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Algebra 1 Marcos De la Cruz Algebra 1(6 th period) Ms.Hardtke5/14/10.

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Presentation on theme: "Algebra 1 Marcos De la Cruz Algebra 1(6 th period) Ms.Hardtke5/14/10."— Presentation transcript:

1 Algebra 1 Marcos De la Cruz Algebra 1(6 th period) Ms.Hardtke5/14/10

2 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

3 Addition Property of Equality If the same number is added to both sides of an equation, both sides will be and remain equal If the same number is added to both sides of an equation, both sides will be and remain equal 3=3 (equation) 3=3 (equation) If 2 is added to both sides If 2 is added to both sides 2+3= =3+2 5=5 5=5 Negative Special Case Negative Special Case y=3x+5 (equation) If (-3) is added to both sides Y-3=3x+5-3 Y-3=3x+2 Its still equal

4 Multiplication Property of Equality States that when both sides of an equal equation is multiplied and the equation remains equal States that when both sides of an equal equation is multiplied and the equation remains equal If 5=5 (equation) If 5=5 (equation) 5x2=2x5 5x2=2x5 You multiply both sides by 2 You multiply both sides by 2 10=10 10=10 Still remains equal Still remains equal

5 Reflexive Property of Equality When something is the exact same on both sides When something is the exact same on both sides A = A A = A 7x = 7x 7x = 7x 3456x = 3456x 3456x = 3456x

6 Symmetric Property of Equality When two variables are different but are the same number/amount (equal symmetry) When two variables are different but are the same number/amount (equal symmetry) If a=b, then b=a If a=b, then b=a If c=d, then d=c If c=d, then d=c If xyp=xyo, then xyo=xyp If xyp=xyo, then xyo=xyp

7 Transitive Property of Equality When numbers or variables are all equal When numbers or variables are all equal If a=b and b=c, then c=a If a=b and b=c, then c=a if 5x=100 and 100=4y, then 4y=5x if 5x=100 and 100=4y, then 4y=5x if 0=2x and 2x=78p, then 78p=0 if 0=2x and 2x=78p, then 78p=0

8 Associative Property of Addition The sum of a set of numbers is the same no matter how the numbers are grouped. Associative property of addition can be summarized algebraically as: The sum of a set of numbers is the same no matter how the numbers are grouped. Associative property of addition can be summarized algebraically as: (a + b) + c = a + (b + c) (3 + 5) + 2 = = 10 (3 + 5) + 2 = = (5 + 2) = = (5 + 2) = = 10 (3 + 5) + 2 = 3 + (5 + 2). (3 + 5) + 2 = 3 + (5 + 2).

9 Associative Property of Multiplication The product of a set of numbers is the same no matter how the numbers are grouped. The associative property of multiplication can be summarized algebraically as: The product of a set of numbers is the same no matter how the numbers are grouped. The associative property of multiplication can be summarized algebraically as: (ab)c = a(bc)

10 Commutative Property of Addition The sum of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, the commutative property of addition states: The sum of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, the commutative property of addition states: a + b = b + a = because = 7 and = = because = 7 and = = =

11 Commutative Property of Multiplication The product of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, commutative property of multiplication can be written as: The product of a group of numbers is the same regardless of the order in which the numbers are arranged. Algebraically, commutative property of multiplication can be written as: ab = ba 8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8 8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8

12 Distributive Property The sum of two addends multiplied by a number is the sum of the product of each addend and the number The sum of two addends multiplied by a number is the sum of the product of each addend and the number A(b+c) A(b+c) Ab + Ac Ab + Ac 3x(2y+4) 3x(2y+4) 6xy + 12x 6xy + 12x

13 Property of Opposites/Inverse Property of Addition When a number is added by itself negative or positive to make zero When a number is added by itself negative or positive to make zero a + (-a) = (-5) = (-5) = 0 -3y + (3y) = 0 -3y + (3y) = 0

14 Property Of Reciprocals/Inverse Property of Multiplication For two ratios, if a/b = c/d, then b/a = d/c For two ratios, if a/b = c/d, then b/a = d/c a(1/a) = 1 a(1/a) = 1 5(1/5) = 1 5(1/5) = 1 8/1 x 1/8 = 1 8/1 x 1/8 = 1 A number times its reciprocal, always equals one A number times its reciprocal, always equals one A Reciprocal is its reverse and opposite (the signs switch from + to — or vice versa) A Reciprocal is its reverse and opposite (the signs switch from + to — or vice versa)

15 Reciprocal Function (continued) The reciprocal function: y = 1⁄x. For every x except 0, y represents its multiplicative inverse The reciprocal function: y = 1⁄x. For every x except 0, y represents its multiplicative inverse

16 Identity Property of Addition A number that can be added to any second number without changing the second number. Identity for addition is 0 (zero) since adding zero to any number will give the number itself: A number that can be added to any second number without changing the second number. Identity for addition is 0 (zero) since adding zero to any number will give the number itself: 0 + a = a + 0 = a 0 + (-3) = (-3) + 0 = (-3) = (-3) + 0 = = = = = 5

17 Identity Property of Multiplication A number that can be multiplied by any second number without changing the second number. Identity for multiplication is "1,“ instead of 0, because multiplying any number by 1 will not change it. A number that can be multiplied by any second number without changing the second number. Identity for multiplication is "1,“ instead of 0, because multiplying any number by 1 will not change it. a x 1 = 1 x a = a (-3) x 1 = 1 x (-3) = -3 (-3) x 1 = 1 x (-3) = -3 1 x 5 = 5 x 1 = 5 1 x 5 = 5 x 1 = 5

18 Multiplicative Property of Zero Anything number or variable multiplied times zero (0), will always equal zero Anything number or variable multiplied times zero (0), will always equal zero 5 x 0=0 5g x 0=0 No matter what number is being multiplied by zero, it will always be zero A really long way to explain the Multiplicative Property of Zero (Proof)

19 Closure Property of Addition Sum (or difference) of 2 real numbers equals a real number Sum (or difference) of 2 real numbers equals a real number 10 – (5)= 5 10 – (5)= 5

20 Closure Property of Multiplication Product (or quotient if denominator 0) of 2 Reals equals a real number Product (or quotient if denominator 0) of 2 Reals equals a real number 5 x 2 = 10 5 x 2 = 10

21 (Exponents) Product of Powers Property Exponents Exponents are the little numbers above numbers, that mean that the number is multiplied by itself that many times Exponents are the little numbers above numbers, that mean that the number is multiplied by itself that many times 7 × 7 = 7 × 7 = (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) When two exponents or numbers with exponents are being multiplied, you add both exponents, but you still multiply the number or variable When two exponents or numbers with exponents are being multiplied, you add both exponents, but you still multiply the number or variable 3x (5x ) = 15x (3+4) 7

22 Power of a Product Property To find a power of a product, find the power of each factor and then multiply. In general: To find a power of a product, find the power of each factor and then multiply. In general: (ab) = a · b Or a · b = (ab) a · b = (ab) (3t) (3t) (3t) = 3 · t = 81t (3t) = 3 · t = 81t mmm mmm

23 Power of a Power Property To find a power of a power, multiply the exponents. (Its basically the same as the Power of a Product Property, if forgotten, go one slide back and review.) To find a power of a power, multiply the exponents. (Its basically the same as the Power of a Product Property, if forgotten, go one slide back and review.) (5 ) (5 ) (5 )(5 )(5 )(5 ) = 5 5 (5 )(5 )(5 )(5 ) = 5 5 Its basically this: Its basically this: (a ) = a (4)12 bcbc

24 Quotient of Powers Property When both the denominator and numerator of a fraction have a common variable, it can be canceled, therefore not usable anymore When both the denominator and numerator of a fraction have a common variable, it can be canceled, therefore not usable anymore Also when a variable is canceled, the exponents are subtracted, instead of added as in the Product of Powers Property Also when a variable is canceled, the exponents are subtracted, instead of added as in the Product of Powers Property a /a a a /a a 5 5x5x5 5 5x5x5 —— —————— —— —————— 5 5x5 5 5x5 = 5 (the canceling of common factors) bcb-c 3 2

25 Power of a Quotient Property This is almost the same as the Quotient of Powers Property, but this time, an entire fraction is multiplied by an exponent This is almost the same as the Quotient of Powers Property, but this time, an entire fraction is multiplied by an exponent You also have to cancel the common factors, if there are any You also have to cancel the common factors, if there are any (a/b) a /b — (a/6) (a/b) a /b — (a/6) (and vice versa) (a /36) ccc2 2

26 Zero Power Property If a variable has an exponent of zero, then it must equal one If a variable has an exponent of zero, then it must equal one a =1 a =1 b =1 b =1 c b a =1 c b a =1 (a ) =1 (a ) =

27 Negative Power Property When a fraction or a number has negative exponents, you must change it to its reciprocal in order to turn the negative exponent into a positive exponent When a fraction or a number has negative exponents, you must change it to its reciprocal in order to turn the negative exponent into a positive exponent 4 ¼ 1/16 4 ¼ 1/ The exponent turned from negative to positive

28 Zero Product Property When both variables equal zero, then one or the other must equal zero When both variables equal zero, then one or the other must equal zero if ab=0, then either a=0 or b=0 if ab=0, then either a=0 or b=0 if xy=0, then either x=0 or y=0 if xy=0, then either x=0 or y=0 if abc=0, then either a=0, b=0, or c=0 if abc=0, then either a=0, b=0, or c=0

29 Product of Roots Property The product is the same as the product of square roots The product is the same as the product of square roots

30 Quotient of Roots Property The square root of the quotient is the same as the quotient of the square roots: The square root of the quotient is the same as the quotient of the square roots: AA B B

31 Root of a Power Property

32 Power of a Root Property

33 Density Property of Rational Numbers Between any two rational numbers, there exists at least one additional rational number Between any two rational numbers, there exists at least one additional rational number or or 4 ½

34 Websites PROPERTIES Ϯ Ϯ Ϯ Ϯ Ϯ — MUHS Ϯ Ϯ Ϯ Ϯ Ϯ Ϯ Ϯ Ϯ Ϯ Hotmath.com

35 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

36 Standard/General Form Standard Form Ax + By = C Ax + By = C The terms A, B, and C are integers (could be either positive or negative numbers or fractions) The terms A, B, and C are integers (could be either positive or negative numbers or fractions) If Fractions: If Fractions: Multiply each term in the equation by its LCD (Lowest Common Denominator) Multiply each term in the equation by its LCD (Lowest Common Denominator) Either add or subtract to get either X or Y isolated, in one side of the = Either add or subtract to get either X or Y isolated, in one side of the = If Decimals: If Decimals: Multiply each term in the equation depending on the decimal with the most numbers (by 10, 100, 1000, etc) 1.23 (multiply times 100) Subtract or add to get X or Y isolated If Normal Numbers (neither fractions or decimals): If Normal Numbers (neither fractions or decimals): Just add or subtract to get X or Y isolated

37 Graph Points A Graph Point contains of an X and a Y A Graph Point contains of an X and a Y(x,y) The X and Y mean where exactly the point is located The X and Y mean where exactly the point is located X line graph Y line graph

38 Standard/General Form Ex. Fractions: Fractions: You multiply by the LCM You multiply by the LCM Which in this case is 20x Then to double check it…

39 Point-Slope Form The Point-Slope form. got its name because it uses a single point in a graph and a on the slope of the line The Point-Slope form. got its name because it uses a single point in a graph and a on the slope of the line It is usually used to find the slope of a graph, if the slope is not given in a certain problem or equation It is usually used to find the slope of a graph, if the slope is not given in a certain problem or equation The Y on the Point-Slope form., doesn’t mean that the Y is multiplied by one, but it means to use the first Y of the two or one point given as a problem (same with X) The Y on the Point-Slope form., doesn’t mean that the Y is multiplied by one, but it means to use the first Y of the two or one point given as a problem (same with X) (4,3) and the slope is 2 M = slope Y—stays the same X —is 4 (because 4 is in the x spot) Y — is 3 X—stays the same If the problem gives you two points and no slope, then you are free to choose what which or the Xs or the Ys you may want to use for your Point-Slope Form. If the problem gives you two points and no slope, then you are free to choose what which or the Xs or the Ys you may want to use for your Point-Slope Form ex

40 Point-Slope Form (4,3) and m=2 (4,3) and m=2 you must convert “it” to a slope-intercept form you must convert “it” to a slope-intercept form Y=Mx + B Y-3 = 2(x-4) Y-3 = 2(x-4) Y-3 = 2x-8 Y-3 = 2x-8 Y = 2x – 11 (slope-intercept form) Y = 2x – 11 (slope-intercept form)

41 Slope-Intercept Explanation y=mx+b y=mx+b Sometimes in the Slope-intercept form, there are fractions as the slope or the y-intercept Sometimes in the Slope-intercept form, there are fractions as the slope or the y-intercept B= y-intercept B= y-intercept Rise/Run Rise/Run When the slope is a fraction, you mark the B in a graph, which is the y-intercept When the slope is a fraction, you mark the B in a graph, which is the y-intercept Then depending on the slope, if its positive than the line will look like this… Then depending on the slope, if its positive than the line will look like this… If its not positive, but negative, it will look like: If its not positive, but negative, it will look like:

42 Point-Slope (Slope-Intercept) Graph Y = 2x – 11 Y = 2x – 11 (0,-11) Rise/Run Go up twice and to the side once (5,0)

43 Websites (for further information) Linear Equations

44 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

45 Linear Systems— Method Explanation Substitution The Substitution Method, is used when, there are two equations, and you pick one (the one that looks the easiest to do) and you isolate either the x or the y The Substitution Method, is used when, there are two equations, and you pick one (the one that looks the easiest to do) and you isolate either the x or the y When x or y is isolated, then you will get something like this: When x or y is isolated, then you will get something like this: Y= ?x + ? Y= ?x + ? X= ?y + ? X= ?y + ? Then, you replace the x or the y in the equation that you didn’t touch yet, and you must insert Then, you replace the x or the y in the equation that you didn’t touch yet, and you must insert If you isolated the y, then you will solve for x If you isolated the y, then you will solve for x If you isolated the x, then you will solve for y If you isolated the x, then you will solve for yElimination The Elimination Method, is used when there are two equations and, it is said to be a lot easier than the Substitution Method The Elimination Method, is used when there are two equations and, it is said to be a lot easier than the Substitution Method First, you will have to decide whether you want to go for the x or the y First, you will have to decide whether you want to go for the x or the y Then, you will multiply and cancel/eliminate either x or y depending, on which one did you chose to do (x or y) Then, you will multiply and cancel/eliminate either x or y depending, on which one did you chose to do (x or y) Then you solve for x or y Then you solve for x or y You will eventually substitute, more like insert your y or x answer into the either problem replacing it with x or y You will eventually substitute, more like insert your y or x answer into the either problem replacing it with x or y Then you solve for either x or y Then you solve for either x or y

46 Substitution Method y = x x + 2(11 - 4x) = 8 Answers Isolate the Y or X Substitute the number, insert it Solve for X and Solve for Y (vice versa)

47 Literal Coefficients Simultaneous equations with literal coefficients and literal constants may be solved for the value of the variables just as the other equations discussed in this chapter, with the exception that the solution will contain literal numbers. For example, find the solution of the system: We proceed as with any other simultaneous linear equation. Using the addition method, we may proceed as follows: To eliminate the y term we multiply the first equation by 3 and the second equation by -4. The equations then become … To eliminate x, we multiply the first equation by 4 and the second equation by -3. The equations then become We may check in the same manner as that used for other equations, by substituting these values in the original equations. 3 Variables !!

48 Elimination Method 2x – 3y = 19 2x – 3y = 19 5x – 2y = 20 5x – 2y = 20 2x – 3y = 19 (2) 2x – 3y = 19 (2) 5x – 2y = 20 (-3) 5x – 2y = 20 (-3) 4x – 6y = 38 4x – 6y = x + 6y = x + 6y = x = x = -22 X = 2 X = 2 2x – 3y = 19 2x – 3y = 19 2(2) – 3y = 19 2(2) – 3y = 19 4 – 3y = 19 4 – 3y = 19 -3y = 15 -3y = 15 Y = -5 Y = -5 The two equations The two equations Now we multiply and then later cancel out a variable, depending which one you chose Now we multiply and then later cancel out a variable, depending which one you chose Now we got one answer—x = 2 Now we got one answer—x = 2 Now we must insert the two, into the either of the equations…(substitution method) Now we must insert the two, into the either of the equations…(substitution method) Now you got the y = -5 Now you got the y = -5

49 Dependent When a system is "dependent," it means that ALL points that work in one of them ALSO work in the other one When a system is "dependent," it means that ALL points that work in one of them ALSO work in the other one Graphically, this means that one line is lying entirely on top of the other one, so that if you graphed both, you would really see only one line on the graph, since they are imposed on top of each other Graphically, this means that one line is lying entirely on top of the other one, so that if you graphed both, you would really see only one line on the graph, since they are imposed on top of each other One of them totally DEPENDS on the other one One of them totally DEPENDS on the other one

50 Independent When a system is "independent," it means that they are not lying on top of each other When a system is "independent," it means that they are not lying on top of each other There is EXACTLY ONE solution, and it is the point of intersection of the two lines There is EXACTLY ONE solution, and it is the point of intersection of the two lines It's as if that one point is "independent" of the others. It's as if that one point is "independent" of the others. To sum up, a dependent system has INFINITELY MANY solutions. An independent system has EXACTLY ONE solution To sum up, a dependent system has INFINITELY MANY solutions. An independent system has EXACTLY ONE solution

51 Consistent We say that a point is a "solution" to the system when it makes BOTH equations true, right? We say that a point is a "solution" to the system when it makes BOTH equations true, right? This is to say that there exists a point (or set of points) that "work" in one equation and also "work" in the other one This is to say that there exists a point (or set of points) that "work" in one equation and also "work" in the other one So we say that this point is CONSISTENT from one equation to the next So we say that this point is CONSISTENT from one equation to the next

52 Inconsistent On the other hand, if there are NO points that work in both, then we say that the equations are INCONSISTENT On the other hand, if there are NO points that work in both, then we say that the equations are INCONSISTENT NO numbers that work in one are consistent with the other NO numbers that work in one are consistent with the other To sum up, a consistent system has at least one solution. An inconsistent system has NO solution at all To sum up, a consistent system has at least one solution. An inconsistent system has NO solution at all

53 Websites Linear Systems:

54 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

55 1 st Power Equations (1 Variable) In order to get the answer, when there is only one variable In order to get the answer, when there is only one variable You must, isolate the variable, and if it has a sign with it (a negative sign) or a number with it, than you can and must divide the number to the other side You must, isolate the variable, and if it has a sign with it (a negative sign) or a number with it, than you can and must divide the number to the other side In order to get the variable completely alone In order to get the variable completely alone Then you get your answer Then you get your answer

56 1 Variable Problems 5x + 3 = 2 (2 – 3x) 5x + 3 = 2 (2 – 3x) 5x + 3 = 4 – 6x 5x + 3 = 4 – 6x 5x = 4 + (-3) – 6x 5x = 4 + (-3) – 6x 5x = 1 – 6x 5x = 1 – 6x 5x + 6x = 1 5x + 6x = 1 11x = 1 11x = 1 X = 1/11 X = 1/11 2x = 8 2x = 8 X = 4 X = 4 -x + 20 = – 3x + 2(5x – 10) -x + 20 = – 3x + 2(5x – 10) -x + 20 = – 3x + 10x – 20 -x = 7x – 40 -8x = -40 X = 5 These 1 variable problems are fairly simple and easy These 1 variable problems are fairly simple and easy All you have to do is isolate the variable All you have to do is isolate the variable Then just add, subtract, or divide and solve the problem Then just add, subtract, or divide and solve the problem

57 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

58 Factoring FOIL FOIL FOIL is a type of factoring that includes two “globs” is a type of factoring that includes two “globs” (3x + 2)(3x – 2) (3x + 2)(3x – 2) this FOIL means that the O and I in FOIL, will be the same number, but one will be negative and one positive, therefore, they will cancel each other out this FOIL means that the O and I in FOIL, will be the same number, but one will be negative and one positive, therefore, they will cancel each other out

59 PST PST, is when two globs are reversed “FOILed” and they equal perfectly PST, is when two globs are reversed “FOILed” and they equal perfectly (x + 2)(x + 2) (x + 2)(x + 2) X + 4x + 4 (PST) X + 4x + 4 (PST) The first number in a PST, to check if you got a PST, the first number has to be squared, and if its not, then take out the GCF The first number in a PST, to check if you got a PST, the first number has to be squared, and if its not, then take out the GCF The First and Last number should have roots, while the middle number should be the double of the roots of both the First and Last number The First and Last number should have roots, while the middle number should be the double of the roots of both the First and Last number 2

60 Factor GCF The GCF stands for the Greatest Common Factor The GCF stands for the Greatest Common Factor Which means, that if you have a binomial or a trinomial with prime numbers in common or more variables than needed, then you can factor them out, and then continue to solve the problem Which means, that if you have a binomial or a trinomial with prime numbers in common or more variables than needed, then you can factor them out, and then continue to solve the problem Whatever you factored out, will still be part of the Answer of the problem Whatever you factored out, will still be part of the Answer of the problem

61 Difference of Squares First take out the GCF (always) First take out the GCF (always) If there are two globs that if FOILed, arent a PST, but they just make a binomial, but it can be divided into two more binomials If there are two globs that if FOILed, arent a PST, but they just make a binomial, but it can be divided into two more binomials Then you have conjugates Then you have conjugates (? + ?) (? - ?) (? + ?) (? - ?) As long as you have a negative glob, that you can still divide into more globs, you can continue to divide, but if one glob is the same as another glob, then your answer will only contain the glob, but only once As long as you have a negative glob, that you can still divide into more globs, you can continue to divide, but if one glob is the same as another glob, then your answer will only contain the glob, but only once

62 Sum or Difference of Cubes The Sum or Difference of Cubes, is when you take variable squares or numbers with roots cubed, and they are separated and into a binomial and a trinomial The Sum or Difference of Cubes, is when you take variable squares or numbers with roots cubed, and they are separated and into a binomial and a trinomial

63 Reverse FOIL This is the same thing as FOIL factoring, but there is a Trial and Error system This is the same thing as FOIL factoring, but there is a Trial and Error system That means, that when given trinomial, you will have to guess and check if it FOILs the correct globs, and you will have to continue to do that, until you get the correct globs That means, that when given trinomial, you will have to guess and check if it FOILs the correct globs, and you will have to continue to do that, until you get the correct globs

64 Factor By Grouping 4x4 4x4 It is a binomial because there are two terms, and a repeated glob, it is a common glob It is a binomial because there are two terms, and a repeated glob, it is a common glob Which means GCF Which means GCF 2x2 2x2 Sometimes you can rearrange the order of the terms, to find the correct glob Sometimes you can rearrange the order of the terms, to find the correct glob

65 Factor By Grouping 3x1 3x1 Rearrange into a PST Rearrange into a PST Then make two perfect globs Then make two perfect globs If conjugates then separate them If conjugates then separate them

66 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

67 Rational Expressions PST PST X + 10x +25 X + 10x +25 (x + 5) (x + 5) We define a Rational Expression as a fraction where the numerator and the denominator are polynomials in one or more variables. We define a Rational Expression as a fraction where the numerator and the denominator are polynomials in one or more variables (2)(4) 2 + = = = (4)(5) (2)(4) 2 + = = = (4)(5) 5 Addition and Subtraction of Rational Expressions 3x - 4x x(3x - 4) 3x - 4 = = 2x - x x(2x - 1) 2x Multiplication and Division of Rational Expressions

68 R.E First - multiply the first term in each set of parenthesis: 4x * x = 4x 2 Outside - multiply the two terms on the outside: 4x * 2 = 8x Inside - multiply both of the inside terms: 6 * x = 6x Last - multiply the last term in each set of parenthesis: 6 x 2 = 12 FOIL

69 Websites Rational Expressions:

70 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

71 Completing the Square (1) (X + n)² = X² + 2nx + n² (X + n)² = X² + 2nx + n² Note the rightmost term (n²) is related to 2n (the x coefficient) by the formula Note the rightmost term (n²) is related to 2n (the x coefficient) by the formula Solving this by "completing the square" is as follows: 1) Move the "non X" term to the right: Solving this by "completing the square" is as follows: 1) Move the "non X" term to the right: 4X² + 12X = 16 4X² + 12X = 16 2) Divide the equation by the coefficient of X² which in this case is 4 2) Divide the equation by the coefficient of X² which in this case is 4 X² + 3X = 4 X² + 3X = 4 3) Now here's the "completing the square" stage in which we: 3) Now here's the "completing the square" stage in which we: take the coefficient of X divide it by 2 square that number then add it to both sides of the equation. take the coefficient of X divide it by 2 square that number then add it to both sides of the equation.

72 Completing the Square (2) In our sample problem the coefficient of X is 3 dividing this by 2 equals 1.5 squaring this number equals (1.5)² = 2.25 Now, adding that to both sides of the equation, we have: In our sample problem the coefficient of X is 3 dividing this by 2 equals 1.5 squaring this number equals (1.5)² = 2.25 Now, adding that to both sides of the equation, we have: X² + 3X = X² + 3X = ) Finally, we can take the square root of both sides of the equation and we have: 4) Finally, we can take the square root of both sides of the equation and we have: X = Square Root ( ) X = Square Root ( ) X = Square Root (6.25) -1.5 X = Square Root (6.25) -1.5 X = X = X = 1.0 X = 1.0 Let's not forget that the other square root of 6.25 is -2.5 and so the other root of the equation is: Let's not forget that the other square root of 6.25 is -2.5 and so the other root of the equation is: ( ) = -4 ( ) = -4

73 Quadratic Formula We can follow precisely the same procedure as above to derive the Quadratic Formula. All Quadratic Equations have the general form: aX² + bX + c = 0

74 Discriminant and the Quadratic Equation The Discriminant is a number that can be calculated from any quadratic equation A quadratic equation is an equation that can be written as The Discriminant is a number that can be calculated from any quadratic equation A quadratic equation is an equation that can be written as ax ² + bx + c where a ≠ 0 ax ² + bx + c where a ≠ 0 The Discriminant in a quadratic equation is found by the following formula and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation. discriminant= b² − 4ac Example of the discriminant The Discriminant in a quadratic equation is found by the following formula and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation. discriminant= b² − 4ac Example of the discriminant Quadratic equation = y = 3x² + 9x + 5 Quadratic equation = y = 3x² + 9x + 5 The discriminant = 9 ² − The discriminant = 9 ² − 4 3 5

75 Quadratic Equation y = x² + 2x + 1 Quadratic Equation: y = x² + 2x + 1 a = 1 b = 2 c = 1 The discriminant for this equation is 2² = 4 − 4 = 0 Since the discriminant of zero, there should be 1 real solution to this equation. Below is a picture representing the graph and one solution of this quadratic equation Graph of y = x² + 2x +1

76 Websites R.E.:

77 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

78 F(x) In Algebra f(x) is another symbol for y In Algebra f(x) is another symbol for y Y = 3 Y = 3 F(x) = 3 F(x) = 3 Its practically the same things, but people use it for confusion Its practically the same things, but people use it for confusion

79 Domain and Range Domain Domain For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given explicitly. For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given explicitly. Range Range The range of f is the set of all values that the function takes when x takes values in the domain The range of f is the set of all values that the function takes when x takes values in the domain

80 Domain Example: The function y = √(x + 4) has the following graph The domain of the function is x ≥ −4, since x cannot take values less than −4. (Try some values in your calculator, some less than −4 and some more than −4. The only ones that "work" and give us an answer are the ones greater than or equal to −4). Note: The enclosed (colored-in) circle on the point (-4, 0). This indicates that the domain "starts" at this point. That x can take any positive value in this example

81 Range Example 1: Let's return to the example above, y = √(x + 4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y ≥ 0. Example 2: The curve of y = sin x shows the range to be betweeen −1 and 1 The domain of the function y = sin x is "all values of x", since there are no restrictions on the values for x. graphs/2a_Domain-and-range.php graphs/2a_Domain-and-range.php

82 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

83 Solving Inequalities Linear inequalities are also called first degree inequalities, as the highest power of the variable in these inequalities is 1. E.g. 4x > 20 is an inequality of the first degree, which is often called a linear inequality. Linear inequalities are also called first degree inequalities, as the highest power of the variable in these inequalities is 1. E.g. 4x > 20 is an inequality of the first degree, which is often called a linear inequality. Many problems can be solved using linear inequalities. Many problems can be solved using linear inequalities. We know that a linear equation with one pronumeral has only one value for the solution that holds true. For example, the linear equation 6x = 24 is a true statement only when x = 4. However, the linear inequality 6x > 24 is satisfied when x > 4. So, there are many values of x which will satisfy the inequality 6x > 24. We know that a linear equation with one pronumeral has only one value for the solution that holds true. For example, the linear equation 6x = 24 is a true statement only when x = 4. However, the linear inequality 6x > 24 is satisfied when x > 4. So, there are many values of x which will satisfy the inequality 6x > 24.

84 Inequalities Recall that: Recall that: the same number can be subtracted from both sides of an inequality the same number can be subtracted from both sides of an inequality the same number can be added to both sides of an inequality the same number can be added to both sides of an inequality both sides of an inequality can be multiplied (or divided) by the same positive number both sides of an inequality can be multiplied (or divided) by the same positive number if an inequality is multiplied (or divided) by the same negative number, then: if an inequality is multiplied (or divided) by the same negative number, then:

85 Inequalities

86 Conjunctions When two inequalities are joined by the word and or the word or, a compound inequality is formed. A compound inequality like -3 < 2x + 5 and 2x + 5 ≤ 7 is called a conjunction, because it uses the word and. The sentence -3 < 2x + 5 ≤ 7 is an abbreviation for the preceding conjunction. Compound inequalities can be solved using the addition and multiplication principles for inequalities. When two inequalities are joined by the word and or the word or, a compound inequality is formed. A compound inequality like -3 < 2x + 5 and 2x + 5 ≤ 7 is called a conjunction, because it uses the word and. The sentence -3 < 2x + 5 ≤ 7 is an abbreviation for the preceding conjunction. Compound inequalities can be solved using the addition and multiplication principles for inequalities.

87 Disjunction A compound inequality like 2x - 5 ≤ -7 or is called a disjunction, because it contains the word or. Unlike some conjunctions, it cannot be abbreviated; that is, it cannot be written without the word or. A compound inequality like 2x - 5 ≤ -7 or is called a disjunction, because it contains the word or. Unlike some conjunctions, it cannot be abbreviated; that is, it cannot be written without the word or.

88 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

89 Word Problems 1) The sum of twice a number plus 13 is 75. Find the number. The word is means equals. The word and means plus. Therefore, you can rewrite the problem like the following: The word is means equals. The word and means plus. Therefore, you can rewrite the problem like the following: The sum of twice a number and 13 equals 75. The sum of twice a number and 13 equals 75. Using numbers and a variable that represents something, N in this case (for number), you can write an equation that means the same thing as the original problem. 2N + 13 = 75 Using numbers and a variable that represents something, N in this case (for number), you can write an equation that means the same thing as the original problem. 2N + 13 = 75 Solve this equation by isolating the variable. 2N + 13 = 75 Equation = -13 Add (-13) to both sides N = 62 Solve this equation by isolating the variable. 2N + 13 = 75 Equation = -13 Add (-13) to both sides N = 62 N = 31 N = 31 Divided both sides by 2 Divided both sides by 2

90 Word Problems 2) Find a number which decreased by 18 is 5 times its opposite. Again, you look for words that describe equal quantities. Is means equals, and decreased by means minus. Also, opposite always means negative. Keeping that information in mind makes it so an equation can be written that describes the problem, just like the following: N - 18 = 5(-N) Equation. N - 18 = -5N Multiplied out. 5N N + 18 Add (5N + 18) to both sides. 6N = 18 N = 3 Divide both sides by 6 to isolate N.

91 Word Problems 3) Julie has $50, which is eight dollars more than twice what John has. How much has John? First, what will you let x represent? The unknown number -- which is how much that John has. The unknown number -- which is how much that John has. What is the equation? What is the equation? 2x + 8 = 50. 2x + 8 = 50. Here is the solution: Here is the solution: x = $21 x = $21

92 Word Problems 4) Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore; how much did she spend there? Here is the equation. Here is the equation. 3x − 7 = 35 3x − 7 = 35 Here is the solution: Here is the solution: x = $14 x = $14

93 Algebra Topics 1- Properties 2- Linear Equations 3- Linear Systems 4- Solving 1 st Power Equations (1 Variable) 5- Factoring 6- Rational Expressions 7- Quadratic Equations 8- Functions 9- Solving 1 st Power Inequalities (1 Variable) 10- Word Problems 11- Extras

94 11 - The END


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