Algebra 1-semester exam review By: Ricardo Blanco

In the next slides you will review: The properties we learned What are they used for and when to recognize them Addition Property (of Equality) Multiplication Property (of Equality) Reflexive Property (of Equality) Symmetric Property (of Equality) Transitive Property (of Equality)

Properties 1.Addition Property (of Equality) 2. Multiplication Property (of Equality) Examples in order 1. if a= b, then a + c = b + c. is added to both sides of an equation, the two sides remain equal. That is, 2.if a= b, then a + c = b + c.. If the same number If a = b then a·c = b·c.

Properties 3. Reflexive Property (of Equality) 4. Symmetric Property (of Equality) 5. Transitive Property (of Equality) 3. a=a 4. if a=b then b=a 5. If a = b and b = c, then a = c.

In the next slides you will review Associative Property of Addition Associative Property of Multiplication Commutative Property of Addition Commutative Property of Multiplication Distributive Property (of Multiplication over Addition)

Properties 6. Associative Property of Addition 7. Associative Property of Multiplication 6. the sum does not change. (2 + 5) + 4 = 11 or 2 + (5 + 4) = answer will still not chage.(3 x 2) x 4 = 24 or 3 x (2 x 4) = 24.

Properties 8. Commutative Property of Addition 9. Commutative Property of Multiplication 8. As per the commutative property of addition, the expression = 19 can be written as = 19. so, = x 2 = 2 x 4

Properties 10. Distributive Property (of Multiplication over Addition) 10. 3( ) = 3(2) + 3(7) + (3)(-5) 3(4) = (4) = = = 12

In the next slides you will review Prop of Opposites or Inverse Property of Addition Prop of Reciprocals or Inverse Prop. of Multiplication Identity Property of Addition Identity Property of Multiplication

Properties 11. Prop of Opposites or Inverse Property of Addition 12. Prop of Reciprocals or Inverse Prop. of Multiplication 11. In other words, when you add a number to its additive inverse, the result is 0. Other terms that are synonymous with additive inverse are negative and opposite. a + (-a) = In other words, when you multiply a number by its multiplicative inverse the result is 1. A more common term used to indicate a multiplicative inverse is the reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is found by "flipping" a upside down. The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.

Properties 13. Identity Property of Addition 14. Identity Property of Multiplication 13. Identity property of addition states that the sum of zero and any number or variable is the number or variable itself = According to identity property of addition, the sum of a number and 0 is the number itself. 4 × 1 = 4

In the next slides you will review Multiplicative Property of Zero Closure Property of Addition Closure Property of Multiplication Product of Powers Property Power of a Product Property Power of a Power Property

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

Properties 18. Product of Powers Property 19. Power of a Product Property 20. Power of a Power Property 18.when you multiply powers having the same amount add the exponents. 72 × 76 (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) (7 × 7) × (7 × 7 × 7 × 7 × 7 × 7) 19. (3t)4 (3t)4 = 34 · t4 = 81t4 (3t)4 = 34 · t4 = 81t4 20. (ab)c = abc

In the next slides you will review Quotient of Powers Property Power of a Quotient Property Zero Power Property Negative Power Property zero product property

Properties 21. Quotient of Powers Property 22. Power of a Quotient Property 21. This property states that to divide powers having the same base, subtract the exponents. (am)n = amn 22. This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them.

Properties 23. Zero Power Property 24. Negative Power Property 23. If a variable has an exponent of zero, then it must equal one 3 0=1 24. 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

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

In the next slides you will review Product of Roots Property Quotient of Roots Property Root of a Power Property Power of a Root Property

Properties 26. Product of Roots Property 26. The product is the same as the product of square roots X =

Properties 27. Quotient of Roots Property 27. the quotient is the same as the quotient of the square roots

Properties 28. Root of a Power Property 29. Power of a Root Property

Property quiz Problems in which you determine the property. You will fill in the answer on the power point when finished go back through the properties to make sure you have the correct answers. when finished go back through the properties to make sure you have the correct answers A. if a= b, then a + c = b +c. B. a=a C. If a = b and b = c, then a = c. D. answer will still not chage.(3 x 2) x 4 = 24 or 3 x (2 x 4) = 24. E. 4 x 2 = 2 x 4

Solving1st power equations In the next slides you will see how to- A. with only one inequality sign B. conjunction C. disjunction

Solving1st power equations- with only one inequality sign This will only be true if x is equal to four The answer will be x > 4 Which on a number line is 6x = 24 6x > 24 6x > 24 x > 4 x > 4

Solving1st power equations- conjunction A conjunction is true only if both the statements in it are true A conjunction is a mathematical operator that returns an output of true if and only if all of its operands are true. -2 < x <= 4

Solving1st power equations- disjunction A disjunction is statement which connects two other statements using the word or. To solve a disjunctions of two open sentences, you find the variables for which at least one of the sentences is true. The graph consists of all points that are in the graph Ex. -3<x or x<4 Line where the lines

Linear equations in two variables Standard form Next determine whether or not the equations is linear or not. Next subtract 5x from both sides Ax + By = C y = 5x - 3 y = 5x - 3 5x + y = -3 This would be -5x + y = -3 it would become a straight line

Linear equations in two variables cont. A graphed linear equation

Linear systems A. substitution B. addition/subtraction C. check for understanding of terms- 1.dependent 2. inconsistent 3. consistent Solving equations in two variables Graphing points Standard/General Form Slope- Intercept Form Point-Slope Form Slopes

Linear systems-substitution 1.looks like it would be easy to solve for x, so we take it and isolate x: 2. Now that we have y, we still need to substitute back in to get x. We could substitute back into any of the previous equations, but notice that equation 3 is already conveniently solved for x: 3. answer is 1 1.2y + x = y + x = 3 3.x=3-2y x=3-2(1) x=3-2 x=1

Linear systems-add/sub (elimination) 1. Note that, if I add down, the y's will cancel out. So I'll draw an "equals" bar under the system, and add down: 2. Now I can divide through to solve for x = 5, and then back-solve, using either of the original equations, to find the value of y. The first equation has smaller numbers, so I'll back- solve in that one: 1. 2x + y = 9 3x – y = x + y = 9 3x – y = 16 5x =25 5x = (5) + y = y = 9 y = –1

Linear systems-understanding terms 1. inconsistent 2. consistent 3. dependent A system is inconsistent if it has no solutions A system is consistent if there is at least one solution A system is dependent if it has many solutions

Factoring-methods and techniques A. Factoring GCF B. Difference of squares C. Sum and difference of cubes D. Reverse of foil E. PST F. Factoring by grouping In the next slides you would learn each.

Factoring GCF EXAMPLE EXAMPLE these are the steps you'll need to go through. 1.3x 2 + 6x - 4x (3x 2 + 6x) - (4x + 8) 3 3x (x + 2) - 4 (x + 2) 4.(3x - 4) (x + 2) grouping is important pulling out the GCF will take one or two times

Difference of squares-binomials you must find out what is a common factor you must find out what is a common factor then make into binomials You must watch squares in case answer might be prime EXAMPLE 1.a 2 -b 2 2.(a+b)(a-b)=a 2 -b 2 Prime example EXAMPLE EXAMPLE 1.a 2 +b 2

Sum and difference of cubes- binomials find difference opposite product in the middle Use parenthesis very important. EXAMPLE 1. x x 3 – (x-2)(x2+2x+22) 4.(x-2)(x2+2x+4)

Reverse of foil-trinomials Just do foil in reverse Trial and error it may take you a couple of tries to find the correct answer. EXAMPLE 1.3x 2 - 6x + x (3x+1)(x-2)

PST-perfect square trinomial The first term and the last term will be perfect squares. The coefficient of the middle term will be double the square root of the last term multiplied by the square root of the coefficient of the first term. There will be many different problems that will be PST EXAMPLE 1.x 2 + 6x + 9 = 0 2.x 2 + 2(3)x + 32= 0 3.(x + 3)2 = 0 4. x+3=0 5.x=-3EXAMPLE (ax) 2 + 2abx + b 2

Factoring by grouping-four or more items remember it is a binomial and make sure you set problem up for globs the key is to find a common factor and keep factoring out the problem EXAMPLE 1. x 3 -4x 2 +3x-12 2.x 3 -4x 2 +3x-12=x 2 (x- 4)+3(x-4) 3.(x-4)(x 2 +3)

Functions A Function is a correspondence between two sets, the domain and the range, that assigns to each member of the domain exactly one member of the range. Each member of the range must be assigned to at least one member of the domain. example of equation h(k)= x 2 - 2x -2

Simplifying expressions with exponents You would use properties when doing 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. x 6 × x 5 x 6 × x 5 = (x 6 )(x 5 ) = (xxxxxx)(xxxxx) (6 times, and then 5 times) = xxxxxxxxxxx (11 times) = x 11

Simplifying expressions with exponents cont. The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the rules again. The " 6 8 " means I have eight copies of 6 on top; the " 6 5 " means I have five copies of 6 underneath. Then you would cancel out the top and bottom then you would have your simplified expression.

Word problems In three more years, Jack's grandmother will be six times as old as Jack was last year. If Jack's present age is added to his grandmother's present age, the total is 68. How old is each one now? Let 'g' be Jack's grandmother's current age Let 'j' be Jack's grandmother's current age If Jack's present age is added to his grandmother's present age, the total is 68 j + g = 68 In six more years, Jack's grandmother will be six times as old as Jack was last year (g+3) = 6 (j-1) If Jack's present age is added to his grandmother's present age, the total is 68 j+g=68 Solving both equations we get Jack's age (j) as 11 and Jack's grandmother's age (g) as 57

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

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