1. Prerequisites: Fundamental Concepts of Algebra
Chapter P
Prerequisites: Fundamental Concepts of Algebra

2. P.1 Algebraic Expressions & Real Numbers

3. Objectives Evaluate algebraic expressions Use mathematical models
Find the intersection of 2 sets
Find the union of 2 sets
Recognize subsets of the real numbers
Use Inequality symbols
Evaluate absolute value
Use absolute value to express distance
Identify properties of the real numbers
Simplify algebraic expressions

4. Intersection of Sets What they have in common A = {all tall children}
B = {all girls}
A intersect B = {all tall girls}
All children that are girls AND are tall

5. Union of Sets Combination of everything in both sets
A = {all tall children}
B = {all girls}
A union B = {all girls OR tall children} = {all girls and all tall boys}

6. A group of biology majors are taking Biology I & Chem. I
A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is the intersection of the 3 groups?
Students in biology, chemistry, & physics.
Students in chemistry.
Students in calculus.
Students in physics.

7. Absolute Value │x│ represents the distance between x and zero
Distance is always a positive quantity, therefore going left or right x units results in a distance of x units
│x - 2│ represents the distance between x and 2
Distance is again always positive. (i.e. the distance between 2 and 3 is 1 and the distance between 2 and 1 is 1) │3 - 2│ = │1 - 2│ = 1

8. Real numbers are a field
Commutative (addition & multiplication)
Associative (addition & multiplication)
Identity (additive = 0 & multiplicative = 1)
Inverse (additive = -x & multiplicative = 1/x)
Distributive (multiplication over addition)
ALL these properties are useful when manipulating algebraic expressions & equations

9. P.2
Exponents and Scientific Notation

10. Objectives Use the product and quotient rules
Use the zero-exponential rule
Use the negative-exponent rule
Use the power rule
Find the power of a product
Find the power of a quotient
Simplify exponential expressions
Use scientific notation

11. Product & Quotient Rules

12. Example
Simplify:

13. Quotient Rule explains zero-exponent rule
Any real number divided by itself (except 0) equals 1
If x is any nonzero number & y is an exponent:

14. Quotient Rule Explains Negative Exponent Rule

15. Working with Negative Exponents
In general, expressions are not considered simplified when negative exponents are present.
A negative exponent in the denominator becomes positive when moved to the numerator
A negative exponent in the numerator becomes positive when moved to the denominator

16. Raising an Exponent to an Exponent (Power Rule)
Exponents are multiplied
WHY?

17. When is your expression simplified?
No negative exponents are present
Each base appears only once
No parentheses remain
Example:

18. Scientific Notation
What is it? A number greater than or equal to 1 & less than 10 (either pos. or neg.) multiplied by 10 raised to an exponent
Example:

19. Why Use Scientific Notation?
It allows us to express very large numbers or very small numbers in a more concise manner.
Diminishes the error in writing very large or small numbers by eliminating the need to have all zeros written. (easy to have one too many or too few zeros)

20. Converting into Scientific Notation

21. Rules of Thumb
Count decimal places you move to place the decimal to the right of one non-zero digit
Large numbers are represented by multiplying by ten raised to a positive exponent
Small numbers are represented by multiplying by ten raised to a negative exponent

22. P.3
Radicals & Rational Exponents

23. Objectives Evaluate square roots Simplify (nth root of nth power)
Use product & quotient rules to simplify square roots
Add & subtract square roots
Rationalize denominators
Evaluate & perform operations with higher roots
Understand & use rational exponents

24. Principal Square Root
It is true that 4 squared and (-4) squared both equal 16, BUT the principal square root of 16 is 4 NOT -4
By convention, the radical symbol represents the positive (or PRINCIPAL) square roots of a number, thus for real numbers, x, greater than or equal to 0:

25. Examples

26. Multiplying & Dividing with Radicals (Roots)
A product or quotient under a radical can be written as the product or quotient of separate radicals
Products or quotients involving square roots can be expressed as a single square root involving products or quotients under the radical

27. Adding & Subtracting Square Roots
ONLY when you’re taking the square root of the same number can you add or subtract square roots

28. Simplify Expressions, then Add/Subtract (if possible)

29. What is a conjugate?
Pairs of expressions that involve the sum & the difference of two terms
The conjugate of a+b is a-b
Why are we interested in conjugates?
When working with terms that involve square roots, the radicals are eliminated when multiplying conjugates

30. Multiplying conjugates which involve square roots

31. Expressions with radicals in the denominator are NOT simplified
Eliminate the radical from the denominator by multiplying by the numerator and the denominator by the conjugate of the denominator
Sometimes the result may not LOOK simpler!

32. Other Roots
The nth root of a number means “what number could you raise to the nth power to get your original number?”
You can take an odd root of a negative number or a positive number.
You can only take an even root of a positive number.

33. Rules for other roots
Add and subtract only same roots of same number (i.e. you can add cube roots of 3 but NOT cube roots of 3 and cube roots of 4)
Multiply & divide same roots following same rules as square roots

34. Subtract, if possible, & simplify:

35. Expressing roots as rational exponents
Any root can be expressed as a rational exponent, then rules of exponents apply

36. Expressions may involve exponents AND roots
If possible, it’s often easier to take the root first (the rational exponent), then raise the value to the other exponent

37. P.4
Polynomials

38. Objectives Understand the vocabulary of polynomials
Add & Subtract polynomials
Multiply polynomials
Use FOIL in polynomial multiplication
Use special products in polynomial multiplication
Perform operations with polynomials in several variables

39. A polynomial in x is many terms added or subtracted with each term involving a constant and x raised to a power.
Only same powers of x can be added/subtracted
When multiplying polynomials, the distributive property holds. (i.e. every term in one polynomial must be multiplied by every term in the other polynomial.

41. Special Products

42. P.5
Factoring Polynomials

43. Objectives Factor out the greatest common factor Factor by grouping
Factor trinomials
Factor difference of squares
Factor perfect square trinomials
Factor sum & difference of cubes
Use a general strategy for factoring
Factor expressions containing fractional & negative exponents

44. Factoring strategies FIRST: Look for greatest common factor
Group terms (if 4 or more) to find common terms between groups
If only 3 terms, rewrite into 4 terms by multiplying leading coefficient by the constant term (a times c), then rewrite bx as the sum of 2 terms whose product of their coefficients is ac (then group as in previous item)

45. EXAMPLE

46. Factor by Recognition Difference of Squares Difference or Sum of Cubes
Opposite signs cause all middle terms to cancel out

47. Factor Completely

48. P.6
Rational Expressions

49. Objectives Specify domain of a rational expression
Simplify rational expressions
Multiply rational expressions
Divide rational expressions
Add & subtract rational expressions
Simplify complex rational expressions
Simplify fractional expressions that occur in calculus
Rationalize numerators

50. Domain restrictions
No values can be substituted in for x that would create a zero denominator or a negative value under a positive root

51. Simplify rational expressions
Factor numerator and denominator to cancel common terms
Do NOT forget that the terms cancelled still were in the original expression, therefore must be considered when stating the domain

52. Adding & Subtracting Rational Expressions
Expressions MUST have a common denominator to be added/subtracted
Remember when creating a common denominator, both the numerator & denominator must be multiplied by the same term, otherwise the resulting expression will NOT be equivalent to the original

53. Simplify

54. Rational expressions that occur in calculus
To simplify this expression, you may have to rationalize the numerator (if f(x) involves a root)

55. P.7 Equations Objectives Solve linear equations in one variable
Solve linear equations containing fractions
Solve rational equations with variables in the denominators
Solve a formula for a variable
Solve equations involving absolute value

56. Objectives continued Solve quadratic equations by: a) Factoring
b) Using the square root property
c) Completing the square
d) Using the quadratic formula
(WHEN TO USE WHICH METHOD?)
Use discriminant to determine # & type of solutions
Solve application problems involving quadratics
Solve radical equations

57. What is a linear equation in one variable and what is “solving it”?
Only one variable (x or y, generally) is in the equation and it is NOT squared or raised to a power other than 1.
To “solve” the equation means to find the value (or values) that would make the equation true.

58. How do we solve an equation?
Eliminate parentheses (distribute!)
Collect like terms (additive identity)
Isolate the variable (multiplicative identity)
Remember: it’s an EQUATION to start with, meaning the left equals the right. It will no longer be equal, if something is done to one side and not the other!
CHECK your solution in the original equation: does it make it true?

59. EXAMPLE 4(2x-3) = 2(x+3) 1)Distribute to eliminate parentheses
2)Collect x’s on one side & constants on the other (use additive identity)
8x(-2x) – 12(+ 12) = 2x(-2x) + 6(+ 12)
6x = 18
3)Isolate the x (use multiplicative identity)
4) Check your solution in the original 4(2(3)-3) = 2(3+3)
4(3) = 2(6) YES!!

60. Rational Equations Equations that involve fractions!
The variable (x) could be in the numerator of the denominator. IF the x is found in the denominator, we must consider values x canNOT take on. (i.e. zero denominator)
EVEN after you’ve simplified an equation to eliminate the fractions, you haven’t eliminated the original restriction that may have been present.
With fractions, EITHER eliminate the fraction OR get a common denominator (if denominators are EQUAL, so are numerators)

61. Solve by getting a common denominator

62. Types of Equations
Conditional: True under certain conditions (could be one or several solutions)
Inconsistent: Inconsistencies between the 2 sides (never true – NO solutions)
Identity: One side of the equation is identical to the other (doesn’t matter what x is, infinitely many solutions)

63. Example
Solve 3x – 6 = 3(x – 2)
Notice, after distributing on the right, 3(x – 2) = 3x – 6
The left side is identical to the right. No matter what values you plug in for x, it will always be true.
The solution set is: {all reals}
THIS IS AN IDENTITY.

64. Example Solve: 4x – 8 = 4(x – 5) Distribute on right = 4x – 20
Think: Can 4 times a number minus 8 possibly equal 4 times the same number minus 20??? NO!!
If you continue to solve, you get:
0x = -12 (Can 0 times a number ever equal -12? NO!
INCONSISTENCIES!! Solution: { }

65. Quadratic equations
Various methods to solve different quadratics

66. What is a quadratic equation?

67. Zero-Product Rule
If the product of two or more numbers is zero, at least one of the numbers must equal zero!
If AB=0, then A=0 and/or B=0
One or both of the terms must equal zero
Why is this important?
It allows us an easy way to solve an equation, but FIRST make certain the expression is a product that equals zero.

68. A product involves FACTORS
(2x-3)(x+2)=0
2x – 3 is a factor of the expression, as is 2+x
Set each factor = 0
2x – 3 = 0, thus x = 3/2
x + 2 = 0, thus x = -2
SO, if EITHER x = 3/2 or x = -2, the original expression = 0
SO, solve by FACTORING if equation, once equal to 0, is FACTORABLE

69. Often, you must get expression into factored form FIRST:

70. Solving with square root property
When would you use this approach?
When one side of the equation is a perfect square
EXAMPLE:

71. Solve by Completing the Square
When can you use this method? ALWAYS
However, if the expression is factorable or is already a perfect square, those methods may be more desirable
HOW does it work?
If you don’t have a perfect square, you create one by adding a “well-chosen” zero (adding the same term to both sides)
Decide what to add by determining what additional term would create a perfect square

72. EXAMPLE

73. Completing the square generalized to any quadratic equation results in the quadratic formula.
When can you use it? ALWAYS. (However, it still may be easier to factor & use zero-factor property or take the square root if it’s already a perfect square.)

74. Solve:
x = -1, 3
x = 1, -3
x = 2,3
x = 2

75. What is the discriminant and why is it useful to us?
The discriminant is the part of the quadratic equation that is under the radical.
Based on what is under that radical, we can determine if our solution will be an integer (is what’s under there a perfect square?), an irrational (is what’s under there a positive number that is NOT a perfect square), or complex (is what’s under there a negative number?)

76. What if your equation involves a variable under a radical?
In order to eliminate an nth root, you must raise both sides of the equation to the nth power.
Be CERTAIN that you isolate the radical (have it on one side of the equation by itself) before you raise both sides to the nth power.

77. What if the variable is found under a radical twice in an equation?
Isolate one radical and raise both sides to the nth power.
Then, isolate the other radical (it will not have disappeared from the other side), and raise both sides to the nth power again.

78. What is x is raised to an exponent that is NOT an integer?
If the variable (or expression involving a variable) is raised to the (m/n) exponent, you must isolate that expression and then raise BOTH sides to the (n/m) power.
WHY?? When you raise one exponent to another, you multiply the 2 exponents.

79. What if the equation involves an expression inside absolute value brackets?
Recall what absolute value means: What is within those brackets could be positive or negative and still have the same overall value.

80. Solve:
No solution.
{7/3}
{10/3, 4/3}
{-7/3, 7/3}

81. P.8
Modeling with Equations

82. Objectives
Use equations to solve problems

83. Solving Word Problems 1) Carefully read the problem
2) Determine what do you know and what do you want to know
3) Identify variables
4) Develop equation relating what you know & what you want to know
5) Solve the equation & check (correct?)
6) Make certain you answered the question you were being asked!

84. EXAMPLE
You need to drive from Chicago to your cousin’s house in Omaha (a distance of 550 miles) at an average 65 mph on the Interstate highway. What time should you leave if you have to be at your cousin’s at 3:30 pm?

85. What is the relationship between known & unknown?
What do you want to know?
How long will it take you to drive? (x = time)
What time must you leave?
What do you know?
Total distance you’ll travel (550 miles)
Speed (65 miles per hour)
What is the relationship between known & unknown?
Distance = Rate x Time
550 miles = 65 mph x (X) (cont. on next slide)

86. Did you answer the question. NO – WHEN should you leave
Did you answer the question? NO – WHEN should you leave? In order to arrive at 3:30pm, you leave 8.5 hrs earlier, which would be at 8:00 am.

87. EXAMPLE
You have been asked to make an aluminum can (cylindrical shape) to hold 300 ml of your product. The can is to be 10 cm high. How much aluminum (in square cm) do you need?

88. -How much material you will need (surface area).
What do you know?
Can holds 300 ml (the volume!)
Height = 10 cm
What do you want to know?
-How much material you will need (surface area).
What relates the known & unknown?
For cylinders:

89. (example continued)

90. Now find surface area. (answer the question
Now find surface area! (answer the question!) (remember, a cylinder is just 2 circles and a rectangle)

91. Ava purchased a new ski jacket, on sale for $66. 50
Ava purchased a new ski jacket, on sale for $ The coat had been advertised as 30% off! What was the original cost?
$95
$86.50
$90
$96.50

92. P.9
Linear Inequalities and Absolute Value Inequalities

93. Objectives Use interval notation.
Find intersections & unions of intervals.
Solve linear inequalities.
Recognize inequalities with no solution or all numbers as solutions.
Solve compound inequalities.
Solve absolute value inequalities.

94. Linear inequalities
For equalities, you are finding specific values that will make your expression EQUAL something. For inequalities, you are looking for values that will make your expression LESS THAN (or equal to), or MORE THAN (or equal to) something.
In general, your solution set will involve an interval of values that will make the equation true, not just specific points.

95. What if you have more than one inequality?
If two inequalities are joined by the word “AND”, you are looking for values that will make BOTH true at the same time. (the INTERSECTION of the 2 sets)
If two inequalities are joined by the word “OR”, you are looking for values that will make one inequality OR the other true (not necessarily both), therefore it is the UNION of the 2 sets.

96. What IS an absolute value inequality?
Recall that absolute value refers to the expression inside the brackets being either positive or negative, therefore the absolute value inequality involves 2 separate inequalities
IF absolute value expression is LESS THAN a value, you’re looking for values that are WITHIN that distance (intersection of the 2 inequalities)
IF absolute value expression is MORE THAN a value, you’re looking for values that are getting further away in both directions (union of the 2 inequalities)

97. If the absolute value is greater than a number, you’re considering getting further away in both directions, therefore an OR. (get further away left OR right)
See next slide for example:

99. If, however, the absolute value was LESS than a number (think of this as a distance problem), you’re getting closer to your value and staying WITHIN a certain range. Therefore, this is an intersection problem (AND)
Same problem as before, but solved as a LESS than inequality. (next slide)

101. Don’t leave common sense at the door!
Remember to use logic!
Can an absolute value ever be less than or equal to a negative value?? NO! (therefore if such an inequality were presented, the solution would be the empty set)
Can an absolute value ever be more than or equal to a negative value?? YES! ALWAYS! (therefore if such an inequality were given, the solution would be all reals)