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Algebra 1 Review Casey Andreski Bryce Lein

Algebra 1 Review Casey Andreski Bryce Lein.

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Algebra 1 Review Casey Andreski Bryce Lein

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  1. Algebra 1 Review Casey Andreski Bryce Lein

  2. In the next slides you will review:Solving 1st power equations in one variableA. Don't forget special cases where variables cancel to get {all reals} or B. Equations containing fractional coefficientsC. Equations with variables in the denominator – remember to throw out answers that cause division by zero

  3. Special cases Cancel variables 3x+2=3(x-1) distribute 3x+2=3x-3 subtract 3x 2=-3 finished

  4. Fractional Coefficient • 1/2x - 3 + 1/3x = 2 multiply by a common denominator • 3x - 18 + 2x = 12 add like terms • 5x = 40 divide by 5 • X = 8 finished

  5. Variables in the denominator • 5/x + 3/4 = 1/2 Multiply by a common denominator • 5 + 3/4x = 1/2x group like terms • 5 = -3/4x + 2/4x add like terms • 5 = -1/4x multiply by common denominator • -20 = x

  6. Properties

  7. Example: a + c = b + c Addition Property (of Equality) Multiplication Property (of Equality) Example: If  a = b  then  a x c = b x c.

  8. Example: a = a Reflexive Property (of Equality) Symmetric Property (of Equality) Example: a = b then b = a Transitive Property (of Equality) Example: If a = b and b = c, then a = c

  9. Example: a + (b + c) = (a + b) + c Associative Property of Addition Associative Property of Multiplication Example: a x (b x c) = (a x b) x c

  10. Example: a + b = b + a Commutative Property of Addition Commutative Property of Multiplication Example: a x b = b x a

  11. Distributive Property (of Multiplication over Addition Example: a x (b + c) = a x b + a x c

  12. Example: a + (-a) = 0 Prop of Opposites or Inverse Property of Addition Prop of Reciprocals or Inverse Prop. of Multiplication Example: (b)1/b=1

  13. Example: y + 0 = y Identity Property of Addition Identity Property of Multiplication Example: b x 1= b

  14. Example: a x 0 = 0 Multiplicative Property of Zero Closure Property of Addition Example: 2 + 5 = 7 Closure Property of Multiplication Example: 4 x 5 = 20

  15. Example: 42 x 44 = 46 Product of Powers Property Power of a Product Property Example: (2b)3 = 23 x b3 = 8b3

  16. Example: 54/53 = 625/125 or 54-3 = 51 = 5 Quotient of Powers Property Power of a Quotient Property Example: (4/2)2 = 42/22 = 4

  17. Example: a0 = 1 Zero Power Property Negative Power Property Example: a-6 = 1/a6

  18. Zero Product Property Example: If ab = 0 , then either a = 0 or b = 0.

  19. Product of Roots Property Quotient of Roots Property

  20. Example: Root of a Power Property Power of a Root Property Example:

  21. Now you will take a quiz!Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 1. 14 + 3 = 3 + 14 Answer: Commutative Property (of Addition)17 = 17

  22. In the next slides you will review:Solving 1st power inequalities in one variable. (Don't forget the special cases of {all reals} and )A. With only one inequality signB. ConjunctionC. Disjunction

  23. With only one inequality sign 3 + x < 3 + 2 Click when ready to see the answerer X < 2 2

  24. 3+5<1+x>-2-1 Conjunction Click when you’re ready to see the answer. 8<1+x>-2-1 7<x>-4 -4 7

  25. Disjunction 3x>(14+4) or x<3-4 Click to see the answer 3x>18 or x<-1 X>6 -1 6

  26. In the next slides you will review:Linear equations in two variables Lots to cover here: slopes of all types of lines; equations of all types of lines, standard/general form, point-slope form, how to graph, how to find intercepts, how and when to use the point-slope formula, etc. Remember you can make lovely graphs in Geometer's Sketchpad and copy and paste them into PPT.

  27. Slope Finding the slope with 2 given points m = Slope Example: (9,-3) (6,2) 2-9 -7 6+3 9 Click for an example

  28. Equations of Lines Slope intercept form- Y = Mx + B Standard form – Ax + By = C Point slope form- Y – Y1 = M (X – X1)

  29. Graphing Lines Point Slope- use this when you only have 2 points. First : find the slope Next put the equation into point slope form: y-y1=m(x-x1) Example: (3,5) (2,1) Slope: = 4 Y-5=4(x-3) = y-5=4x-12 = y=4x-7

  30. Graphing Lines Slope intercept - y=-3x+7 7= y intercept -3 = slope

  31. Graphing Lines Standard form - 3x + 2y = 6 Set x to zero to find y Set y to zero to find x Points : (2,0) (0,3)

  32. In the next slides you will review:Linear Systems A. Substitution Method B. Addition/Subtraction Method (Elimination ) C. Check for understanding of the terms dependent, inconsistent and consistent

  33. Substitution Method 4x-5y=12 Y=2x-8 Put (2x-8) in for y for the top equation Click for solution 4x-5(2x-8)=12 Distribute 4x-10x+40=12 add/subtract common terms -6x=28 Divide X= -3/14

  34. Addition/Subtraction Method (Elimination ) 3x+5y=7 2x-4y=5 Multiply both equations to get either x or y to cancel 2(3x+5y)=7 = 6x+10y=14 3(2x-4y)=5 = 6x-12y=15 Subtract 22y=-1 Divide by 22 y= -1/22

  35. Terms Dependent- both same line (Infinite solutions) Inconsistent- parallel lines (No solutions) Consistent- Intersecting lines (One solution)

  36. In the next slides you will review:Factoring – since we just completed the Inspiration Project on this topic, just summarize all the factoring methods quickly. Note that you will be using your factoring methods in areas 7 & 8 below so no need to include extra practice problems here.

  37. Factoring Binomials difference of squares 49x4-9y2 (7x2+3y) (7x2-3y) sum and diff of squares a3-27 (a-3) (a2+3a+9) click for answers

  38. Factoring Trinomials GCF 2b+4b2+8b 2b(1+2b+4) Reverse foil x2+5x+6 (x+3) (x+2) PST 4x2-20x+25 (2x-5)2 Click for answers

  39. 4 or More Click for answers 3 by [(x1 x2+8x+16-3y2 (x+4)2-3y2 [(x+4)-3y] +4)-3y] 2 by 2 c3+bc+2c2+2b c2(c+2)+b(c+2) (c2+b) (c+2)

  40. In the next slides you will review:Rational expressions – try to use all your factoring methods somewhere in these practice problems A. Simplify by factor and cancel B. Addition and subtraction of rational expressions C. Multiplication and division of rational expressions

  41. Factor and Cancel =

  42. Addition and subtraction of rational expressions Click to see steps

  43. Multiplication and division of rational expressions Click to see answer Division is multiplication of the reciprocal

  44. In the next slides you will review:Functions A. What does f(x) mean? Are all relations function? B. Find the domain and range of a function. C. Given two ordered pairs of data, find a linear function that contains those points. D. Quadratic functions – explain everything we know about how to graph a parabola

  45. Functions f(x) means that f is a function of x All functions are relations but not all relations are functions A function is 1 to 1 which means for each input there is exactly one output

  46. Functions Domain- Set of inputs Range- Set of outputs f(x)=2x-1 Domain – all real numbers Range – all real numbers

  47. Functions (1,1) and (0,-1) Are two ordered pairs of the linear function f(x)=2x-1

  48. Quadratic functions f(x)=ax2+bx+c Vertex x= , then solve for f(x) X-intercepts set f(x) equal to zero factor and solve for x y-intercepts Set x to zero and solve for f(x) line of symmetry the line of

  49. In the next slides you will review:Simplifying expressions with exponents – try to use all the power properties and don't forget zero and negative powers.

  50. Exponents Property #1x0 = 1 Example: 40 = 1 and (2500000000000000000000)0 = 1 Property #2xn × xm = xn + mExample: 46 × 45 = 46 + 5 = 411Property #3xn ÷ xm = xn − mExample: 46 ÷ 45 = 46 − 5 = 41Property #4(xn)m = xn × m(52)4 = 52 × 4 = 58 Property #5(x × y)n = xn × yn(6 × 7)5 = 65 × 7 Property #6x-n = 1 ÷(xn) = 1/(xn)8-4 = 1 ÷ (84) = 1 / (84)Property #7(x/y)n = xn / yn(8/5)4 = 84 / 54 Property #8 www.basic-mathematics.com

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