1-A2 Identity and Equality Properties Alg 1H

Transcription

1 -A Identity and Equality Properties Alg H Identify the IDENTITY OR EQUALITY PROPERTY that is shown in each statement. You may write on this handout. Put the code on the answer line = 8. If = +, then + =. 0 = x = (-)x. If = 6 +, then 6 + = = ( ) = = 8 9. a + b + 0 = a + b 0. x x. If + a = a + b, then a + b = + a. If 0 = and = 7 7, then 0 = () = 6 4. If = and =, then =. x y = ()x + (-)y 6. t + r + 0 = t + r 7. r = r = (x y + 6) = x y (x + y + z) = 0 8x. 8x. If x = 0 and y =, then xy = 0. If 7 = 4(x ) then 4(x ) = 7 4. x + = x +. If x = 0 then 0 = x 6. If x =, and x + = y, then + = y 7. If x = y, then y = x 8. If x = a + b and a + b = y, then x = y Choose from the following codes: Property Codes A Id Additive Identity M Id Multiplicative Identity MPZ Multiplicative Property of Zero PNO Property of Negative One A Inv Additive Inverse Property M Inv Multiplicative Inverse Prop. Rflx Reflexive Property Sym Symmetric Property Sub. Substitution Property T Transitive Property Explain whether or not each set is closed under the given operation. Use examples. 9. Is the set of odd integers closed under addition? 0. Is the set of positive integers closed under multiplication?. Is the set of even integers closed under multiplication?. Is the set of odd integers closed under multiplication?

2 -A cont. Identify the COMMUTATIVE, ASSOCIATIVE OR DISTRIBUTIVE PROPERTY shown = = 7 6. ( + 4) + = + (4 + ) 6. (7 ) = 7( ) 7. ( + 7) = (9 + 4) = (4 + 9) 9. a(b + c) = (b + c)a 40. ( ) = 6 4. ( )4 = 4( - ) 4. (x + y) + c = c + (x + y) 4. ( + a) + b + c = ( + a + b) + c 44. a + b + c = c + b + a 4. a b c = c a b 46. ( + 4) + = + ( + 4) 47. x + (y + z) = (x + y) + z 48. (x ) + 4(x ) = 4(x ) + (x ) 49. (a + b) (a + b) = a + b a b 0. (a + b) = (a + b). a + b + x + y = (a + b) + (x + y). a(b + ) + (c + ) = (c + ) + a(b + ). a(b + c)(c + ) = a(c + )(b + c) 4. (a + )( b) + = ( b)(a + ) +. 0(a + b) + = + 0(a + b) = (8 ) 7. (a + )(b + )x = x(a + )(b + ) 8. ( + c+ d) = + c + d 9. a ( b + )(c) = a( b + )(ac) 60. (x + y) + (r + s) = x + (y + r) + s Choose from the following codes: Property Codes CA Commutative Property of Add. AA Associative Property of Add. CM Commutative Property of Mult. AM Associative Property of Mult. D Distributive Property Explain whether or not each set is closed under the given operation. Use examples. 6. Is the set of even natural numbers closed under addition? 6. Is the set of odd integers closed under division? 6. Is the set of positive integers closed under subtraction? 64. Is the set of negative integers closed under addition?

3 -A Distributive Property and Combining Like Terms DIRECTIONS: Simplify each expression by using the distributive property and then combining like terms. Follow the format given in the example below. Example Problem: - (x - ) - 4 x 4x x 6 4x 6x or 6x step write out the problem as shown optional step change all subtraction to addition of a negative step perform distributive property step 4 combine the "like" terms step circle or box your simplified expression x x x x x x or 6x x Do in Spiral or on Binder Paper.) 7 + (y + 4) + 6y 9.) x - (x - ) + 8x +.) - 4(x - ) + x 0.) y(y - 4) + (y + ) + y.) x(x - ) + (x + 4).) u(x - y) + u(x - z) + uy 4.) (x - y) - (x - y).) 7 - x( + z) + 7x - + 8xz.) u - (u - 4) - u.) y - 4(x - y) + (x - 6) - 6.) 8 - (z - ) + 6z 4.) z(z - 4) - (z - ) + - z 7.) + x(z - ) + xz.) u(u - 4) - 4(u - 8) - 6u ) - (y - ) - (y + ) 6.) 4x(y + ) - (x - y) + 6x - 8xy

4 -A Review Lessons 6 Chapter Alg. H Do on binder paper. Show Support work! Begin each problem by copying it down. Simplify:.) ( 7) ( 4).) + (-) (-).) - 4 [- (-7)] 4) ) -[ 4(-6) +(-)] - (-6 + ) 6) -[ 4(-6 -) (-6 + 7)] 7) (-) + 4[(-4) + ( + (-))] 8) {[(-) + (-)] (-4) + 7} 9) [6(-) + (- + )9] + [7(-4) 4] Evaluate each expression for a = -, b =, and c = -. Show support work with parentheses for all substitution. 0.) a b c.) c - a b.) b - a - c.) c - b - a Translate each word phrase into an expression, then simplify it. 4.) The difference of 7 and, decreased by..) Five times the sum of a and b, decreased by three times b. 6.) Twelve less than four times the sum of x and two. If the problem is a word problem, you do not have to copy down the problem onto your own paper. 7.) Six more than the sum of negative eleven and y, increased by one half the difference of 4y and two. Supply the answer for each blank. Write your answers on your binder paper. You do not have to copy down the problems. 8.) The multiplicative property of zero states that any number times zero equals. 9.) The product of negative one and a number will give you the of the number. 0.) The product of a positive number and a negative number is..) The product of two negative numbers is..) The product of a number and it's multiplicative inverse is always..) In the expression x, the coefficient of x is. 4.) Number above is an example of what property?..) The product of a real number and its reciprocal is.

5 6.) The quotient of two negative numbers is. 7.) The reciprocal of a real number is sometimes called its inverse. 8.) The opposite of x + 7 is. 9.) a b is the same as. Choose your answer from these choices: b a, a b, a b 0.) Explain the 4 rules of order of operations.. Simplify:.) x 6 x +.) (a + b) 4(a b).) 4.) (a b).) ( 7a 4b) 7 6.) 4 4x 6) 7.) t + t 7 8.) -(x ) ( 4x) 9.) 4(x ) (x + ) 40.) (-a)(-a)(b) 4.) (-)(-x + ) ) -(- y) (-y + ) 4.) n (n + ) + n 44.) x + (x ) Write the reciprocal of each number is simplest form. 4.) 46.) 47.) 0.4 Evaluate each expression if a = -, b = 4, and c = -. Show support work with parentheses for all substitution. 48.) b( a c) a 49.) ( a c) 0.) ( a b c) b

6 -A4 Review Handout # for Lessons 6 Chapter Alg. H Do on binder paper. Show support work. Simplify:.).) 4 4.) ) 6 4.) ) 7.) 8 Evaluate each expression if a = -, b = 4, and c = ) 0.) x 9.) ab bc a a b c What value for x would result in an impossible division? How is that answer written? Name the property shown:.) h + (i + j) = h + (j + i).) a + (0)(b + 4) = a.) xy + z yx = z 4.) (x + ) = + ( )(x + ) Write an algebraic expression for the following problems:.) Twice the sum of b and 8, decreased by 9. 6.) The product of j and k decreased by twice the difference of x and y. 7.) The quotient of and y, minus four times their sum. 8.) 0 less than the product of and w. 9.) The sum of a number and, divided by. Simplify each expression. Show all steps. 0.) y x (-y).) -x (- + x).) -a (a + b).) -x (- + y) 4.) -(-y).) -b 6b 6.) 0 x 7.) -(-x 9) 8.) 0 [(w ) w] 9.) 9[7x + (4 + x)] 6x 8 0.) y + [6y + (y + )].) [(m + 4n) n] m.) 0(q + p) + 7[q + 4(q + 9p + )].) 4(d + e) (d e) 4.) (t + s) + 7(t s).) [9(v + 4w + ) 4(v + 7)] (8v + w) 6.) [6p 9q] F HG I K J 7.) (y 7) F HG I K J 8.) 7 (c + 4d) 9.) (6a 4b) 8 40.) (4u + 8v) 9

7 -A Writing Equations for Word Problems Review Handout Alg H Write let statements for each. The second unknown must use the same variable as the first.. Reina is 8cm taller than Patricia. a. Let r = Reina s height in centimeters. Then let = Patricia s height. or b. Let p = Patricia s height in centimeters. Then let = Reina s height.. Devin ran times as many laps as Claire did. a. Let d = the numb. of laps Devin ran. Then let = the numb. of laps Claire ran. or b. Let c = the numb. of laps Claire ran. Then let = the numb. of laps Devin ran.. Emily skied a slalom course.67 seconds slower than Lauren. a. Let e seconds = Emily s time. Then let seconds = Lauren s time. or b. Let L seconds = Lauren s time. Then let seconds = Emily s time. Write the missing let statement. 4. Let c = Catherine s age. Rachael is years older than Catherine.. Let g = the amount that Greg earned last week. Greg earned $4 more this week than last week. 6. Let p be the number of points the Panthers scored. The Lions scored double the number of points the Panthers scored. 7. Let a be the total amount Amanda spent for blouses. Each of the blouses cost the same. 8. Let b be the amount of money Bryan had before he went shopping. Bryan spent $4.0 while shopping.

8 -A cont. For each of the following, write let statements, verbal sentneces and equations. Do not solve. 9. When 0 is subtracted from the quotient of a number and 4, the difference is. What is the number? 0. A carpenter cut a board that was 0 feet long into two pieces. The longer piece was three times the length of the shorter piece. What is the length of each piece?. Joseph is times as old as Lisa. Their combined age is. How old is each person?. When is added to a certain number and the sum is divided by 4, the quotient is. What is the number?. On a street that is 7 m long, a fire hydrant is placed so that it is m farther from one end of the street than from the other end. How far is the hydrant from each end of the street? 4. A hockey team played 86 games and finished only games in a tie. It lost one-sixth as many games as it won. How many games did it win?. One of the longer sides of Ann Wong s rectangular lot borders on a straight river. She needs 00 m of fence to fence the three sides not on the river. If the length of the lot is 6 m more than the width, what are the dimensions (length and width) of the lot? 6. The sum of three numbers is 0. The second of the numbers is eight less than the first, and the third number is four more than the first. Find the third number.

9 -A6 Handout Write an algebraic expression for each verbal expression.. Ten less than the product of 4 and a number cubed.. Three-fourths the difference of a number and. Evaluate each expression. c h c h a f L NM a f O. 4 7 QP a f 7 Evaluate each expression if x =, y = 8, and z =. 6. xy z z 7. F HG x y I KJ z ax yf Evaluate each expression for the given values ab for a =, b = 9. ab 4 b ab = b for a =, 4 = x. x + x x for 4 = b, = a 0. ab + 4 for. Name the property shown in the example at right. Simplify.. x (x + 7) 0 0c) (6a + c) (a (b ) 8(b + ) a f Give the definition for a solution to an equation. (hint: See your notes, not your book.) 7. Prove or disprove the following statement (show all work steps): is a solution to the equation 7 ax f 4 8

10 -A0 Alg H DO ON BINDER PAPER It is required that you show all work steps! CLEARING EQUATIONS OF FRACTIONS & PRACTICE WITH COMPLICATED EQUATIONS.) a a.) x 6 x.) w w ) (0 4y) = 6 y 4.) 6x ( x) = 4(x ) 4.).) x x 4 7 x x 0 6.) (x + ) x = (x + ) 7.) 8.) x 4 x 7x 7 x 4 9.) (y + ) y = (y ) 0.) x x 0 4.) (m + ) 6 = (m + ).) x 4 x y.) y ) y y ) 4(n + ) = 4 ( n) x x 8.) x 4 9.) 0.).) 6 x 4x 4y y x 4 x x 7 7.) Write an equation that represents the problem, then solve it to find the number. One fourth of a number is 8 more than one sixth of the number. Find the number.

11 -A Algebra H Lesson -8 Name Solving for a Specific Variable Solve the equation for the indicated variable. (Remember that this means to isolate the indicated variable on one side of the equal sign using inverse operations.) Show all your work steps as you do inverse operations. You may write on this handout, circle final answers..) rt = d ; for t.) x y = y ; for x.) x + y = z ; for y 4.) 4x + 7 = y ; for x.) (x ) = y ; for x 6.) x = (y + 6) ; for y 7.) bh = A ; for h 8.) x y = 4 ; for y 9.) x = y + 4 ; for y 0.) a(x + b) = c ; for x **Don t forget that this assignment also includes the following problems in your HEATH book: HEATH Pg. 7, # 8,, 7-9.

12 -A6 PRACTICE ON LESSONS - TO -8 Alg H DO ON BINDER PAPER, SHOW ALL WORK STEPS. Translate each sentence into an equation:. Three times the sum of a and b is nine.. The product of two and the sum of a and three is equal to five less than the quotient of b and four. Solve the equations:. 6(k + ) = 4(k + 7) 4. d + 6(d + ) = 0. ( x) = (4 x) 6. x + (x ) = x 7. (x ) (x ) = x 8. 6x ( x) = -6 + x 9. x 7 = (x ) + 0. x + (x 4) = (x + ) 6x +. m = m. (x + 4) = 8. (4 6x) = 6 4. (x + ) = (x + 4). (6 y) = 8 6. x x + ( + x) = 0 7. (x + 4) = 8 Solve the proportions: x 6 x 4 x 8. 8 a a + = a a 0. (8 4y) = x 4. 4 x. x x x x y 8 4y 7 6. x x 4 7. x 4 x 8. p p 4p 6. A bottle of perfume costs $4.0. If your local sales tax is 7.% what will be the total cost to buy the perfume?. A CD player normally costs $89.9. It's on sale for a % discount. What is the final cost to you? Don't forget to apply 7% sales tax to your purchase. Solve for the indicated variable: x y 4x. ; for y 4. x + 4y = 0 ; for x. l + w = P ; for l 6. P + Prt = A ; for r 7. A = P( + rt) ; for t 8. ax + bx = c ; for a x 4y 9. ; for y 40. xy + 7 = y ; for x