1. MATH 010
JIM DAWSON

2. 1.1 INTRODUCTION TO INTEGERS
This section is an introduction to:
Positive Integers
Negative Integers
Opposites
Additive Inverse
Absolute Value

3. 1.2 ADDING AND SUBTRACTING INTEGERS
If the signs are the same : ADD the absolute values and place the common sign in the answer.
6+7=13
- 13+(-5)= -18

4. If the signs are different : SUBTRACT the absolute value of the smaller number from the absolute value of the larger number. Place the sign of the larger number using absolute value in the answer.
14+(-6)= 8
-21+10= -11

5. 1.3 MULTIPICATION AND DIVISION OF INTEGERS
Determine the sign of the answer first:
Count the negative signs: even number of negative signs – the answer is positive
An odd number of negative signs- the answer is negative

6. 2.Multiply or Divide using the absolute values
-5 x (-6)=30
7 x (-4)= -28

7. 1.4 REVIEW OF FRACTIONS AND DECIMALS WITH SIGNS
The rules for sign are the same for fractions and decimals as they were for integers.

8. CONVERTING BETWEEN FRACTIONS, DECIMALS, AND PERCENTS
Change a percent to a decimal.
Move the decimal point TWO places from right to left.
Change a decimal to percent.
Move the decimal point TWO places from left to right.

9. FRACTION TO A PERCENT
Change the fraction to a decimal(numerator divided by denominator) and move the decimal point TWO places from left to right.

10. CHANGE A PERCENT WITH A FRACTION TO A FRACTION
Drop the % and multiply by 1 over 100.

12. EXPONENTIAL NOTATION AND SOLVING EXPONENTS
If the base is negative and does not have parentheses around it the sign of the answer is ALWAYS negative.
If the base is negative and has parentheses around it; look at the exponent to find the sign of the answer
Even numbered exponent: positive answer
Odd numbered exponent: negative answer

13. ORDER OF OPERATIONS AGREEMENT
Priority #1-GROUPING SYMBOLS
Priority #2- EXPONENTS
Priority #3- MULTIPLY AND DIVIDE AS THEY OCCUR FROM LEFT TO RIGHT
Priority #4- ADD AND SUBTRACT AS THEY OCCUR FROM LEFT TO RIGHT

14. TRANSLATE AND SIMPLIFY
The translation must be done first and then simplify using the rules learned previously in the chapter.
The answer must be in descending order.

15. 2.1 EVALUATING VARIABLE EXPRESSIONS
COMBINING LIKE TERMS
Add or subtract the terms with the same variable part
Place the answer in descending order
2a+3b-4a+7b=2a-4a+3b+7b
-2a+10b

16. 2.2 SIMPLIFYING VARIABLE EXPRESSIONS
Combining like terms
Combine the terms with the same variable part or the constants
-3a+7+5a-9=-3a+5a and7-9
2a-2

17. MULTIPLYING VARIABLE TERMS
Multiply the number parts and bring the variable into the answer.
-3x(5)=-3(5)=-15x
7(-4a)=-7(4)=-28a
(-2b)(-6)=-2(-6)=12b

18. MULTIPLYING VARIABLE TERMS
Multiply the number parts and bring the variable into the answer.
-6(-4a)=-6(-4)= 24a
(-5x)(-3)=(-5)(-3)=15x

19. APPLYING THE DISTRIBUTIVE PROPERTY
The Distributive Property is used to remove parentheses.
If the terms inside the parentheses are different, multiply the term on the outside by every term on the inside.
Place the answer in descending order.
3(2x-4)=3(2x) and 3(-4)=6x-12

20. If you cannot combine like terms inside the parentheses, multiply the outside term by each term inside the parentheses.
-3(4x+2)=-3(4x) and –3(2)
-12x-6
Place the answer in descending order

21. SIMPLIFYING A GENERAL VARIABLE EXPRESSION
Use the Distributive Property to remove parentheses and brackets
Combine like terms when possible
Place the answer in descending order
Be careful with sign!

22. 2.3 TRANSLATING VERBAL EXPRESSIONS
Memorize the expressions on p.67
Rules for Parentheses
Use parentheses to infer multiplication when needed
Use parentheses to separate two processes together not separated by a number or a variable

23. Use parentheses to separate a more than or less than phrase with a number and letter next to the phrase from any other phrase in the expression

24. EXAMPLES OF TRANSLATING VERBAL EXPRESSIONS
7 ADDED TO 3 LESS THAN A NUMBER
7+(n-3)
4 TIMES THE DIFFERENCE BETWEEN A NUMBER AND 4
4(n-4)
THE SUM OF 2 AND THE PRODUCT OF A NUMBER AND 9

25. EXAMPLES OF TRANSLATING
6 TIMES THE TOTAL OF A NUMBER AND 8
6(n+8)
5 INCREASED BY THE DIFFERENCE BETWEEN 10 TIMES a AND THREE
5+(10a-3)

26. TRANSLATE AND SIMPLIFY
Translate the verbal expression FIRST and then simplify using the rules that were applied earlier in the chapter.
The answer must be in descending order if the expression was able to be simplified.

27. DEFINING THE UNKNOWNS
In order to define an unknown quantity, assign a variable to that quantity, and then attempt to express other unknown quantities in terms of the same variable.
These are equations of one variable; therefore, the same variable must be used when defining any unknowns.

28. 3.1 SOLVING EQUATIONS OF THE FORM x+a=b
The Addition Property of Equations
The goal is to solve for the unknown
VARIABLE = CONSTANT
Find the number that is on the same side of the equation as the variable and use the opposite process on both sides of the equation to solve the unknown quantity.

29. SOLVING EQUATIONS USING THE ADDITION PROPERTY x+a=b
Find the number that is on the same side of the equation as x and use the opposite process to remove the number from the x side . The number to be removed is 4. It`s opposite is –4. The unknown can be solved by the following; x+4-4=12-4;x=8
You must do the same thing on both sides of the equation; -4 on both sides

30. SOLVE AN EQUATION OF THE FORM ax=b
Use the Multiplication Property of Equations to solve the unknown
Find the number that is on the same side of the equation as the unknown and multiply both sides by the reciprocal and the sign that is with the number.
Apply the Division Principle as a shortcut with integers and decimals when possible.

31. THE BASIC PERCENT EQUATION
Percent x Base = Amount
P x B =A
20% of what number is 30?
Translate and solve the verbal expression. Change the percent to a decimal or fraction.
0.20 x n = 30; Solve for n; 30 divided 0.20 = n; n = 150

32. MORE EXAMPLES OF THE BASIC PERCENT EQUATION
70 is what percent of 80?
Translate and solve. Change the answer to a percent.
70 = n x 80; 70 divided by 80 = which is 87.5%; n = 87.5%
What is 40% of 80?
Translate and solve. Change the percent to a decimal.
40% = 0.40;n = 0.40 x 80;n =32

33. 3.2 GENERAL EQUATION PART 1 Solve an equation of the form ax+b=c
The goal is to write the equation as variable=constant.
5x+6=26 ; solve for x by applying the Addition Property of equations to +6
5x+6-6=26-6
5x=20; divide both sides by 5; x=4
Check by replacing x with 4

34. 3.3 GENERAL EQUATION PART 2 To solve an equation of the form ax+b=cx+d
Apply the Addition Property of Equations twice and then the Multiplication Property of Equations to solve the unknown.
7a-5=2a-20; subtract 2a from both sides ; 5a-5=-20; add 5 to both sides; 5a=-15; divide both sides by 5; a=-3 and check.

35. 3.4 TRANSLATE AND SOLVE
Use the translation rules from chapter 2 and solve the equations of one variable.
Consecutive Integer Formulas
Consecutive Integers:n,n+1,n+2
Consecutive Even Integers; n,n+2,n+4
Consecutive Odd Integers; n,n+2,n+4

36. SUM OF TWO NUMBERS WORD PROBLEMS
Define the unknowns first.
Smaller number is x; Larger number is the sum minus x (the smaller number).
Translate and solve for the smaller number first and then the larger number. Each problem must have two answers and add to equal the original sum.

37. ADDITION AND SUBTRACTION OF POLYNOMIALS
Monomial- a polynomial of one term.
Binomial- a polynomial of two terms.
Trinomial- a polynomial of three terms.
Quadnomial- a polynomial of four terms
Descending Order- the exponents of the variable decrease from left to right in the answer.

38. 4.1 ADDING AND SUBTRACTING POLYNOMIALS
Addition of polynomials
Combine the like terms inside both sets of parentheses(same sign-ADD; different signs-SUBTRACT).
Subtraction of polynomials
Multiply each term in the second polynomial by –1(there is a minus sign in front of the parenthese) then combine the like terms in both polynomials.

39. 4.2 MULTIPLYING MONOMIALS
Multiply the coefficients and add the like variable exponents.
Simplifying powers of monomials
Distribute the outside exponent to each exponent in the monomial. Simplify the coefficient completely in the answer. This is the only time exponents are actually multiplied.

40. 4.3 MULTIPLICATION OF POLYNOMIALS
Monomial times a polynomial.
Multiply the monomial by applying the distributive property to each term inside the parentheses( the polynomial)
Multiplying two polynomials.
Apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial and then combine the like terms. Place the answer un descending order.

41. TO MULTIPLY TWO BINOMIALS
Use the FOIL method to multiply two binomials.
This is the simple application of the distributive property in an ordered method.
F0IL METHOD;F- first terms are to be multiplied;O- outside terms are multiplied; I-inside terms are multiplied;L-last terms are multiplied. Combine the middle two terms if possible.

42. MULTIPLYING BINOMIALS WITH SPECIAL PRODUCTS
The Sum and Difference of two terms.
Do FOIL; the middle two terms will cancel; the answer will be a binomial with a minus sign between the terms.
The Square of a binomial.
Do FOIL; the middle two terms will be the same so add them; the answer will be a trinomial.

43. 4.4 NEGATIVE EXPONENTS Division of monomials.
To divide two monomials with the same base, subtract the smaller exponent from the larger exponent.
Zero as an exponent.
If zero is the dominant exponent the answer is always 1.

44. RULES FOR SIMPLIFYING NEGATIVE EXPONENTS
The negative exponent must be made positive by moving it to the opposite place in the fraction. This may be done first in the problem, but especially in the answer.
If there is a like base in the numerator and denominator and both exponents are negative they must be switched and made positive; then use division rules to simplify.

45. MIORE RULES FOR NEGATIVE EXPONENTS
If the bases are the same and one of the exponents is negative and one is positive, move the negative exponent to the positive exponent and ADD the exponents together.
When multiplying negative exponents, combine the like base`s exponents together using sign rules for addition and subtraction. Make neg. exponents positive.

46. SCIENTIFIC NOTATION
In scientific notation, a number is expressed as the product of two factors, the first number must be a number between one and ten(use of a decimal point may be needed), and the other number a power of ten.
To find the exponent in a number greater than one, count the place values after the first number.

47. MORE ON SCIENTIFIC NOTATION
To write a decimal in scientific notation.
Place a decimal point after the first number in the decimal.
To write the power of ten, count the place values from the decimal point to the first number in the decimal, this is the exponent.

48. 4.5 DIVISION OF POLYNOMIALS
TO divide a polynomial by a monomial.
Divide each term of the polynomial(numerator) by the monomial.Simplify the expression.

49. TO DIVIDE POLYNOMIALS
The process for dividing polynomials is similar to the one for dividing whole numbers. The use of long division is the method.
Steps: Divide the like variable terms and place the answer in the quotient. Multiply the quotient by each term on the outside of the problem.

50. STEPS FOR DIVISION
Step 3 is to subtract the products( change the sign of the second term and combine the like terms).
The process starts over; divide, multiply, and subtract.
If there is a remainder, write it as a fraction.

51. 5.1 GREATEST COMMON FACTOR
Find the GCF of the coefficients which is the largest number the numbers are divisible by evenly.
Find the GCF of the variable parts by choosing the variable part with the smallest exponent, but the variables must be in common.

52. FACTORING BY GCF
Find the GCF of each term in the polynomial and write it outside a parentheses.
Divide each term in the polynomial by the GCF and write it inside the parentheses. This is factoring by GCF.

53. FACTOR BY GROUPING The polynomial must be a quadnomial(four terms).
Steps for factoring by grouping:
Group the first two terms and the second two terms with parentheses. The sign in front of the third term is not inside the parentheses.

54. STEPS FOR GROUPING
Find the GCF of each set of terms and factor it out.
To write the answer; write the common binomial factor once and combine the GCF`s into one binomial and check the sign of this binomial to make sure it is right.

55. 5.2 FACTOR BY EASY METHOD Factor out a GCF first, if possible.
Find the signs of the binomials and place the correct variable in each binomial.

56. EASY METHOD
Find the factors of the last term whose sum or difference equals the middle term. Write the correct factors in the binomials.

57. 5.3 TRIAL FACTORING Factor out a GCF first, if possible.
If the first term is 2 or greater, factor by trial factors.
Find the signs of the binomials using the same rules as easy method.

58. TRIAL FACTORING
Find the factors of the first and last terms and place them in a chart.
Do outer and inner FOIL with the factors. The answer must match the middle term. Write the factors in the correct binomials and check.

59. 5.4 SPECIAL FACTORING To factor the difference of two squares.
The problem must be a binomial with a negative sign.
The signs of the binomials will be (+) and (-).

60. THE DIFFERENCE OF TWO SQUARES
Find the perfect squares of both terms and set the up as the difference of two squares.
Write the terms twice, once in each binomial.

61. PERFECT- SQUARE TRINOMIALS
This method may be used as a shortcut to trial factoring.
Criteria:
Must be a trinomial with a (+) sign in front of the last term.

62. CRITERIA FOR SPECIAL FACTORING
The first and last terms must have perfect squares.
Multiply the perfect squares together twice and add them. The answer must match the middle term or factor by another method.

63. 5.5 SOLVING EQUATIONS BY FACTORING
The Principle of Zero Products states that if the product of two factors is zero, then at least one of the factors must be zero.
If a x b = 0, then a =0 or b =0.

64. QUADRATIC EQUATION
A Quadratic Equation is in standard form when the polynomial is in descending order AND equal to zero.
Factor and solve. Each problem will have two answers.

65. 6.1 TO SIMPLIFY A RATIONAL EXPRESSION
Factor the numerator and denominator.
Divide by the common factors.
Be careful with the sign of the simplified answer.

66. TO MULTIPLY RATIONAL EXPRESSIONS
Factor ALL numerators and denominators.
Divide by the common factors.
Multiply the numerators.
Multiply the denominators.

67. TO DIVIDE RATIONAL EXPRESSIONS
Change division to multiplication and invert the second fraction.
Follow the steps for multiplication to simplify the problem.
Be careful with the sign of the answer.(Multiplying and dividing)

68. 6.2 FINDING THE LCM Factor the denominators first. To find the LCM:
What is the greatest number of times a term(monomial) occurs or a set of terms(binomial) occurs?

69. ADDITION AND SUBTRACTION
Factor the denominators.
Find the LCM of the denominators and place them under one fraction bar.
Place the fractions in higher terms.(Divide and Multiply)

70. ADDITION AND SUBTRACTION STEPS
Combine like terms in the numerator.
Simplify the answer(factor and cancel).

71. 6.4 COMPLEX FRACTIONS
Find the LCM of the denominators of the fractions in the numerator and denominator.
Multiply the LCM by every term in the numerator and denominator.
Simplify the answer(factor and cancel).

72. 6.5SOLVING EQUATIONS WITH FRACTIONS
Find the LCM of the denominators.
Multiply the LCM by every term in the problem.
Solve and check( if any denominators equal zero the answer is NO SOLUTION).

73. 6.6 RATIO AND PROPORTION Cross multiply and solve for the unknown.
In the word problems, make sure that the rates are set up with like units on top and like units on the bottom.

74. 6.7 LITERAL EQUATIONS
A literal equation is an equation that has more than one variable.
Use the Addition and Multiplication Properties to help solve for one of the variables.

75. 10.1 RADICAL EXPRESSIONS
A square root of a number x is a number whose square is x.
The square root of 16 is 4.
The number 4 is considered the perfect square of 16.

76. PRODUCT PROPERTY
If the number under the radical does not have a perfect square, apply the Product Property of Square Roots.
Find the first number that has a perfect square that goes into the number evenly.

77. = 25= =
Find the square root of 36 which is 6 and leave 10 under the radical sign.

78. MORE ON THE PRODUCT PROPERTY
Write the perfect square on the outside of the radical and the number that does not have a perfect square on the inside the radical.

79. TO SIMPLIFY VARIABLE RADICAL EXPRESSIONS
If there is a variable under the radical, find the perfect square by dividing the exponent by 2 and write the answer.

80. 10.2 ADDITION AND SUBTRACTION OF RADICALS
Apply the Product Property to each term. If the terms under the radical are the same, combine the like terms outside the radical.

81. 10.3 MULTIPLICATION AND DIVISION OF RADICALS
Multiply the terms under the radicals and place the answer under one radical.
Apply the Product Property.

82. MULTIPLYING RADICALS
If there are parentheses then apply the Distributive Property or FOIL and then combine like terms if possible.
Conjugates are the sum and difference of two terms.

83. DIVISION OF RADICALS
Rewrite the radical expression as the quotient of the square roots.
Apply the Product to each term.
Simplify and write the answer.
The answer cannot have radical in the denominator.

84. DIVISION OF RADICALS
If the denominator has a radical, the process is called rationalizing the denominator.
Multiply the denominator by both the numerator and denominator.
Simplify what is left.

85. 7.1 GRAPHING
To graph the ordered pair (2,3), the 2 is plotted along the x-axis and 3 is plotted on the y-axis.
The origin is 0 on the graph.
2 is the x-coordinate and 3 is the y-coordinate.

86. RELATIONS AND FUNCTIONS
A relation is any set of ordered pairs.
The domain is the set of first coordinates.
The range is the set of second coordinates.

87. FUNCTIONS
A function is a relation in which no two ordered pairs that have the same first coordinate have different second coordinates.

88. GRAPHING EQUATIONS OF THE FORM Y=MX+B
In this section we learn to graph equations three different ways.
Define a set of three values for x and solve the equation, graph the three sets of ordered pairs and connect them with a straight line.

89. 9.2 ADDITION AND MULTIPLICATION PROPERTIES
To solve an inequality, apply the same properties that were used to solve equations of one variable.