Real Numbers



Name________________________________________________ Date_________________________ Algebra I – Pd ____Real Numbers and Properties 1AReal NumbersReal numbers are divided into two types, rational numbers and irrational numbersRational Numbers:Any number that can be expressed as the quotient of two integers. (fraction).Any number with a decimal that repeats or terminates.Subsets of Rational Numbers:Integers: rational numbers that contain no fractions or decimals.{…,-2, -1, 0, 1, 2, …}Whole Numbers: all positive integers and the number 0.{0, 1, 2, 3, … }Natural Numbers (counting numbers): all positive integers (not 0).{1, 2, 3, … }Irrational Numbers:Any number that cannot be expressed as a quotient of two integers (fraction).Any number with a decimal that is non-repeating and non-terminal (doesn’t repeat and doesn’t end).Most common example is π.PropertiesCommutative Properties of Addition and Multiplication:The order in which you add or multiply does not matter.a + b = b + a and a ? b = b ? aSymmetric Property:If a + b = c, then c = a + bIf , thenReflexive Property:a + b = a + bNothing changesAssociative Properties of Addition and Multiplication.The grouping of addition and multiplication does not matter. (Parenthesis)Transitive Property:If a = b and b = c, then a = c.If, and, thenDistributive Property:a (b + c) = ab + ac and a(b – c) = ab – acAdditive Identity:When zero is added to any number or variable, the sum is the number or variable.a + 0 = aMultiplicative Identity:When any number or variable is multiplied by 1, the product is the number or variable.a ? 1 = aMultiplicative Property of Zero:When any number or variable is multiplied by zero, the product is 0.a ? 0 = 0Additive Inverse:The opposite of a number. The sum of a number and its additive inverse is 0.The additive inverse of 5 is negative 5 because 5 + (-5) = 0x + (-x) = 0Multiplicative Inverse:The reciprocal of a number. The product of a number and its multiplicative inverse is 1.The multiplicative inverse of 5 is 15 because 515=1 x1x=1Name___________________________________________ Date_________________________ Algebra I – Pd ____Real Numbers and Properties 1BNumbers WorksheetPart 1 – Use an integer to express the number(s) in each application below:1) Erin discovers that she has spent $53 more than she has in her checking account.________2) The record high Fahrenheit temperature in the United States was 134? on July 10th, 1913. ________3) A football team gained 5 yards _________, then lost 10 yards on the next play ________4) The shore surrounding the Dead Sea is 1348 feet below sea level. ___________Part 2 – Tell whether each statement is true or false. (write the entire word)5) - 2 < 4 ____________6) 6 > - 3 ____________7) - 9 < - 12 ____________8) - 4 ≥ - 1 ____________9) - 6 ≤ 0 ____________10) - 15 > - 5 ____________Part 3 – Write an example of a number that satisfies each given condition.11) An integer between 3.6 and 4.6 ____________12) A rational number between 2.8 and 2.9 ____________13) A whole number that is not positive and is less than 1 ____________14) A whole number that is greater than 3.5 ___________15) A real number that is neither negative nor positive ____________Part 4 – Circle the correct answer to the following questions.______16) Which number is a whole number but not a natural number? a) – 2?????????????? b) 3????????????????? c) ? ??????????????? d) 0______17) Which number is an integer but not a whole number? a) – 5?????????????? b) ? ??????????????? c) 3? ??????????????? d) 2.5______18) Which number is irrational? a) π????????????????? b) 4????????????????? c) .1875 ????????? d) .33Part 5 – Write an example down for the following questions. 19) Give an example of a number that is rational, but not an integer.? ??????????? 20) Give an example of a number that is an integer, but not a whole number.??????? 21) Give an example of a number that is a whole number, but not a natural number.? ????___________ 22) Give an example of a number that is a natural number, but not an integer. Properties Worksheet A. Complete the Matching Column (put the corresponding letter next to the number)1)?????? If 18 = 13 + 5, then 13+5 = 18a) Reflexive2)?????? 6 · (2 · 5) = (6 · 2) · 5??????????????????????????????????????b) Additive Identity???????????? 3)?????? 3(9 + 2) = 3(9) + 3(2)?????????????????????????????????????c) Multiplicative identity4)???????15 + (10 + 3) = (15 + 10) + 3?????????????????????????d) Associative Property of Mult.5)?????? 50 · 1 = 50????????????????????????????????????????????????????e) Transitive6)?????? 7 ? 4 = 4 ? 7???????????????????????????????f) Associative Property of Add.7)?????? 13 + 0 = 13 g) Symmetric8)?????? 11 + 8 = 11 + 8????????????????????????????????? h) Commutative?Property of Mult.9)?????? If 30 + 34 = 64 and 64 = 82, then 30 + 34 = 82????????I) Multiplicative property of zero10)? 11 ? 0 = 0???????????????????????????????j)?Distributive______ 11) Which property is represented by:5+ (4 + 7x) = (5 + 4) + 7x? a) Associative Property of Add.c) Distributive Propertyb) Commutative Property of Add.d) Symmetric Property______ 12) Which property is illustrated by 5(a + 6) = 5(a) + 5(6)a) associative prop. of add.b) distributivec) transitived) symmetric ______ 13) What is the formula for area of a rhombus?a) A = lhb) A = ? h(b1 + b2)c) A = ? d1d2d) A = lwh______ 14) What property is represented by: If 4 + 14 = 18 and 18 = 6 ? 3, then 14 + 4 = 6 ? 3 ?a) Symmetric Propertyc) Commutative Property of Add.b) Transitive Propertyd) Awesome Property______ 15) Which property is represented by:3 + 9 = 9 + 3?a) Transitive Propertyc) Reflexive Propertyb) Symmetric Propertyd) Commutative Property of Add.______ 16) Which property is represented by:If 3 + 11 = 14, then 14 = 3 + 11?a) Transitive Propertyc) Reflexive Propertyb) Commutative Property of Add.d) Symmetric Property17) Write a statement that illustrates the Additive Identity Property:______________________________18) Write a statement that illustrates the Multiplicative Identity Property:________________________19) Write a statement that illustrates the Symmetric Property:______________________________20) Write a statement that illustrates the Associative Property of Add.:______________________________Name___________________________________________ Date_________________________ Algebra I – Pd ____Real Numbers and Properties 1CDirections: You may answer of all the following questions on this paper.1) State which property is being showna) 3(x + 5) = 3x + 3(5)__________________________________________b) (9 + 4) + 5 = 9 + (4 + 5)__________________________________________c) 36 ? 0 = 0 __________________________________________d) 8m + 6m = m(8 + 6)__________________________________________e) (m ? 14) ? n = m ? (14 ? n)__________________________________________f) 162 ? 1 = 162__________________________________________g) 4 ?14 = 1__________________________________________h) 5 + 8 = 8 + 5__________________________________________2) Complete the matching column_____1) 5(6 + 2) = 5(6) + 5(2)??????????????????????????? a) Additive Identity????????????_____ 2) If 40 = 4(10), then 4(10) = 40 b) Associative Property of Mult._____ 3)5 + 18 = 18 + 5????????????????????????????????????????????????c) Commutative?Property of Add._____ 4) 14 · 1 = 14????????????????????????????????????????????????????d) Distributive Property_____ 5) 15 + 2 = 15 + 2e) Transitive Property_____ 6)(5 + 6) + 8 = 5 + (6 + 8)???????????????????????f) Multiplicative Identity_____ 7) If 62 = 36 and 36 = 4(9), then 62 = 4(9)???????????g) Associative Property of Add._____ 8) 23 · 0 = 0h) Multiplicative property of ZeroI) Commutative Property of Mult.???????????????????????????????j)?Symmetric Propertyk) Reflexive Property3) Is the following operation true?If true, which property is being show, if false, explain why.(8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4) Which is an illustration of the commutative property of addition?a) a + 0 = ab) a(b + c) = ab + acc) a + b = b + ad) (a + b) + c = a + (b + c)_____ 5) Which statement illustrates the associative property of multiplication?a) 212=1b) 2(3 + 4) = 2(3) + 2(4)c) 2(3 · 4) = (2 · 3)4d) 2(1) = 26) Give an example of each:a) a number that is whole, but not a natural number___________________________b) a number that is rational, but not a whole number___________________________c) a number that is rational, but not an integer___________________________d) a number that is irrational___________________________e) a number that is a natural number, but not an integer___________________________f) a number that is an integer, but not a natural number___________________________7) Write True or False (do not write T or F… you must write out the whole word!)a) all whole number are rational_____________________b) all integers are natural numbers_____________________c) all real numbers are rational_____________________d) all irrational number are real_____________________e) all natural numbers are irrational_____________________Name____ANSWERS___________________________ Date_________________________ Algebra I – Pd ____Real Numbers and Properties 1C1) State which property is being showna) Distributive propertyb) Assoc. Prop. Of Add.c) Multi. Prop of Zerod) Distributive Propertye) Assoc. Prop. Of Mult.f) Multi. Identityg) Multiplicative Inverseh) Comm. Prop of Add.2) Complete the matching column1) D2) J3) C4) F5) K6) G7) E8) H3) (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2) False4) C5) C6) Give an example of each:a) a number that is whole, but not a natural numberZerob) a number that is rational, but not a whole numberNegatives or Decimalsc) a number that is rational, but not an integerDecimals or Fractionsd) a number that is irrationalπ 2 3e) a number that is a natural number, but not an integerNot possiblef) a number that is an integer, but not a natural numberNegatives or Zero7) a) Trueb) Falsec) Falsed) Truee) FalseName___________________________________________ Date_________________________ Algebra I – Pd ____Evaluating Expressions 1DWhen given an algebraic expression with one or more unknown variables, we cannot find the exact value of the expression. If we are told what the value(s) or the variable(s) are then we can find the exact value.For Example:13x + 2ySince we do not know the values of x and y we cannot simplifyIf we are told to evaluate 13x + 2y when x = 3 and y = 4, we can substitute these values in.First step:Write the expression13x + 2ySecond step:Replace the variables13(3) + 2(4) ** Always use parenthesisThird step: Simplify using PEMDAS39 + 8 = 47Errors often occur due to sloppiness and laziness, so be careful and be neat!!Directions: Evaluate the following1) 50 – 3x, when x = 71) _______________2) 2x2- 5x+4 when x = 72) _______________3) 2a5+ n-1d when a = 40, n = 10 and d = 33) _______________4) 2x2- 2x2 when x = 44) _______________5) x2-8y when x = 5 and y = 125) _______________6) r2+4s when r = 3 and s = 0.56) _______________7) a2+b2- d2 when a = 8, b = 6, and d = 37) _______________8) 3w-2x2 when w = 10 and x = 88) _______________9) 3w2- 2x2 when w = 10 and x = 89) _______________10) 59F-32 when F = 8610) _______________11) 12x(y+z) when x = 8, y = 5, and z = 211) _______________Extra Examples:Part 1 – Using your order of operations evaluate the expression.1) 5 + 2 – 32) 12 – 6 + 13) 10 · 2 ÷ 44) 4 + 3 · 25) 8 · 3 – 106) 5 – 14 ÷ 7 7) 2 + 36 ÷ 48) 10 ÷ 5 + 3 · 29) 4 – 20 ÷ 10 + 710) 3 · 22 + 111) 2 · 32 ÷ 312) 4(2 + 3) – 8 Part 2 – Using the given information, evaluate the expression.13) 3 + 2x2, when x = 214) 30 – 3x2, when x = 315) 3a – 2b, when a = 2 and b = 316) 5x2 – y, when x = 3 and y = 517) 2x + x2 – 4, when x = 418) a3 – 3a + 5, when a = 219) a2 ÷ 5 + 3, when a = 520) x · y – 8, when x = 3 and y = 421) a2 – b ÷ 4, when a = 5 and b = 822) 3y – x2 · 4, when x = 2 and y = 623) 2a+b3 when a = 4 and b = 124) 7 – xy · 2, when x = 15 and y = 525)(2y + 6) – 4y, when y = 326) 5c – (2 + c), when c = 227) 2b(7 + b), when b = 128) (3x + 1)x, when x = 329) 3x(2y – 3), when x = 5 and y = 230) 3x + 2(2x + 5), when x = 131) 15 ÷ (2a + 1), when a = 132) (7x + 4) ÷ 2, when x = 233) x ÷ (2y + 1), when x = 21 and y = 134) b ÷ (3a – 2), when a = 2 and b = 1635) (6a + 2) ÷ b, when a = 3 and b = 236) [10 – (2x ÷ 3)] + y, when x = 3 and y = 437) 8a2- 10b ÷3c+ 42b2÷(3a-9) when a = -3, b = 6, and c = 238) 2y2+8x2÷-10÷3z4x+6y+8y+2 when x = 5, y = 7, and z = -439) a2+ b2÷5c10b-7a-1 when a = -8, b = -6 and c = 4Name________ANSWERS_______________________ Date_________________________ Algebra I – Pd ____Evaluating Expressions 1DDirections: Evaluate the following1) 292) 673) 434)315) 216) 9.27) 918) 1969) 17210) 3011) 28Extra Examples:Part 1 – Using your order of operations evaluate the expression.1) 42) 73) 54) 105) 146) 37) 118) 89) 910) 1311) 612) 12Part 2 – Using the given information, evaluate the expression.13) 1114) 315) 016) 4017) 2018)719) 820) 421) 2322) 223) 324) 125) 026) 627) 1628) 3029) 1530)1731) 532) 933) 734) 435) 1036) 1237) 8a2- 10b ÷3c+ 42b2÷3a-9 when a = -3, b = 6, and c = 2-33238) 2y2+8x2÷-10÷3z4x+6y+8y+2 when x = 5, y = 7, and z = -4-131539) a2+ b2÷5c10b-7a-1 when a = -8, b = -6 and c = 4-1Name___________________________________________ Date_________________________ Algebra I – Pd ____Combining Like Terms 1ESimplifying and Combining Like Terms ExponentCoefficient 4x2 Variable (or Base)* Write the coefficients, variables, and exponents of the following:?CoefficientsVariablesExponents8c2???9x???y8???12a2b3???Like Terms: Terms that have identical variable parts (same variable(s) and same exponent(s)). When simplifying using addition and subtraction, you combine “like terms” by keeping the "like term" and adding or subtracting the numerical coefficients.Examples: 3x + 4x = 7x13xy – 9xy = 4xy12x3y2 - 5x3y2 = 7x3y2 Why can’t you simplify? 4x3 + 4y311x2 – 7x 6x3y + 5xy3Simplify the following:1) 7x + 5 – 3x2) 6w2 + 11w + 8w2 – 15w3) 6x + 4 + 15 – 7x4) (12x – 5) – (7x – 11)5) (2x2 - 3x + 7) – (-3x2 + 4x – 7)6) 11a2b – 12ab2WORKING WITH THE DISTRIBUTIVE PROPERTYExample: 3(2x – 5) + 5(3x +6) = Since in the order of operations, multiplication comes before addition and subtraction, we must get rid of the multiplication before you can combine like terms. We do this by using the distributive property:3(2x – 5) + 5(3x +6) =3(2x) – 3(5) + 5(3x) + 5(6) = 6x - 15 + 15x + 30 =Now you can combine the like terms:Final answer: 6x + 15x = 21x 3(2x – 5) + 5(3x + 6) = 21x + 15-15 + 30 = 15Combining Like Terms Examples:1) 7x+ 8+ 5x-102) 2x-5+ 6+3x3) 9-3x- -6x+74) 7x+9- 8-2x5) (13u-9)+(8u+17)6) 19a-5- 6a-117) 7c2- 5c+10- 7c2- 9c8) 16c2- 8+13c+ 9-4c2+ 7c9) 7n-13- 19n2- 15+12n10) 22x2- 13x+7+ -9x2+21x-711) 4x2- 8x+3+ 6x2- 4x+1112) x2- 13x+7- 3x2- 4-20x13) 7+x2- 4x+ 10x2+9x-514) 15+10x+7x2- -6x-10x2+ 915) 3y2- 5y+10+ 7y2- 13y+2416) -6a2+4-9a- 5-4a2+ 7a17) 3b4+2b2-8b+14+3b4+16b-1118) 25z2- 9- 15z2+7z-2019) 6k3-4k2+7k+1- 4k3-3k2+6k+120) 6x2+13xy-5y2+9xy-2x2+3y221) 5xy+6x2y-4y2x+9x2y-5xy+10xy222) 2x+8-3x2+5x-1623) 3x+7y-5- 2x2-9x+1224) 3y-9x+7- 15+8y-13x25) 15xy-6x2y+9y2x- (9x2y+11xy-4xy2)26) 5z2+10z- -15z2-18+9zDistributive Property Examples: 1) 4(7x - 8) + 6(5x + 10)2) 6(4x2 – 5x + 2) + 3(-8x2 + 11x + 4)3) 5(4x2 – 8x + 3) – 7(6x2 – 4x + 11)4) 4(6x3 – 4x2 + 1) – 9(4x3 – 2x2 + 1)5) 10(4x2 + 8x + 7) – 8(5x2 + 10x –9)6) 6(4x2 – 3x +2) + 5(3x – 6)7) 9(4x2 – 7x + 12) – 12(3x2 – 5x – 9)8) 4(6x3 – 4x2 + 11) – 7(5x2 + 9)9) 7(9x + 3y) + 4(2x + 6)10) 8( 4x2 – 3x) + 5 (6x – 7)11) 6(3x + 8) + 4 ( 2x2 + 9)12) 5(3 + 7y) + 6(8y – 4y2)13) 7( 2x + 8) – 4(3x2 + 5x – 6)14) 9(3xy + 7y – 5) + 5(3y2 + 6)15) 5( 3y2+ 7y – 10) + 6(2y2 – 8y + 6)16) 3(7x + 2y – 8) – 5(9x + 4y – 11)17) 3(12x4 – 16x3 + 4x2 – 8x + 24) – 4(9x4 – 12x3 –3x2 –6x + 18)Combining Like Terms Examples: ANSWERS1) 12x -22) 5x +13) 3x + 24) 9x + 15)21u + 8 6) 13a + 6 7) 4c + 108) 12c2 + 20c + 19) -19n2 - 5n + 210) 13x2 + 8x 11) 10x2 – 12x + 1412) -2x2 + 7x + 1113) 11x2 + 5x + 214) 17x2 + 16x + 615) 10y2 - 18y + 3416) -2a2 - 16a – 1 17) 6b4 + 2b2 +8b + 318) 10z2 – 7z + 1119) 2k3 – k2 + k 20) 4x2 – 2y2 + 22xy21) 15x2y + 6xy2 22) -3x2 – 3x + 2423) -2x2 + 12x + 7y -17 24) 4x – 5y – 8 25) -15x2y + 13xy2 4xy26) 20z2 + z + 18Distributive Property Examples: ANSWERS1) 58x+282) 3x2+243) -22x2-12x-624) -12x3+2x2- 55) 1426) 24x2- 3x-187) -3x+2168) 24x3- 51x2- 199) 71x+21y+2410) 32x2+6x-35)11) 8x2+18x+8412) -24y2+77y+1513) -12x2-6x+8014) 27xy+15y2+63y-1515) 27y2-13y-1416) -24x-14y+3117) 24x2Name __________________________________________ Date ________________________ Algebra I – Pd ______ Solving Basic Equations 1FSolving EquationsGolden Rule of Algebra: “Do unto one side of the equal sign as you will do to the other…”**Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other side. If you multiply by -2 on the left side, you have to multiply by -2 on the other. If you subtract 15 from one side, you must subtract 15 from the other. You can do whatever you want (to get the x by itself) as long as you do it on both sides of the equal sign.Solving Single Step Equations:To solve single step equations, you do the opposite of whatever the operation is. The opposite of addition is subtraction and the opposite of multiplication is division. Solve the following equations for x:1)x + 5 = 122)x – 11 = 193)22 – x = 174)5x = -305)x-5 = 36)23 x = - 87)x + 15 = 288)15 – x = 219)x4 = 510)6 + x = 3411)9x = 4512) 7 + x = 19Solving Multi-Step Equations:3x – 5 = 22????????????????? To get the x by itself, you will need to get rid of the 5 and the 3.???? +5??? +5?????????????????? Get rid of addition and subtraction first.? ***Use the opposite order of PEMDAS*** 3x???= 27??????? ???? ??Then, we get rid of multiplication and division.? 3?????????3?? ?? ? x?? = ?9We check the answer by putting it back in the original equation:Check: 3x – 5 = 22 We have that x = 93(9) - 5 = 22 27 - 5 = 22 22 = 22 (It checks!)Directions: Solve and check the Multi-Step Equations on looseleaf.1) 9x - 11 = -38???????????????????2) 160 = 7x + 6???????????????????3) -x5 +3 = 74) ?x - 11 = 165) 4x – 7 = -236) 26 = 60 – 2x7) 21 – 4x = 45 8) x7 - 4 = 4 9) ? x – 5 = 910) 2x + 8 = 3411) 15 = 3x – 9 12) 33 = 3 – 10x13) 6x + 15 = 5114) 19 + 8x = 4315) x3 + 15 = 48Name __________________________________________ Date ________________________ Algebra I – Pd ______ Solving Basic Equations 1FSolving EquationsFront ANSWERS:1) x = 72) x = 303) x = 54) x = -65) x = -156) x = -127) x = 138) x = -69) x = 2010) x = 2811) x = 512) x = 12Back ANSWERS:1) x = -32) x = 223) x = -204) x = 365) x = -46) x = 177) x = -68) x = 569) x = 2810) x = 1311) x = 812) x = -313) x = 614) x = 315) x = 99Name___________________________________________ Date_________________________ Algebra I – Pd ____Like Terms & Basic Equations Directions: Do all problems on looseleaf.Part 1 – Simplify the following expression. (Remember to show your RAINBOW!!)1) 3(5x – 3) + 6(2x + 4)2) 7(2x2 – 6x + 2) + 3(-5x2 + 14x – 4)3) 9(7x2 – 5x + 9) – 7(8x2 – 3x + 12)4) 6(3x3 – 4x2 + 11x – 5) – 10(-2x3 – 6x2 + 6x +7)5) 4(2x2 + 6x + 5) – 8(x2 + 3x – 5) 6) 7(4x2 – 3x + 2) + 9(3x – 6)7) 12(3x2 – 6x + 9) – 9(4x2 – 8x – 12) 8) 5(6x3 – 4x2 + 11) – 6(5x2 + 9)9) 10(3x4 – 5x3 + 7x2 – 10x + 6) – 5(6x4 – 10x3 – 14x2 – 20x + 12)Part 2 – Solve for x and check your answers.1) 2x + 9 = 152) 7x – 6 = 293) 24 – 2x = -124) x5 -6 = - 85) 12x + 2 = -346) 62 – 4x = 707) 88 – 11x = 08) 15 = 24 + 3x9) 13.5x + 2 = -2510) 6x + 2 = 3011) 2.5x – 9 = 18.512) 4x – 12 + 2x = 42 ................
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