 , giving 4 x = 180 , or x = 45 Thus c = x = 45° c Problems 1-4: Given x = 127°, find the measures of the other angles: 1. t 2. y 3. z 4. w t x y z w 17 5. Find x: 2x 3x B. Formulas for perimeter P and area A of triangles, squares, rectangles, and parallelograms: Rectangle, base b, altitude (height) h: P = 2b + 2h A = bh P = 2b + 2 h = 2 • 7 + 2 • 8 = 14 + 16 = 30 units A = bh = 7 • 8 = 56 sq. units A square is a rectangle with all sides equal, so the formulas are the same (and simpler if the side length is s): s s example: Square with side 11cm has P = 4s = 4 •11 = 44cm A = s 2 = 112 = 121cm 2 (sq. cm) A parallelogram with base b and height h has A = bh . If the other side length is a, then P = 2 a + 2b . example: Parallelogram has sides 4 and 6, and 5 is the length of the altitude perpendicular 5 6 to the side 4. 4 P = 2 a + 2b = 2 • 6 + 2 • 4 = 12 + 8 = 20 units A = bh = 4 • 5 = 20 sq. units In a triangle with side lengths a, b, c and h is the altitude to side b, P = a+b+c A = 12 bh = bh 2 example: P = a+b+c = 6 + 8 + 10 = 24 units A = 12 bh = 12 (10)(4.8) = 24 sq. units Problems 6-13: Find P and A for each of the following figures: 6. 7. 8. 9. 5 4 If a wire is bent in the shape, the perimeter is the length of the wire, and the area is the number of square units enclosed by the wire. example: Rectangle with b = 7 and h = 8 : P = 4s A = s2 10. Parallelogram with sides 36 and 24, and height 10 (on side 36). 11. Parallelogram, all sides 12, altitude 6. 12. Triangle with sides 5, 12, 13, 13 and 5 is the height on side 12. 12 13. The triangle shown: 5 3 Rectangle with sides 5 and 10. Rectangle, sides 1.5 and 4. Square with side 3 mi. Square, side 43 yd. C. Formulas for circle area A and circumference C: A circle with radius r (and diameter d = 2 r ) has distance around (circumference) r C = πd or C = 2πr (If a piece of wire is bent into a circular shape, the circumference is the length of wire.) example: A circle with radius r = 70 has d = 2 r = 170 and exact circumference C = 2πr = 2 • π • 70 = 140π units. If π is approximated by 22 , 7 C = 140π = 140(22 = 440 units approximately. 7) If π is approximated by 3.1, the approximate C = 140(3.1) = 434 units. The area of a circle is A = πr 2 : example: If r = 8 A = πr 2 = π • 8 2 = 64 π sq. units Problems 14-16: Find C and A for each circle: 14. r = 5 units 15. r = 10 feet 16. d = 4 km D. Formulas for volume V: A rectangular solid (box) with length l, width w, and height h, has volume V = lwh . example: A box with dimensions 3, 7, and 11 has what volume? V = lwh = 3• 7 •11 = 231 cu. units A cube is a box with all edges equal. If the edge is e the volume V = e 3 example: A cube has edge 4 cm. V = e 3 = 4 3 = 64 cm 3 (cu. cm) A (right circular) cylinder with radius r and altitude h has V = πr 2 h example: A cylinder has r = 10 and h = 14 . 18 The exact volume is V = πr 2 h = π •10 2 •14 = 1400π cu. units If π is approximated by 22 , 7 r example: The exact volume of a sphere with radius 6 in. is V = 43 πr 3 = 43 • π • 6 3 = 43 π (216) = 288π in 3 Problems 17-24: Find the exact volume of each of the following solids: 17. 18. 19. 20. 21. 22. 23. 24. C 28. 82° , 82° 29. 68° , 44° C ∠C = 30° . 37. Find measures of ∠B and ∠C , if ∠A = 30° . B B 6 6 A C B 8 C 8 If a triangle has equal angles, the sides opposite these angles also have equal measures. example: Find the measures of ∠B , AB and AC , A B 70° given this figure, and ∠C = 40° : 16 ∠B = 70° (because all angles add to 180° ) Since ∠A = ∠B , AC = BC = 16 . AB can be found with trig -- later. C 41. Can a triangle be isosceles and have a 90° ? 42. Given ∠D = ∠E = 68° D and DF = 6 . Find the measure of ∠F and F length of FE : E An isosceles triangle is defined to have at least two sides with equal measure. The equal sides may be marked: 12 G. Similar triangles: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. example: ∆ABC and ∆FED are similar: C 7 12 Problems 30-35: Is the triangle isosceles? 30. Sides 3, 4, 5 A A F. Isosceles triangles: Or the measures may be given: 36. Find measures of ∠A and ∠B , if 39. If the angles of a triangle are 30°, 60° , and 90° , can it be isosceles? 40. If two angles of a triangle are 45° and 60° , can it be isosceles? Problems 25-29: Given two angles of a triangle, find the measure of the third angle: 25. 30°, 60° 26. 115° , 36° 27. 90° , 17° 35. 8 E. Sum of the interior angles of a triangle: The three angles of any triangle add to 180° . example: Find the measures of angles C and A: A ∠C (angle C) is marked to show its measure is 90° . 36° ∠B + ∠C = 36 + 90 = 126 , so ∠A = 180 −126 = 54° 33. 38. Find measure of ∠A . Box, 6 by 8 by 9. Box, 1 23 by 56 by 2 25 . Cube with edge 10. Cube, edge .5 . Cylinder with r = 5 , h = 10 Cylinder, r = 3 , h = 2 Sphere with radius r = 2 . Sphere with radius r = 43 . B 34. The angles which are opposite the equal sides also have equal measures (and all three angles add to 180° ). example: Find the measures of C B ∠A and ∠C , 7 given ∠B = 65° : 7 ∠A + ∠B + ∠C = 180 , and A ∠A = ∠B = 65 , so ∠C = 50° V = 1400 • 227 = 4400 cu. units If π is approximated by 3.14, V = 1400(3.14) = 4396 cu. units A sphere (ball) with radius r has volume V = 43 πr 3 32. Sides 8, 8, 8 31. Sides 7, 4, 7 53 The pairs of 36 corresponding A B F sides are AB and FE BC and ED , and AC and FD . D 53 36 E 19 43. Name two similar triangles and list the pairs of corresponding sides. If two triangles are similar, any two corresponding sides have the same ratio (fraction value): example: the ratio a to x, or is the same as b and cz . Thus, y a x a , x a = cz , x b = cz . y = by , and Each of these equations is called a proportion Problems 44-45: Write proportions for the two similar triangles: 44. 32 + x 2 = 5 2 9 + x 2 = 25 x 2 = 25 − 9 = 16 x = 16 = 4 45. Problems 51-54: Each line of the chart lists two sides of a right triangle. Find the length of the third side: leg leg hypotenuse 51. 15 17 52. 8 10 53. 5 12 54. 2 3 example: Find x: Write and solve a proportion: 2 5 the sum of the squares of the legs equals the square of the hypotenuse. (The legs are the two shorter sides; the hypotenuse is the longest side.) If the legs have lengths a and b, and the hypotenuse length is c, then a 2 + b 2 = c 2 (In words, ‘In a right triangle, leg squared plus leg squared equals hypotenuse squared.’) example: A right triangle has hypotenuse 5 and one leg 3. Find the other leg. Since leg 2 + leg 2 = hyp 2 , = x3 , so 2 x = 15 , x = 7 12 Problems 55-56: Find x: 55. 56. Problems 46-49: Find x: 46. 48. 47. 49. If the sum of the squares of two sides of a triangle is the same as the square of the third side, the triangle is a right triangle. example: Is a triangle with sides 20, 29, 21 a right triangle? 20 2 + 212 = 29 2 , so it is a right triangle. 50. Find x and y: Problems 57-59: Is a triangle right, if it has sides: 57. 17, 8, 15 58. 4, 5, 6 H. Pythagorean theorem: In any triangle with a 90° (right) angle, Answers: 1. 127° 2. 53° 3. 53° 4. 127° 5. 36° 6. 7. 8. 9. 10. 11. 12. 13. P 30 un 11 un 12 mi 3 yd 120 un 48 un 30 un 12 un A 50 un2 6 un 2 9 mi2 9 yd2 16 360 un2 72 un2 30 un2 6 un2 59. 60, 61, 11 14. 15. 16. 17. 18. 19. 20. C 10 π un 20 π ft 4 π km 432 10 3 1000 .125 A 25 π un2 100 π ft2 2 4 π km 20 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 250π 6π 32 π 9π 3 16 90° 29° 73° 16° 68° no yes yes yes yes can’t tell 36. 37. 38. 39. 40. 75° each 120°, 30° 60° no no 41. yes: 42. 44°, 6 43. ∆ABE , ∆ACD AB, AC AE , AD BE , CD 45. 3 9 d c 46. 14 44. = 155 = 124 = a a +b = f f +e 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 9 4 14 5 45 2 40 , 16 7 3 8 6 13 5 9 41 yes no yes 5 TOPIC 9: WORD PROBLEMS A. Arithmetic, percent, and average: 1. What is the number, which when multiplied by 32, gives 32 • 46 ? 2. If you square a certain number, you get 9 2 . What is the number? 3. What is the power or 36 that gives 36 2 ? 4. Find 3% of 36. 5. 55 is what percent of 88? 6. What percent of 55 is 88? 7. 45 is 80% of what number? 8. What is 8.3% of $7000? 9. If you get 36 on a 40-question test, what percent is this? 10. The 3200 people who vote in an election are 40% of the people registered to vote. How many are registered? Problems 11-13: Your wage is increased by 20%, then the new amount is cut by 20% (of the new amount): 11. Will this result in a wage which is higher than, lower than, or the same as the original wage? 12. What percent of the original wage is this final wage? 13. If the above steps were reversed (20% cut followed by 20% increase), the final wage would be what percent of the original wage? Problems 14-16: If A is increased by 25%, it equals B: 14. Which is larger, B or the original A? 15. B is what percent of A? 16. A is what percent of B? 17. What is the average of 87, 36, 48, 59, and 95? 18. If two test scores are 85 and 60, what minimum score on the next test would be needed for an overall average of 80? 19. The average height of 49 people is 68 inches. What is the new average height if a 78-inch person joins the group? B. Algebraic substitution and evaluation: Problems 20-24: A certain TV uses 75 watts of power, and operates on 120 volts: 20. Find how many amps of current it uses, from the relationship: volts times amps equals watts. 21. 1000 watts = 1 kilowatt (kw). How many kilowatts does the TV use? 22. Kw times hours = kilowatt-hours (kwh). If the TV is on for six hours a day, how many kwh of electricity are used? 23. If the set is on for six hours every day of a 30-day month, how many kwh are used for the month? 24. If the electric company charges 8¢ per kwh, what amount of the month’s bill is for TV power? Problems 25-33: A plane has a certain speed in still air, where it goes 1350 miles in three hours: 25. What is its (still air) speed? 26. How far does the plane go in 5 hours? 27. How far does it go in x hours? 28. How long does it take to fly 2000 miles? 29. How long does it take to fly y miles? 30. If the plane flies against a 50 mph headwind, what is its ground speed? 21 31. If the plane flies against a headwind of z mph, what is its ground speed? 32. If it has fuel for 7.5 hours of flying time, how far can it go against the headwind of 50 mph? 33. If the plane has fuel for t hours of flying time, how far can it go against the headwind of z mph? Problems 53-55: A gathering has twice as many women as men. W is the number of women and M is the number of men: C. Ratio and proportion: 56. $12,000 is divided into equal shares. Babs gets four shares, and Ben gets the one remaining share. What is the value of one share? Problems 34-35: x is to y as 3 is to 5: 34. Find y when x is 7. 35. Find x when y is 7. Problems 36-37: s is proportional to P, and P = 56 when s = 14 : 36. Find s when P = 144 . 37. Find P when s = 144 . Problems 38-39: Given 3 x = 4 y : 38. Write the ratio x : y as the ratio of two integers 39. If x = 3 , find y. Problems 40-41: x and y are numbers, and two x’s equal three y’s: 40. Which of x or y must be larger? 41. What is the ratio of x to y? Problems 42-44: Half of x is the same as one-third of y: 42. Which of x and y is the larger? 43. Write the ratio x : y as the ratio of two integers. 44. How many x’s equal 30 y’s? 53. Which is correct: 2M = W or M = 2W ? 54. If there are 12 women, how many men are there? 55. If the total number of men and women present is 54, how many of each are there? E. Problems leading to two linear equations: 57. Two science fiction coins have values x and y. Three x’s and five y’s have of 75¢, and one x and two y’s have a value of 27¢. What is the value of each? 58. In mixing x gm of 3% and y gm of 8% solutions to get 10 gm of 5% solution, these equations are used: .03x + .08y = .05(10), and x + y = 10 How many gm of 3% solution are needed? F. Geometry: 59. Point x is on each of two given intersecting lines. How many such points x are there? 60. On the number line, points P and Q are two units apart. Q has coordinate x. What are the possible coordinates of P? Problems 61-62: B C D. Problems leading to one linear equation: 45. 36 is three-fourths of what number? 46. What number is 3 4 of 36? 47. What fraction of 36 is 15? 48. 2 3 of 1 6 of 3 4 of a number is 12. What is the number? 49. Half the square of a number is 18. What is the number? 50. 81 is the square of twice what number? 51. Given a positive number x. Two times a positive number y is at least four times x. How small can y be? 52. Twice the square root of half a number is 2x. What is the number? A O 61. If the length of chord AB is x and the length of CB is 16, what is AC? 62. If AC = y and CB = z , how long is AB (in terms of y and z)? Problems 63-64: The base of a rectangle is three times the height: 63. Find the height if the base is 20. 64. Find the perimeter and area. 65. In order to construct a square with an area which is 100 times the area of a given square, how long a side should be used? 22 2 cm, it is a square. If the original area was 40cm 2 , what is the square area? 70. This square is cut into two a b smaller squares and two non-square a rectangles as shown. Before being cut, the b large square had area Problems 66-67: The length of a rectangle is increased by 25% and its width is decreased by 40%. 66. Its new area is what percent of its old area? 67. By what percent has the old area increased or decreased? 68. The length of a rectangle is twice the width. If both dimensions are increased by 2 cm, the resulting rectangle has 84cm 2 more area. What was the original width? 69. After a rectangular piece of knitted fabric shrinks in length 1 cm and stretches in width Answers: 1. 46 2. 9 3. 2 4. 1.08 5. 62.5% 6. 160% 7. 56.25 8. $581 9. 90% 10. 8000 11. lower 12. 96% 13. same (96%) 14. B 15. 125% 16. 80% 17. 65 18. 95 19. 68.2 20. .625 amps 21. .075 kw 22. .45 kwh 23. 13.5 kwh 24. $1.08 25. 450 mph 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. (a + b) . The two smaller 2 squares have areas a 2 and b 2 . Find the total area of the two non-square rectangles. Show that the areas of the 4 parts add up to the area of the original square. 2250 mi 450x mi 40 hr 9 y 450 hr 400 mph 450 − z mph 3000 mi (450 − z )t mi 35 21 3 5 36 576 4:3 9 4 x 3:2 y 3:2 45 48 27 5 12 144 6 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 9 2 2x 2x2 2M = W 6 18 men, 36 women $1500 x :15 ¢, y : 6 ¢ 6 gm 1 x − 2, x + 2 x −16 y+z 20 3 P = 160 3 , A = 400 3 10 times the original side 75% 25% decrease 5cm 49 2ab 2 a 2 + 2ab + b 2 = (a + b)