Math Review Packet for Pre-Algebra to Algebra 1 Epressions, Equations, Eponents, Scientific Notation, Linear Functions, Proportions, Pythagorean Theorem 2016 Math in the Middle
Evaluating Algebraic Epressions 1. Substitute the given values for the variables in the epression 2. Evaluate the epression using the order of operations Parentheses/Brackets (inside to outside) Eponents Multiplication/Division (left to right) Addition/Subtraction (left to right) e: 9 2 4(y + z) for = -, y = 2, z = 5 9(-) 2 4(2 + 5) 9(-) 2 4(2 + 15) 9(-) 2 4 17 9 9 4 17 81 4 17 81 68 = 1 The Distributive Property 1. Multiply the number outside the parentheses by each term in the parentheses. 2. Keep the addition/subtraction sign between each term. e: 5(8 ) 5(8 ) 5(8) 5() 40 15 Simplifying Algebraic Epressions 1. Clear any parentheses using the Distributive Property 2. Add or subtract like terms (use the sign in front of each term to determine whether to add or subtract) e: 2( 4) 12 + 9 2( 4) 12 + 9 6 8 12 + 9-6 + 1
Evaluate each epression for a = 9, b = -, c = -2, d = 7. Show your work. 1. a - cd 2. 2b + c 2. a + d c b 4. (a b) 2 + d(a + c) 5. 4c (b a) 6. a b - 5a 7. 2bc + d(12 5) 8. b + 0.5[8 (2c + a)] Simplify each epression using the Distributive Property. 9. 5(2g 8) 10. 7(y + ) 11. -(4w ) 12. (6r + )2 Simplify each epression, showing all work. 1. 8( + 1) 12 14. 6w 7 + 12w z 15. 9n 8 + (2n 11) 16. (7 + 4y) 2(2 + y) 17. (15 + 8d)(-5) 24d + d 18. 9(b 1) c + b + c 19. 20f 4(5f + 4) + 16 20. 8(h 4) h (h + 7)
Solving One-Step Equations 1. Cancel out the number on the same side of the equal sign as the variable using inverse operations (addition/subtraction; multiplication/division) 2. Be sure to do the same thing to both sides of the equation! e: -18 = 6j -18 = 6j 6 6 - = j j = - Solving Two-Step Equations 1. Undo operations one at a time with inverse operations, using the order of operations in reverse (i.e. undo addition/subtraction before multiplication/division) 2. Be sure to always do the same thing to both sides of the equation! e: a 7 12 = 9 a 7 7 X 12 = 9 + 12 + 12 a 7 = X 7 a = 21 Solving Multi-Step Equations 1. Clear any parentheses using the Distributive Property 2. Combine like terms on each side of the equal sign. Get the variable terms on the same side of the equation by adding/subtracting a variable term to/from both sides of the equation to cancel it out on one side 4. The equation is now a two-step equation, so finish solving it as described above e: 5(2 1) = + 4-1 10-5 = + 4-1 10-5 = 7-1 - 7-7 - 5 = - 1 + 5 + 5 = 4 = 4
Solve each equation, showing all work. 21. f 64 = -2 22. -7 = 2d b 12 = 6 24. 1 = m + 21 2. 25. 5 = -28 w + 8 = -9-8 + h 4 = 1 28. 22 = 6y + 7 26. 27. 29. 8 4 = + 1 0. -2(5d 8) = 20 1. 7r + 21 = 49r 2. -9g = -(g + 2). 5( 2) = 5(4 + 1) 4. d 4 + d = 8d (-12) 5. f 6 = -2f + (f 2) 6. -2(y 1) = 4y (y + 2)
Scientific Notation Standard Form to Scientific Notation: move the decimal after the first non-zero digit and eliminate any trailing zeros. Multiply by 10 to the power equal to the number of places you moved the decimal point. If the original number was greater than 1, the eponent is positive. If the number was less than 1, the eponent is negative. Scientific Notation to Standard Form: move the decimal point the number of places indicated by the eponent. If the eponent is positive, move the decimal right. If negative, move left. e: 0.0000571 0.0000571 Original number < 1, so negative eponent = 5.71 10-5 e:.5 10 Positive eponent, so move decimal right.500 =,500 Negative Eponents & Simplifying Monomials Zero Eponent: power equals 1 Any number raised to the zero Negative Eponent: Move the base to the opposite side of the fraction line and make the eponent positive Monomial Monomial: Multiply the coefficients and add the eponents of like bases Monomial Monomial: Divide the coefficients and subtract the eponents of like bases Power of a Monomial: Raise each base (including the coefficient) to that power. If a base already has an eponent, multiply the two eponents Power of a Quotient: Raise each base (including the coefficient) to that power. If a base already has an eponent, multiply the two eponents e: y 0 = 1 e: -4 = 1 4 e: (4 )(2 5 ) = 8 8 e: a a 6 = a 5 = 1 a 5 e: (-2fg 5 ) = -8f g 15 e: 5d c 2 = 25d6 c 2
Convert each number to Scientific Notation. 7. 67,000,000,000 8. 0.000921 9. 0.00000000004 40.,201,000,000,000,000 Convert each number to Standard Form. 41. 5.92 10-5 42. 1.1 10 7 4. 6.7 10-8 44..27 10 2 Simplify each epression. Write your answers using only positive eponents. 45. w -9 46. m 5 47. f5 f 48. m 2 h 2 g 49. (a 5 ) 2 50. 1 b 51. z 0 52. 4r 6 r 2r 2 5. 9p 2 q 54. 8d 2cd 2 55. (g 4 h) 2 (2g h -1 ) 2 56. (6a) 0 57. (-n 2 k 4 ) 2 58. w 5 2 59. y w 2 y 4 6 10 7 2 10 60. (1.5 10-6 ) (4 10 9 )
Slope & Rate of Change Finding the Slope Given Two Points: Use the coordinates from the points in the slope formula: Slope (m) = y 2 y 1 2 1 Finding the Rate of Change From a Table: Determine the amount the dependent variable (y) is changing and the amount the independent variable () is changing. Rate of Change = change in y change in Finding the Slope From a Graph: Choose 2 points on the graph. Find the vertical change (rise) and horizontal change (run) between the 2 points and write it as a fraction rise run. (Up is positive, down is negative, right is positive, and left is negative). y e: (4, -2), (-, 8) m = 8 ( 2) 4 = 10 7 = 10 7 e: m 1 y 1 2 y 2 # months 5 7 9 Cost ($) 80 10 180 20 = 50 2 +2 +2 +2 +50 +50 +50 = 25 dollars/month rise = +1 run = -2 m = 1 2 = 1 2 Graphing Linear Equations Slope-Intercept Form: y = m + b e: y = 2-4 slope y-intercept y-intercept: -4 How To Graph: 1. Make a point on the y-ais at the y-intercept. 2. Use the slope to determine where to make the net point. The numerator tells you the rise (how far up/down) and the denominator tells you the run (how far right/left) to make the net point.. Repeat to make more points and then connect the points with a line. slope: 2 = 2 1 rise run
Find the slope of the line that passes through the points. Show your work. 61. (-5, ), (2, 1) 62. (8, 4), (11, 6) 6. (9, ), (9, -1) 64. (-4, -2), (-6, 4) Find the rate of change. Show your work. 65. 66. Number of Hours 6 9 12 Distance (in miles) 15 270 405 540 Number of Weeks 1 5 7 Pounds 17 169 165 161 Find the slope of the line. 67. 68. 69. Graph the line. 70. y = - 71. y = 1 + 2 72. y = - 1 7. y = - 2 2 74. y = 2 + 1 75. y = 1 4
Solving Proportions 1. Set the two cross-products equal to each other 2. Solve the equation for the variable e: m 4 = 5 5m = 12 5 5 m = 2.4 Similar Figures 1. To find a missing side length, set up a proportion, matching up corresponding sides. 2. Solve the proportion using the steps above. e: 9 mm 1.5 = 9 5.5 5.5 mm = 2.45 mm 1.5 mm The Pythagorean Theorem *** The Pythagorean Theorem applies to right triangles only ** The sides net to the right angle (a & b) are legs e: 12 cm The side across from the right angle (c) is the hypotenuse b = leg a = leg Pythagorean Theorem: a 2 + b 2 = c 2 To find the hypotenuse: add the squares of the legs and then find the square root of the sum To find a leg: subtract the square of the given leg from the square of the hypotenuse and then find the square root of the difference 15 cm is the hypotenuse 12 2 + 15 2 = 2 144 + 225 = 2 69 = 2 = 69 19.2 cm e: a =?, b =, c = 6 a is a leg a 2 + 2 = 6 2 a 2 + 9 = 6 a 2 = 6 9 = 27 a = 27 5.2
Solve each proportion, showing all work. 76. 6 7 = 4 77. 12 m 5 = k 78. h 7 = 8 79. 22 2 n = 9 80. 6 4 21 = c Assume each pair of figures is similar. Find the missing side length, showing all work. 81. 7 cm 82. 1.5 in 8. ft 2 cm 1 in n 14 cm 6 in 8 ft r 2 ft 84. 5 mm 85. 86. 20 mm y 18 in 12 in 9 mm 18 in j 2 mm 24 in f 10 cm 5 cm 8 cm Find the missing side length in each right triangle to the nearest tenth. Show your work! 87. a = 6, b = 8, c =? 88. a =?, b = 9 cm, c = 1 cm 89. a = 7, b =?, c = 14 90. a = 14 in, b = 14 in, c =? 91. 92. 9. 7 in 94. 5 10 mm 5 in 10 mm 20 18 95. 15 96. 104 in 97. 5 ft 98. 1 10 ft 52 in 20 cm 24 cm Determine whether or not you can form a right triangle from the given side lengths. Eplain. 99. 18, 22, 26 100. 5, 12, 1
Evaluate each epression for a = 9, b = -, c = -2, d = 7. Show your work. 1. a - cd 2. 2b + c 2. a + d c b 2-50 -6 19 4. (a b) 2 + d(a + c) 5. 4c (b a) 6. a b - 5a 7. 2bc + d(12 5) 8. b + 0.5[8 (2c + a)] 4-48 61-1.5 Simplify each epression using the Distributive Property. 9. 5(2g 8) 10. 7(y + ) 11. -(4w ) 12. (6r + )2 10g - 40 7y + 21-12w + 9 12r + 6 Simplify each epression, showing all work. 1. 8( + 1) 12 14. 6w 7 + 12w z 15. 9n 8 + (2n 11) 16. (7 + 4y) 2(2 + y) -4 + 8 18w - z - 7 15n - 41 17 + 10y 17. (15 + 8d)(-5) 24d + d 18. 9(b 1) c + b + c 19. 20f 4(5f + 4) + 16 20. 8(h 4) h (h + 7) -6d - 75 12b - 9 0 6h - 9
Solve each equation, showing all work. 21. f 64 = -2 22. -7 = 2d 2. b 12 = 6 24. 1 = m + 21 f = 41 d = - 7 / 2 = -.5 b = 72 m = -8 25. 5 = -28 26. w + 8 = -9 27. -8 + h 4 = 1 28. 22 = 6y + 7 = -5 w = 19 h = 84 y = 5 / 2 = 2.5 29. 8 4 = + 1 0. -2(5d 8) = 20 1. 7r + 21 = 49r 2. -9g = -(g + 2) = 1 = - 2 / 5 = -0.4 r = 1 / 2 = 0.5 no solution. 5( 2) = 5(4 + 1) 4. d 4 + d = 8d (-12) 5. f 6 = -2f + (f 2) 6. -2(y 1) = 4y (y + 2) = - d = -4 all real numbers y = 4 / 5 = 0.8
Convert each number to Scientific Notation. 7. 67,000,000,000 8. 0.000921 9. 0.00000000004 40.,201,000,000,000,000 6.7 10 10 9.21 10-4 4 10-11.201 10 15 Convert each number to Standard Form. 41. 5.92 10-5 42. 1.1 10 7 4. 6.7 10-8 44..27 10 2 0.0000592 11,000,000 0.000000067 27 Simplify each epression. Write your answers using only positive eponents. 45. w -9 46. m 5 47. f5 f 48. m 2 1 w 9 m f 8 h 6 h 2 g g 49. (a 5 ) 2 50. 1 b 51. z 0 52. 4r 6 r 2r 2 a 10 b 1 24r 9 5. 9p 2 q q 54. 8d 2cd 2 p 2 4d 5 c 55. (g 4 h) 2 (2g h -1 ) 2 56. (6a) 0 4g 14 1 57. (-n 2 k 4 ) 2 58. w 5 2 y 59. w 2 y 4 9n 4 k 8 w 9 6 10 7 2 10 60. (1.5 10-6 ) (4 10 9 ) 9 y 9 10 4 6 10
Find the slope of the line that passes through the points. Show your work. 61. (-5, ), (2, 1) 62. (8, 4), (11, 6) 6. (9, ), (9, -1) 64. (-4, -2), (-6, 4) m = - 2 7 m = 2 m = undefined m = - Find the rate of change. Show your work. 65. 66. Number of Hours 6 9 12 Distance (in miles) 15 270 405 540 Number of Weeks 1 5 7 Pounds 17 169 165 161 45 miles per hour -2 pounds per week Find the slope of the line. 67. 68. 69. m = - m = 0 m = 2 Graph the line. 70. y = - 71. y = 1 + 2 72. y = - 1 7. y = - 2 2 74. y = 2 + 1 75. y = 1 4
Solve each proportion, showing all work. 76. 6 7 = 4 77. 12 m 5 = k 78. h 7 = 8 79. 22 2 n = 9 80. 6 4 21 = c m = 4.6 k = 7.2 h = 28 h = 88 h = 15.75 Assume each pair of figures is similar. Find the missing side length to the nearest tenth. 81. 7 cm 82. 1.5 in 8. ft 2 cm 1 in n 14 cm 6 in 8 ft = 4 cm = 4 in = 5. ft r 2 ft 84. 5 mm 85. 86. 20 mm y 10 cm 18 in 12 in 9 mm 18 in f j 2 mm 24 in j 18. mm y = 1.5 in f = 6.25 cm 5 cm 8 cm Find the missing side length in each right triangle to the nearest tenth. Show your work! 87. a = 6, b = 8, c =? 88. a =?, b = 9 cm, c = 1 cm 89. a = 7, b =?, c = 14 90. a = 14 in, b = 14 in, c =? c = 10 units a 9.4 cm b 12.1 units c 19.8 in 91. 92. 9. 7 in 94. 5 10 mm 5 in 10 mm 20 18 = 4 units 14.1 mm 8.6 in 8.7 units 95. 15 96. 104 in 97. 5 ft 98. 1 10 ft 52 in 19.8 units 116. in.5 ft 24 cm 20 cm 1. cm Determine whether or not you can form a right triangle from the given side lengths. Eplain. 99. 18, 22, 26 100. 5, 12, 1 No; 18 2 + 22 2 26 2 Yes; 5 2 + 12 2 = 1 2