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Algebra 1 A Review and Summary Gabriel Grahek

Algebra 1 A Review and Summary Gabriel Grahek.

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Algebra 1 A Review and Summary Gabriel Grahek

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  1. Algebra 1 A Review and SummaryGabriel Grahek

  2. In the next slides you will review:Solving 1st power equations in one variableA. Special cases where variables cancel to get {all reals} or B. Equations containing fractional coefficientsC. Equations with variables in the denominator –(throw out answers that cause division by zero)

  3. 1st Power Equations • Any type of equation that has only one variable. x+3=8 4(x2-2)+8=36 x+3-3=8 (4x2-8)+8=36 x=5 4x2-8=28 4x2=36 x2=9 x=3 Note that the variable can be on both sides.

  4. 1st Power Equations 3x-4=12+x 4y+16=24 3x-4+4=12+4+x 4y+16-16=24-16 3x=16+x 4y=8 3x-x=16+x-x y=2 2x=16 x=8

  5. 1st Power Equations • Special cases- Ø and {all reals} x-(4-3)=x 4x+4=2(2x+2) x-1=x 4x=4x -1=0 x=x Ø

  6. 1st Power Equations • Equations containing fractional coefficients and with variables in the denominator.

  7. In the next slides you will review:Review all the Properties and then take a Quiz on identifying the Property Names

  8. Example:If a, b, and c, are any real numbers, and a=b, then a+c=b+c and c+a=c+bIf the same number is added to equal numbers, the sums are equal. Addition Property (of Equality) Multiplication Property (of Equality) Example:If a, b, and c are real numbers, and a=b, then ca=cb and ac=bc. If equal numbers are multiplied by the same number, the products are equal.

  9. Example:For all real numbers a, b, and c:a=a Reflexive Property (of Equality) Symmetric Property (of Equality) Example:For all real numbers a, b, and c:If a=b, then b=a. Transitive Property (of Equality) Example: For all real numbers a, b, and c:If a=b and b=c, then a=c.

  10. Example:For all real numbers a, b, and c:(a+b)+c=a+(b+c) Example: (5+6)+7=5+(6+7) Associative Property of Addition AssociativeProperty of Multiplication Example:For all real numbers a, b, and c: (ab)c=a(bc) Example:

  11. Commutative Property of Addition Example: For all real numbers a and b:a+b=b+a Example: 2+3=3+2 Commutative Property of Multiplication Example:For all real numbers a, b, and c:ab=ba Example:

  12. Distributive Property (of Multiplication over Addition) Example: For all real numbers a, b, and c:a(b+c)=ab+acand(b+c)a=ba+ca

  13. Prop of Opposites or Inverse Property of Addition Example:For every real number a, there is a real number -a such thata+(-a)=0 and (-a)+a=0 Prop of Reciprocals or Inverse Prop. of Multiplication Example: If we multiply a number times its reciprocal, it will equal one. For example:

  14. Identity Property of Addition Example: There is a unique real number 0 such that for every real number a, a + 0 = a and 0 + a = a Zero is called the identity element of addition. Identity Property of Multiplication Example: There is a unique real number 1 such that for every real number a, a · 1 = a and 1 · a = a One is called the identity element of multiplication.

  15. Example: For every real number a, a · 0 = 0 and 0 · a = 0 Multiplicative Property of Zero Closure Property of Addition Example:Closure property of real number addition states that the sum of any two real numbers equals another real number. Closure Property of Multiplication Example: Closure property of real number multiplication states that the product of any two real numbers equals another real number.

  16. Example: This property states that to multiply powers having the same base, add the exponents.That is, for a real number non-zero a and two integers m and n, am × an= am+n. Product of Powers Property Power of a Product Property Example:This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them. That is, for any two non-zero real numbers a and b and any integer m, (ab)m = am × bm. Power of a Power Property Example:This property states that the power of a power can be found by multiplying the exponents.That is, for a non-zero real number a and two integers m and n, (am)n = amn.

  17. Quotient of Powers Property Example: This property states that to divide powers having the same base, subtract the exponents.That is, for a non-zero real number a and two integers m and n, . Power of a Quotient Property Example: This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. That is, for any two non-zero real numbers a and b and any integer m,

  18. Zero Power Property Example: Any number raised to the zero power is equal to “1”. Negative Power Property Example: Change the number to its reciprocal.

  19. Zero Product Property Example: Zero - Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. If xy = 0, then x = 0 or y = 0.

  20. Product of Roots Property For all positive real numbers a and b, That is, the square root of the product is the same as the product of the square roots. Quotient of Roots Property For all positive real numbers a and b, b ≠ 0: The square root of the quotient is the same as the quotient of the square roots.

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

  22. 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. a + b = b + a Answer: Commutative Property (of Addition)

  23. 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. 2. am × an = am+n Answer: Product of Powers

  24. 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. 3. For every real number a, a · 0 = 0 and 0 · a = 0 Answer: Multiplicative Property of Zero

  25. 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. 4. the sum of any two real numbers equals another real number. Answer: Closure Property of Addition

  26. 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. 5. There is a unique real number 1 such that for every real number a, a · 1 = a and 1 · a = a Answer: Identity property of Multiplication

  27. 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. 6. Answer: Zero Power Property

  28. 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. 7. Answer: Quotient of Powers Property

  29. 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. 8. (ab)m = am Answer: Power of a Product

  30. 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. 9. Answer: Negative Power Property

  31. 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. 10. Answer: Prop of Reciprocals or Inverse Prop. of Multiplication

  32. 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. 11. (ab)c=a(bc) Answer: Associative Property of Multiplication

  33. Solving Inequalities

  34. -4 Solving Inequalities • Remember the Multiplication Property of Inequality! If you multiply or divide by a negative, you must reverse the inequality sign. -2x < 8 x > -4 Solution Set: {x: x > -4} Graph of the Solution:

  35. 8 -5 Solving Inequalities • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or≤ • Conjunction must satisfy both conditions • Conjunction = “AND” {x: -5 < x ≤ 8} Click to see solution graph

  36. 8 -6 Solving Inequalities • Open endpoint for these symbols: > < • Closed endpoint for these symbols: ≥ or≤ • Disjunction must satisfy either one or both of the conditions • Disjunction = “OR” {x: x < -6 or x ≥ 8} Click to see solution graph

  37. 8 -6 Solving Inequalities – Special Cases • Watch for special cases • No solutions that work: Answer is Ø • Every number works: Answer is {reals} • When the disjunction goes the same way you use one arrow. {x: x > -6 or x ≥ 8} Click to see solution graph

  38. Ø Solving Inequalities – Special Cases • Watch for special cases: • No solutions that work: Answer is Ø • Every number works: Answer is {reals} {x: -2x < -4 and -9x ≥ 18} Click to see solution

  39. 3 -2 Solving Inequalities • Now you try this problem 2x > 6 or -16x ≤ 32 Click to see solution and graph -2 < x or x ≤ 3

  40. 5 -6 Solving Inequalities • Now you try this problem. 4x-8 < 12 and -x < 10-4 Click to see solution and graph -6 < x < 5

  41. Type a sample problem here. Blah blah blah. You can duplicate this slide. Type the answer here. Set to fade-in on click Click when ready to see the answer. Type any needed explanation or tips here. Set to fade-in 3 seconds after the answer appears above.

  42. 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.

  43. Linear Equations • Slope= • Point-Slope Formula= • Slope-Intercept Formula= • Midpoint Formula= • Standard/ General Form= Ax+Bx=C • Distance Between Two Points Formula=

  44. Slope Pt-Slope Formula (9,12) and (13, 20) Use when you only have solution points. • (9,12) and (13, 20) • Would be negative if it had a negative sign in front of it. It would then be a falling line and not a rising line.

  45. Midpoint Distance (9,12) and (13, 20) Use to find the Distance between to points. • (9,12) and (13, 20) • Use to find the middle point on a line.

  46. Equations in Two Variables • The pairs of numbers that come out for each variable can be written as an (x,y) value. (ordered pair) • You give the solutions in alphabetical order of the variables. So, it would be (a,b) and not (b,a).

  47. Standard Form • ax+by=c • All linear equations can be written in this form. • A, b, and c are real numbers and a and b are non-zero. A, b, and c are integers. • To change to slope intercept: Ax+bx=c bx=ax+c

  48. How to graph • To graph the slope-intercept form: you can take the y intercept and use the slope to determine the points on the line. • To graph the standard form you have to change it to slope-intercept, explained in the last slide, and then graph it.

  49. To find the x-intercept To find the y-intercept F(x)=mx+b Set f(x)=0 0=mx+b Divide out Example: F(x)=4x-8 0=4x-8 8=4x 2=x • F(x)=mx+b Set x to 0 F=b Example: F(x)=4x+6 F(0)=4(0)+6 F=6

  50. 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

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