Math 95--Review Prealgebra--page 1 Name Date In order to do well in algebra, there are many different ideas from prealgebra that you MUST know. Some of the main ideas follow. If these ideas are not just a review for you (if these are brand-new ideas), you should take Math 93 Prealgebra before you take this course. Positive/Negative Rules Add/Subtract Rules= Combining Rules same sign = add, keep the sign different signs = subtract, keep the bigger sign Multiply/Divide Rules two same sign = positive answer two different signs = negative answer With a subtraction symbol, we think of the subtraction symbol as a negative and link it to the next number. However, if we see a subtract a negative, that immediately becomes a positive. Then you still use the combining rules from above. If you multiply/divide with a string of more than two numbers, then If you multiply with an odd number of negatives, the answer is negative. If you multiply with an even number of negatives, the answer is positive. On ALL examples throughout this worksheet, be sure you look at the problem and know how the answer was obtained. Then follow the examples to do the homework. a. 8 + ( 14) = 6 b. 18 30 = 1 c. 7 + ( 30) = 37 d. 3 = 54 e. 1 ( 14) = 1 + 14 = f. 5( 4) = 0 g. 30 3 = 10 h. 40 ( 8) = 5 i. (3)( 4) = 4 j. 5( 3)(1)( 7)( ) = 10 k. 8 + 14 30 ( 7) l. 3 ( 4) = ( 4)( 4)( 4) = 64 8 + 14 30 + 7 m. 4 ( 5) = ( 5)( 5)( 5)( 5) = 65 8 30 + 14 + 7 n. 7 = (7)(7) = (49) = 49 38 + 1 17 1. 8 5. 8( 5) 3. 9 0 4. 9( 0) 5. 9 ( 0) 6. 14+ ( 6) + 8 ( 4) 18 7. ( 6)(3)( 4)(5) 8. 8 ( 4) 9. a. ( 6) 10. ( 7) 3 b. 6
Math 95--Review Prealgebra--page Combine Like Terms: Algebraically, you add together the variable terms (you add the coefficients) and you add together the constant terms. Combine like terms means to add similar items. Actually, combine like terms also includes subtracting since add/subtract use the same rules from above. o. 5y + 8 + 9y + 14 p. m + 6 14m 18 5y + 9y + 8 + 14 m 14m + 6 18 14y + 36m 1 11. 5z + 19 + 3z + 1. 39a 8 4a 3 Distributive property of multiplication over addition. Most of the time, we ll shorten that and say we re using the distributive property. The full name is important, though, because it does indicate that you need to have an addition (or subtraction) symbol on the inside of the parentheses; it also indicates the number on the outside of the parentheses is being multiplied to both separate items inside the parentheses. You also use rules for multiplying positive/negative numbers. See the examples. q. (3y + 9) r. 6(5z + 8) s. 9(a + 3) 6y + 18 30z + 48 9a + 7 t. 8(7m 5) u. 4(x 3) v. 5(5k 1) 56m 40 4x 1 5k 60 w. (5m + 8) x. 9(4y 7) y. ( z + 3) 10m 16 36y + 63 z 3 13. (5x + 3) 14. 3(7m + 1) 15. 4(z + ) 16. 5(k 6) 17. 9(3y 1) 18. 10(5a 6) 19. 8(5x + ) 0. (5m 7) 1. 3( z + 4) Distribute and Combine. Sometimes, we use both distributive and combine like terms. Generally, use the distributive property to multiply to remove the parentheses. Then combine like terms. z. 5(3k + 6) + 9(4k + 7) aa. 4(3x + 7) + 5(x + ) 15k + 30 + 36k + 63 1x + 8 + 5x + 10 15k + 36k + 30 + 63 1x + 5x + 8 + 10 51k + 93 17x + 38 bb. 6(4z + 9) + 3(z 4) cc. 3(5k 11) (4k 5) ( k + 3) 4z + 54 + 6z 1 15k 33 8k + 10 + 1k 3 4z + 6z + 54 1 15k 8k + 1k 33 + 10 3 30z + 4 8k 6
Math 95--Review Prealgebra--page 3. 5(x + 9) + 8(x + 7) 3. 6(y 8) 3(4y 9) 4. 3(6m + 3) (m 6) 9( 4m + 5) Order of Operations. Order of operations is a way for everyone to get the same result on lengthy problems with multiple operations. Order of operations includes parentheses (work inside to outside), exponents, multiply or divide whichever is first from left to right, and finally add or subtract whichever is first from left to right (however, when all you have left are add/subtract symbols, you can combine in any order as long as you follow positive/negative rules). dd. 4 ( 8) 5 + 3(7 8 ) ee. 5 9(8 5) + (3 + 1) 3 5 + 3(7 64) 5 9(3) + (4) 15 + 3( 57) 5 9(9) + 16 15 171 5 81 + 8 186 68 5. 18( 3) + 5(16 0) 6. 48 3(1 8) 7. 4 5 8. 18(0 10) + 8(6 9) 8 Evaluate. When you evaluate an expression, you take out the variable, put parentheses in the place of the variable, and place a number inside the parentheses. Then you use order of operations to simplify. Remember to use any understood 1 coefficients when that is helpful. ff. 3 4m 3n when m = 5 and n = gg. a + 3a + 8 when a = 4 3 4(5) 3( ) 1a + 3a + 8 4(5)(5) 3( )( )( ) 1( 4) + 3( 4) + 8 4(5) 3( 8) 1( 4)( 4) + 3( 4) + 8 100 + 4 1(16) 1 + 8 14 16 1 + 8 0 Homework. Evaluate. 9. 7m 5n for m = 11 and n = 3 30. z 5z when z = 8
Math 95--Review Prealgebra--page 4 Solving Equations. To solve an equation means to totally isolate the variable on one side of the equal sign and to find the solution of the problem. When you are solving, you are performing a series of steps that leads you to simpler and simpler equations until you finally have the variable totally isolated. In general, you do the following to solve: i) simplify first by clearing any fractions; also by using distributive and/or combining like terms ii) collect like terms by moving variable terms to the left of the equal sign (add or subtract to undo what is written) and by moving constant terms to the right of the equal sign (add or subtract to undo what is written) iii) totally isolate the variable, usually by dividing by the exact coefficient of the variable iv) check the solution in the original equation to see if the solution checks See the examples. Make sure you understand each example and the manipulation. Be aware that during class, I ll show most of my work up and down even though on this worksheet I m showing the work in a horizontal fashion and using bold to show the manipulation. hh. 9m + 14 = m + 49 ii. 4(y + 1) 7y + 8 = 5(y + 4) 9m + m = 49 14 8y + 48 7y + 8 = 5y + 0 7m = 35 1y + 56 = 5y + 0 m = 35 7 1y 5y = 0 56 m = 5 4y = 36 y = 36 4 y = 9 jj. Multiply by LCD 4 kk. Multiply by LCD 15 1x + 9 = 0 10x 60 = 3 1x = 0 9 10x = 3 + 60 1x = 11 10x = 63 Homework. Solve. 31. z + 9 = 50 3. k + 30 = 4 33. m 18 = 44 34. 5b = 0 35. 1d = 30 36. 4z + 4 = 8 37. 6a 54 = 96 38. 4z + 4 = 0 39. 8z + 15 = 5z + 63 40. x 11 = 7x 91 41. 8(4m + 16) = 5(6m + 8) 4. 5y + 8 6y + 10 = 10
Math 95--Review Prealgebra--page 5 43. 4f 3 + 5f 1 = 3(f 5) 44. (8a + 6) 3(4a + 16) = 144 45. 46. 47. 48. Answer Key. 1. 13. 40 3. 11 4. 180 5. 9 6. 7. 70 8. 7 9. a. 36 10. 343 11. 8z + 1 1. 3a 60 b. 36 13. 10x + 6 14. 1m + 3 15. 4z + 8 16. 5k 30 17. 7y 108 18. 50a 60 19. 40x 16 0. 5m + 7 1. 6z 1. 1x + 101 3. 1 4. 53m 30 5. 74 6. 99 7. 1 8. 88 9. 3 30. 4 31. z = 1 3. k = 7 33. m = 6 34. b = 4 35. d = 36. z = 1 37. a = 7 38. z = 1 39. z = 16 40. x = 16 41. m = 44 4. y = 8 43. f = 0 44. a = 45 45. Multiply by LCD 0 46. Multiply by LCD 7 5x + 8 = 10 7x 16 = 60 47. Multiply by LCD 6 48. Multiply by LCD 1 x + 30 = 3 15x 7 = 4