Algebra 1-semester exam review By: Ricardo Blanco.

Algebra 1-semester exam Algebra 1-semester exam reviewreview

By: Ricardo BlancoBy: Ricardo Blanco

In the next slides you will review:In the next slides you will review:

The properties we The properties we learnedlearned

What are they used for What are they used for and when to recognize and when to recognize themthem

Addition Property (of Addition Property (of Equality)Equality)

Multiplication Property (of Multiplication Property (of Equality)Equality)

Reflexive Property (of Reflexive Property (of Equality)Equality)

Symmetric Property (of Symmetric Property (of Equality)Equality)

Transitive Property (of Transitive Property (of Equality)Equality)

PropertiesProperties

1.Addition Property (of 1.Addition Property (of Equality)Equality)

2. Multiplication Property 2. Multiplication Property (of Equality)(of Equality)

Examples in orderExamples in order

1. if a= 1. if a= bb, then , then aa + + c = b + c.c = b + c.

is added to both sides of an is added to both sides of an equation, the two sides equation, the two sides remain equal. That is, remain equal. That is,

2.if a= 2.if a= bb, then , then aa + + c = b + c.c = b + c.

. If the same number If . If the same number If a = ba = b then then a·c = b·ca·c = b·c. .


3. Reflexive Property (of 3. Reflexive Property (of Equality)Equality)

4. Symmetric Property (of 4. Symmetric Property (of Equality)Equality)

5. Transitive Property (of 5. Transitive Property (of Equality)Equality)

3. a=a3. a=a

4. if a=b then b=a4. if a=b then b=a

5. If 5. If aa = = bb and and bb = = cc, , then then aa = = cc. .

In the next slides you will reviewIn the next slides you will review

Associative Property of AdditionAssociative Property of Addition

Associative Property of MultiplicationAssociative Property of Multiplication

Commutative Property of AdditionCommutative Property of Addition

Commutative Property of MultiplicationCommutative Property of Multiplication

Distributive Property (of Multiplication over Distributive Property (of Multiplication over Addition)Addition)


6. Associative Property of 6. Associative Property of AdditionAddition

7. Associative Property of 7. Associative Property of MultiplicationMultiplication

6. the sum does not 6. the sum does not change. (2 + 5) + 4 = change. (2 + 5) + 4 = 11 or 2 + (5 + 4) = 1111 or 2 + (5 + 4) = 11

7. answer will still not 7. answer will still not chage.(3 x 2) x 4 = 24 chage.(3 x 2) x 4 = 24 or 3 x (2 x 4) = 24.or 3 x (2 x 4) = 24.


8. Commutative Property 8. Commutative Property of Additionof Addition

9. Commutative Property 9. Commutative Property of Multiplicationof Multiplication

8. As per the 8. As per the commutative property commutative property of addition, the of addition, the expression 5 + 14 = expression 5 + 14 = 19 can be written as 19 can be written as 14 + 5 = 19. so, 5 + 14 + 5 = 19. so, 5 + 14 = 14 + 5. 14 = 14 + 5.

9. 4 x 2 = 2 x 4 9. 4 x 2 = 2 x 4


10. Distributive Property 10. Distributive Property (of Multiplication over (of Multiplication over Addition)Addition)

10. 10. 3(2 + 7 - 5)3(2 + 7 - 5) = = 3(2) + 3(7) + (3)(-5)3(2) + 3(7) + (3)(-5)

3(4)3(4) = = 6 + 21 - 156 + 21 - 15

1212 = = 1212


Prop of Opposites or Inverse Property of AdditionProp of Opposites or Inverse Property of Addition

Prop of Reciprocals or Inverse Prop. of MultiplicationProp of Reciprocals or Inverse Prop. of Multiplication

Identity Property of AdditionIdentity Property of Addition

Identity Property of MultiplicationIdentity Property of Multiplication


11. Prop of Opposites or 11. Prop of Opposites or Inverse Property of Addition Inverse Property of Addition

12. Prop of Reciprocals or 12. Prop of Reciprocals or

Inverse Prop. of MultiplicationInverse Prop. of Multiplication

11. In other words, when you add a 11. In other words, when you add a number to its additive inverse, the number to its additive inverse, the result is 0. Other terms that are result is 0. Other terms that are synonymous with additive inverse are synonymous with additive inverse are negative and opposite. negative and opposite. a + (-a) = 0.a + (-a) = 0.

12. In other words, when you multiply 12. In other words, when you multiply a number by its multiplicative inverse a number by its multiplicative inverse the result is 1. A more common term the result is 1. A more common term used to indicate a multiplicative used to indicate a multiplicative inverse is the inverse is the reciprocalreciprocal. A . A multiplicative inverse or reciprocal of a multiplicative inverse or reciprocal of a real number real number aa (except 0) is found by (except 0) is found by "flipping" "flipping" a a upside down. The upside down. The numerator of numerator of a a becomes the becomes the denominator of the reciprocal of denominator of the reciprocal of aa and and the denominator of the denominator of aa becomes the becomes the numerator of the reciprocal of numerator of the reciprocal of aa..


13. Identity Property of 13. Identity Property of Addition Addition

14. Identity Property of 14. Identity Property of

MultiplicationMultiplication

13. Identity property of 13. Identity property of addition states that the addition states that the sum of zero and any sum of zero and any number or variable is the number or variable is the number or variable itself. number or variable itself. 4 + 0 = 4 4 + 0 = 4

14. According to identity 14. According to identity property of addition, the property of addition, the sum of a number and 0 is sum of a number and 0 is the number itself. 4 × 1 the number itself. 4 × 1 = 4 = 4


Multiplicative Property of ZeroMultiplicative Property of Zero

Closure Property of Addition Closure Property of Addition

Closure Property of MultiplicationClosure Property of Multiplication

Product of Powers PropertyProduct of Powers Property

Power of a Product Property Power of a Product Property

Power of a Power PropertyPower of a Power Property


15. Multiplicative Property 15. Multiplicative Property of Zero of Zero

16. Closure Property of 16. Closure Property of Addition Addition

17. Closure Property of 17. Closure Property of Multiplication Multiplication

15. The product of any 15. The product of any number and zero is zero- number and zero is zero- a × 0 = 0 a × 0 = 0 16. Closure property of 16. Closure property of addition states that the addition states that the sum of any two real sum of any two real numbers equals another numbers equals another real number.real number. 17. 17. Closure property Closure property of multiplication states of multiplication states that the product of any that the product of any two real numbers equals two real numbers equals another real number.another real number.


18. Product of Powers 18. Product of Powers PropertyProperty

19. Power of a Product 19. Power of a Product Property Property

20. Power of a Power 20. Power of a Power Property Property

18.when you multiply 18.when you multiply powers having the same powers having the same amount add the amount add the exponents.exponents.72 × 76 72 × 76

(7 × 7) × (7 × 7 × 7 × 7 × 7 (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) × 7) 19. (319. (3tt)4 )4

(3(3tt)4 = 34 · )4 = 34 · tt4 = 814 = 81tt44

20. 20. ((abab))cc = = abcabc


Quotient of Powers PropertyQuotient of Powers Property

Power of a Quotient PropertyPower of a Quotient Property Zero Power PropertyZero Power Property

Negative Power PropertyNegative Power Property

zero product propertyzero product property


21. Quotient of Powers 21. Quotient of Powers PropertyProperty

22. Power of a Quotient 22. Power of a Quotient

PropertyProperty

21. This property states 21. This property states that to divide powers that to divide powers having the same base, having the same base, subtract the exponents.subtract the exponents.((amam))nn = = amnamn 22. This property states that 22. This property states that the power of a quotient can the power of a quotient can be obtained by finding the be obtained by finding the powers of numerator and powers of numerator and denominator and dividing denominator and dividing them.them.


23. Zero Power Property 23. Zero Power Property

24. Negative Power 24. Negative Power

PropertyProperty

23. If a variable has an 23. If a variable has an exponent of zero, then it must exponent of zero, then it must equal one 3equal one 30=10=1

24. When a fraction or a 24. When a fraction or a number has negative number has negative exponents, you must exponents, you must change it to its reciprocal change it to its reciprocal in order to turn the in order to turn the negative exponent into a negative exponent into a positive exponentpositive exponent


25. zero product property25. zero product property 25. when your 25. when your variables are equal to variables are equal to zero then one or the zero then one or the other must be zero.other must be zero.


Product of Roots PropertyProduct of Roots Property Quotient of Roots Property Quotient of Roots Property

Root of a Power PropertyRoot of a Power Property

Power of a Root Property Power of a Root Property


26. Product of Roots 26. Product of Roots

PropertyProperty 26. The product is the 26. The product is the same as the product of same as the product of square rootssquare roots

XX = =

a b AB


27. Quotient of Roots 27. Quotient of Roots Property Property

27. the quotient is the 27. the quotient is the same as the quotient same as the quotient of the square rootsof the square roots


28. Root of a Power 28. Root of a Power Property Property

29. Power of a Root 29. Power of a Root Property Property

28.28.

29.29.

Property quizProperty quiz

Problems in which you Problems in which you determine the property.determine the property.You will fill in the answer You will fill in the answer on the power pointon the power point when finished go back when finished go back through the properties to through the properties to make sure you have the make sure you have the correct answers.correct answers.1. 1. 3.3.4.4.5.5.

A. if a= A. if a= bb, then , then aa + + c = b c = b +c.+c.

B. B. a=aa=aC. C. If If aa = = bb and and bb = = cc, , then then aa = = cc. . D. D. answer will still not answer will still not chage.(3 x 2) x 4 = 24 chage.(3 x 2) x 4 = 24 or 3 x (2 x 4) = 24.or 3 x (2 x 4) = 24.E. E. 4 x 2 = 2 x 4 4 x 2 = 2 x 4

Solving1st power equations Solving1st power equations

In the next slides you will see how to-In the next slides you will see how to-

A. with only one inequality signA. with only one inequality sign

B. conjunctionB. conjunction

C. disjunctionC. disjunction

Solving1st power equations-Solving1st power equations-with only one inequality signwith only one inequality sign

This will only be true if This will only be true if x is equal to fourx is equal to four

The answer will be The answer will be x > 4x > 4

Which on a number Which on a number line isline is

66xx = = 2424

66x > x > 2424

x > 4x > 4

Solving1st power equations- Solving1st power equations- conjunctionconjunction

A conjunction is true A conjunction is true only if both the only if both the statements in it are statements in it are truetrue

A conjunction is a A conjunction is a mathematical operator mathematical operator that returns an output that returns an output of true if and only of true if and only if all of if all of its operands are true.its operands are true.

-2 < x <= 4 -2 < x <= 4

Solving1st power equations-Solving1st power equations-disjunctiondisjunction

A disjunction is statement A disjunction is statement which connects two other which connects two other statements using the word statements using the word or.or.

To solve a disjunctions of To solve a disjunctions of two open sentences, you two open sentences, you find the variables for which find the variables for which at least one of the at least one of the sentences is true. The sentences is true. The graph consists of all points graph consists of all points that are in the graphthat are in the graph

Ex. -3<x or x<4Ex. -3<x or x<4

Line where the linesLine where the lines

Linear equations in two variablesLinear equations in two variables

Standard formStandard form

Next determine Next determine whether or not the whether or not the equations is linear or equations is linear or not.not.

Next subtract 5x from Next subtract 5x from both sidesboth sides

AAxx + B + By y = C= C

yy = 5 = 5xx - 3 - 3

55xx + + yy = -3 = -3

This would be -5This would be -5xx + + yy = -3 it would become = -3 it would become a straight linea straight line

Linear equations in two variables Linear equations in two variables cont.cont.

A graphed linear A graphed linear equationequation

Linear systemsLinear systems

A. substitutionA. substitution

B. addition/subtractionB. addition/subtraction

C. check for C. check for understanding of terms-understanding of terms-

1.dependent1.dependent

2. inconsistent2. inconsistent

3. consistent3. consistent

Solving equations in two Solving equations in two variablesvariables

Graphing pointsGraphing points

Standard/General FormStandard/General Form

Slope- Intercept FormSlope- Intercept Form

Point-Slope FormPoint-Slope Form

SlopesSlopes

Linear systems-substitutionLinear systems-substitution

1.looks like it would be 1.looks like it would be easy to solve for easy to solve for xx, so we , so we take it and isolate take it and isolate xx::2. Now that we have 2. Now that we have yy, , we still need to substitute we still need to substitute back in to get back in to get xx. We could . We could substitute back into any substitute back into any of the previous equations, of the previous equations, but notice that equation 3 but notice that equation 3 is already conveniently is already conveniently solved for solved for xx: : 3. answer is 13. answer is 1

1.21.2yy + + xx = 3 = 32. 22. 2yy + + xx = 3 = 33.x=3-2y3.x=3-2y x=3-2(1)x=3-2(1) x=3-2x=3-2 x=1x=1

Linear systems-add/sub Linear systems-add/sub (elimination)(elimination)

1. Note that, if I add 1. Note that, if I add down, the down, the yy's will cancel 's will cancel out. So I'll draw an out. So I'll draw an "equals" bar under the "equals" bar under the system, and add down:system, and add down:2. Now I can divide 2. Now I can divide through to solve for through to solve for xx = 5, = 5, and then back-solve, and then back-solve, using either of the original using either of the original equations, to find the equations, to find the value of value of yy. The first . The first equation has smaller equation has smaller numbers, so I'll back-numbers, so I'll back-solve in that one: solve in that one:

1. 21. 2xx + + yy = 9 = 9 3 3xx – – yy = 16 = 16

2. 22. 2xx + + yy = 9 = 9 33xx – – yy = 16 = 16

5x =255x =25

3. 2(5) + 3. 2(5) + yy = 9 = 9 10 + 10 + yy = 9 = 9 yy = –1 = –1

Linear systems-understanding Linear systems-understanding termsterms

1. inconsistent1. inconsistent

2. consistent2. consistent

3. dependent3. dependent

A system is A system is inconsistentinconsistent

if it has no solutionsif it has no solutions A system is A system is consistentconsistent if if there is at least one there is at least one

solutionsolution A system is dependent if A system is dependent if it has many solutionsit has many solutions

Factoring-methods and techniquesFactoring-methods and techniques

A. Factoring GCFA. Factoring GCF

B. Difference of squaresB. Difference of squares

C. Sum and difference of C. Sum and difference of cubescubes

D. Reverse of foilD. Reverse of foil

E. PSTE. PST

F. Factoring by grouping F. Factoring by grouping

In the next slides you In the next slides you would learn each.would learn each.

Factoring GCFFactoring GCF EXAMPLEEXAMPLE

these are the steps you'll these are the steps you'll need to go through.need to go through.

1.3x1.3x22 + 6x - 4x - 8 + 6x - 4x - 8

2. (3x2. (3x22 + 6x) - (4x + 8) + 6x) - (4x + 8)

3 3x (x + 2) - 4 (x + 2) 3 3x (x + 2) - 4 (x + 2)

4.(3x - 4) (x + 2)4.(3x - 4) (x + 2)

grouping is important grouping is important

pulling out the GCF pulling out the GCF will take one or two will take one or two timestimes

Difference of squares-binomialsDifference of squares-binomials

you must find out you must find out what is a common what is a common factorfactor

then make into then make into binomialsbinomials

You must watch You must watch squares in case squares in case answer might be answer might be primeprime

EXAMPLEEXAMPLE

1.a1.a22-b-b22

2.(a+b)(a-b)=a2.(a+b)(a-b)=a22-b-b22

Prime examplePrime example

EXAMPLEEXAMPLE

1.a1.a22+b+b22

Sum and difference of cubes-Sum and difference of cubes-binomialsbinomials

find differencefind difference

opposite product in opposite product in the middlethe middle

Use parenthesis very Use parenthesis very important.important.

EXAMPLEEXAMPLE

1. x1. x33 -8 -8

2.x2.x33 – 23 – 23

3. (x-2)(x2+2x+22)3. (x-2)(x2+2x+22)

4.(x-2)(x2+2x+4)4.(x-2)(x2+2x+4)

Reverse of foil-trinomialsReverse of foil-trinomials

Just do foil in reverseJust do foil in reverse

Trial and error it may Trial and error it may take you a couple of take you a couple of tries to find the tries to find the correct answer.correct answer.

EXAMPLEEXAMPLE

1.3x1.3x22 - 6x + x - 2 - 6x + x - 2

2.(3x+1)(x-2)2.(3x+1)(x-2)

PST-perfect square trinomialPST-perfect square trinomial

The first term and the last The first term and the last term will be perfect term will be perfect squares.squares.The coefficient of the The coefficient of the middle term will be middle term will be double the square root of double the square root of the last term multiplied by the last term multiplied by the square root of the the square root of the coefficient of the first coefficient of the first term.term.There will be many There will be many different problems that different problems that will be PSTwill be PST

EXAMPLEEXAMPLE

1.x1.x22 + 6x + 9 = 0 + 6x + 9 = 0

2.x2.x22 + 2(3)x + 32= 0 + 2(3)x + 32= 0

3.(x + 3)2 = 03.(x + 3)2 = 0

4. x+3=04. x+3=0

5.x=-35.x=-3

EXAMPLEEXAMPLE

(ax)(ax)22 + 2abx + b + 2abx + b22

Factoring by grouping-four or more Factoring by grouping-four or more itemsitems

remember it is a binomial remember it is a binomial and make sure you set and make sure you set problem up for globsproblem up for globs

the key is to find a the key is to find a common factor and keep common factor and keep factoring out the problemfactoring out the problem

EXAMPLEEXAMPLE

1. x1. x33-4x-4x22+3x-12+3x-12

2.x2.x33-4x-4x22+3x-12=x+3x-12=x22(x-(x-4)+3(x-4)4)+3(x-4)

3.(x-4)(x3.(x-4)(x22+3)+3)

FunctionsFunctions

A Function is a A Function is a correspondence between correspondence between two sets, the domain and two sets, the domain and the range, that assigns to the range, that assigns to each member of the each member of the domain exactly one domain exactly one member of the range. member of the range. Each member of the Each member of the range must be assigned range must be assigned to at least one member of to at least one member of the domain.the domain.

example of equation h(k)= x2 - 2x -2

Simplifying expressions with Simplifying expressions with exponentsexponents

You would use You would use properties when doing properties when doing this.this.The x6 means six copies of x multiplied together and the x5 means five copies of x multiplied together. So if I multiply those two expressions together, I will get eleven copies of x multiplied together.

xx66 × × xx55

x6 × x5 = (x6)(x5) = (xxxxxx)(xxxxx)

(6 times, and then 5 times)

= xxxxxxxxxxx (11 times)

= x11

Simplifying expressions with Simplifying expressions with exponents cont.exponents cont.

The exponent rules tell me to subtract the exponents. But let's The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the rules again. The " 6suppose that I've forgotten the rules again. The " 688 " means I have " means I have eight copies of 6 on top; the " 6eight copies of 6 on top; the " 655 " means I have five copies of 6 " means I have five copies of 6 underneath.underneath.

Then you would cancel out the top and bottom then you would have Then you would cancel out the top and bottom then you would have your simplified expression.your simplified expression.

Word problemsWord problemsIn three more years, In three more years, Jack's grandmother Jack's grandmother will be six times as will be six times as old as Jack was last old as Jack was last year. If Jack's present year. If Jack's present age is added to his age is added to his grandmother's grandmother's present age, the total present age, the total is 68. How old is each is 68. How old is each one now?one now?

Let 'g' be Jack's grandmother's current Let 'g' be Jack's grandmother's current ageageLet 'j' be Jack's grandmother's current Let 'j' be Jack's grandmother's current ageageIf Jack's present age is added to his If Jack's present age is added to his grandmother's present age, the total is grandmother's present age, the total is 6868j + g = 68j + g = 68In six more years, Jack's grandmother In six more years, Jack's grandmother will be six times as old as Jack was will be six times as old as Jack was last yearlast year(g+3) = 6 (j-1)(g+3) = 6 (j-1)If Jack's present age is added to his If Jack's present age is added to his grandmother's present age, the total is grandmother's present age, the total is 6868j+g=68j+g=68Solving both equations we get Jack's Solving both equations we get Jack's age (j) as 11 and Jack's grandmother's age (j) as 11 and Jack's grandmother's age (g) as 57age (g) as 57

Lines best fit or regressionLines best fit or regression

A Regression line is a line A Regression line is a line draw through and scatter-draw through and scatter-plot of two variables. The plot of two variables. The line is chosen so that it line is chosen so that it comes as close to the comes as close to the points as possible.points as possible.When asked to draw a When asked to draw a linear regression line or linear regression line or best-fit line, you have to best-fit line, you have to to draw a line through to draw a line through data point on a scatter data point on a scatter plot. In order to solve plot. In order to solve these problems a these problems a calculator will be neededcalculator will be needed

Lines best fit or Lines best fit or regressionregression

ConclusionConclusion

These slides should have gave you These slides should have gave you information on what we worked on during information on what we worked on during semester two and what you will have to semester two and what you will have to know for the test.know for the test.

Algebra 1-semester exam review By: Ricardo Blanco.

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