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Math Review

Math Review. By: Agustin Guerrero, and Peter Von Rueden. Math Review. This Math Review is review of all that we have done in the 2009-2010 school year. Every thing that the book talks about is in this PowerPoint.

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Math Review

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  1. Math Review By: Agustin Guerrero, and Peter Von Rueden

  2. Math Review • This Math Review is review of all that we have done in the 2009-2010 school year. • Every thing that the book talks about is in this PowerPoint. • The objective of this PowerPoint is to help you remember or relearn the material.

  3. Review of Factoring Methods • Factor GCF  for any # terms • Difference of Squares  binomials • Sum or Difference of Cubes  binomials • PST (Perfect Square Trinomial)  trinomials • Reverse of FOIL  trinomials • Factor by Grouping  usually for 4 or more terms

  4. Solving 1st power inequalities in one variable • X1 y1 equals 1 • ≤ , ≥ , <, > • X> 7 • X<0 • X>2 • 0<x<2 conjunction, x is greater than 2 and 1

  5. Addition Property (of equality)

  6. Multiplication Property (of Equality) • X/2=5 • (2)X/2=5(2) – Mult. Both sides by 2 • X=2- Final Answer

  7. Reflexive Property (of Equality) • Reflexive- Directed back on itself. Of, relating to, or being a verb having an identical subject and direct object • X=X, 2=2 • 2+2=2+2

  8. Symmetric Property (of Equality) • If A=B, Then B must equal A. • Examples. • A+B=4 , A=2 • 2+B=4 – If A equals 2 then B must equal 2 • 2+2=4 • Example 2. • 5+4 = 4+5 • X+2 = 2+X

  9. Transitive Property (of equality) • If A=B, and B=C, then A MUST equal C.

  10. Associative Property of Addition • For all real numbers A,B, and C. • (a+b)+c= a + (b+c) • Example: (5+6) + 7 = 5 + (6+7) • When you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result.

  11. Associative Property of Multiplication • (ab)c = a (bc) • Example 2: • (2 x 3) = 2(3 x 4) • When you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result.

  12. Commutative Property of Addition and Multiplication • For all real numbers A and B • A + B = B + A • AB = BA • Example 1: • 2 + 3 = 3 + 2 • Example 2: • 4 x 5 = 5 x 4

  13. Distributive Property • For all real numbers a, b, and c: • A( B + C) = AB + AC and (B + C) a= BA +CA • Example 1: • 5 x 83 = 5(80 + 3) • = (5 x 80) + ( 5 x 3) • = 400 + 15 • = 415answer

  14. Distributive Prop. Cont. • A( b-c) = ab - ac and (b-c) a = ba – ca • Example: • 8y – 6y • 8y – 6y = ( 8 – 6 )y • = 2y answer

  15. Prop. Of opposites or inverse Property of Addition • For every real number a, there is a unique real number –a such that; • a + (-a) = 0 and (-a) + a = 0

  16. Closure Property of Multiplication • For all real numbers a and b. • Ab= unique real number • There is one and only one possible answer when you multiply two real numbers • Example: • 5(4)= 20… correct • 5(4) ≠ 30 • 5(4) ≠ 10

  17. Power of a root the square root of x Ex. = 3 or 3*3*3= 27 9*3= 27 27=27.. true

  18. Root of a power • xa*xb= x a+b • Example: 22* 24= 26, add the exponents of a same base.

  19. Power of a Quotient • (a/b)m = am/ bm • The is distributed to both the numerator and the denominator.

  20. Prop of Reciprocals or Inverse prop. Of Multiplication • a/b ÷ c/d = a/b • d/c • Example: • 2/1 ÷ 2/1 = 2/1 • ½ = 2/2 or 1 • 1 = ½(2)

  21. Quadratic Formula • A polynomial equation to the second degree. • ax2 + bx + c = 0, and b2 – 4ac ≥ 0, then .. • Example 1: 9x2 + 12x- 1= 0 • X= -b ± √ b2 – 4ac ÷ 2(a) , where a= 9, b= 12, and c= -1 • X= -(12) ± √ (12)2 – 4(9)(-1) ÷ 2(9) = -12 ± √144+36÷ 18 • = -12 ± √180÷ 18= -12 ± √36 x 5 ÷ 18= -12 ± 6 √5 ÷ 18 • = 6(-2 ± √5) ÷ 18 = -2 ± √5 ÷ 3 • Since √5 ≈ 2.24, x≈ -2+ 2.24 ÷ 3= 0.24 ÷ 3≈ 0.1

  22. Linear Equations in two variables • A linear equation in two variables is of the form. • ax + by = c, « ---- Standard Form • where a ≠ 0, b ≠ 0 • Example 1: • 2x + 3y = 5 • X- 2y= 6 • -6x+y=8, A pair of values of x and y that satisfy a given linear equation in two variables is said to be its solution.

  23. Linear Equations in two variables (cont.) • Example 2: • x = 1, y = 1 is a solution of 2x + 3y = 5 • 2(1) + 3(1) = 5 Plug in the Numbers • x = -2, y = 3 is also a solution of 2x + 3y = 5 • 2(-2) + 3(3) = 5 • -4 + 9 = 5 • Similarly, we can find many more solutions for 2x + 3y = 5

  24. Linear Equations in Two variables (cont.) • Slope of a line: • Plot the points on a graph • Then find the slope using this equation m= y2-y1/ x2-x1 ( Rise over Run) • Example: Find the slope through; • (1,2) (3,5) • Plug in the numbers • M= 5-2 / 3-1= 3 / 2 «-------------------------- Final answer

  25. Linear Systems • Substitution Method- A method of solving a system of linear equations in two variable. • Substitution Method helps solve a system of linear equations in two variables. • 1. Solve one equation for one of the variables. • 2. Substitute this expression in the other equation and solve for the other variable. • 3. Substitute this value in the equation in step 1 and solve. • 4. Check the values in both equations. • Example 1: Next slide…….

  26. Linear Systems (cont.) • 2x-8y= 6 • X – 4y= 8 • X- 4y = 8 • x = 8 + 4y • 2x-8y = 6 • 2(8 + 4y) – 8y = 6 • 16 + 8y – 8y = 6 • 16= 6 False… The system has no solutions

  27. Linear Systems (cont.) • Addition/ Subtraction Method (Elimination) • Helps when solving a system of two equations, so that you can create a new equation with just one variable. • Example 1: • solve: 5x – y= 12 • 3x+ y= 4 • 1. Add similar terms of the two equations. 5x- y = 12 • 3x+y = 4y’s eliminated • 2. solve the resulting equation. X= 2 • 3. Substitute 2 for x in either of the original equations to find y.

  28. Linear Systems (cont.) • 3x + y = 4 • 3(2) + y = 4 • y = -2 • 4. Check x = 2 and y = -2 in both original equations. • 3x + y = 4 • 3(2) + (-2) = 4 • 4=4….. True equation

  29. Rational Expressions • An expression for a rational number. • 0= 0/1 , 7= 7/1, 5 2/3= 17/3 • Example: 3= 6/2= 12/4= -15/-5

  30. Functions • F(x) means “y”. Remember not all relations are functions. • The domain and the range that assigns to each member of the domain exactly one member of the range.

  31. Identity property of addition & Mult. • There is a unique real number 0 such that for every real number a, • A+ 0 = a and 0 + a = a • Multiplication – • There is a unique real number such that for every real number a, a*1= a and 1*a = a

  32. Multiplicative property of zero • Fore every real number a, a* 0 = 0 and 0*a= 0. • Example: • 1*0= 0 • 2 *0= 0

  33. Closure property of addition • Closure property of real number addition states that the sum of any two real numbers equals another real number. • Closure property of real number multiplication states that the product of any two real numbers equals another real number.

  34. Example 1: GCF 10x3 – 20x2 – 5x

  35. Example 1: GCF 10x3 – 20x2 – 5x 5x(2x2 – 4x – 1)

  36. Example 2: GCF 3(x + 1)3 – 6(x + 1)2 Hint: Remember that a “glob” can be part of your GCF. Do you see a parenthetical expression repeated here?

  37. Example 3:Difference of Squares 75x4 – 100y2 GCF first! 25(3x4 – 4y2) 25(3x2 – 4y) (3x2 + 4y) Recall these binomials are called conjugates.

  38. Factoring • (3x +3)+(6x+6) • 2x – 6 + 3x – 9 + -3 • Y2 + 8xy + 2y + 16x • (24 + 6) + (36 + 6x)

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