//Index Terms -- to be used in searches // MISCELLANEOUS //SETS //NUMBERS //ABSOLUTE VALUE //OPERATIONS var sequence_test= 'NOW is the time for all good men to come to the ' +' matrix' +' aid of their country.'; //OPERATIONS var unary_operations= '

Each of the following is a unary operation.

' +'

Forming the opposite of a number.
Forming the reciprocal of a number.
Calculating the square root of a non-negative real number.
Calculating the absolute value of an algebraic expression.
Raising a real number to the fifth power.

'; var binary_operations= '

Each of the following is a binary operation.

' +'

Addition
Subtraction
Multiplication
Division

'; var binary_relations= '

Each of the following is a binary relation.

' +'

Equal
Less than
Greater than

'; var discussion_objects_relations_operations= '

One of the things we do in mathematices is use deductive reasoning to study ' +'relations and operations on mathematical objects. It will help your understanding ' +'if you identify each "thing" as either a relation, an operation, or a mathematical ' +'object. Try to make that identification as soon as you are introduced to a new ' +'mathematical "thing".

' +'

Some familiar mathematical objects are numbers, variables, algebraic expressions, ' +'formulas, and geometric figures.

' +'

Examples of unary operations, binary operations, and binary relations ' +'that are of interest in this course are presented elsewhere. You will be introduced ' +'to additional objects, relations, and operations as the course progresses.

'; //NUMBERS var prime_number= '

Each of the following is a prime number.

' +'

2 3 5 7 11 13 17 19 23 29 31 37 41

' +'Each of the following is not a prime number.

' +'

0 1 4 6 8 9 10 12 14 15 16 18 -4 -5 -3 -29 three over four thirteen over two square root of two square root of five

'; var composite_number= '

Each of the following is a composite number.

' +'

4 6 8 10 12 14 16 18 20 22 24 26 28

' +'Each of the following is not a composite number.

' +'

0 1 2 3 5 7 11 13 17 19 23 29 31 37 41 -4 -5 -3 -29 three over four thirteen over two square root of two square root of five

'; var rational_number= '

Each of the following is a rational number.

' +'

0 1 2 3 4 5 6 7 8 10 568749 -4 -5 -3 -29 -98762 three over four thirteen over two seven over eight cube root of 8 square root of 25

' +'Each of the following is not a rational number.

' +'

square root of two square root of five the irrational number pi cube root of seven cube root of four square toot of five divided by square root of 2 square root of four sevens

'; var irrational_number= 'Each of the following is an irrational number.

' +'

square root of two square root of five the irrational number pi cube root of seven cube root of four square toot of five divided by square root of 2 square root of four sevens

' +'

Each of the following is not an irrational number.

' +'

0 1 2 3 4 5 6 7 8 10 568749 -4 -5 -3 -29 -98762 three over four thirteen over two seven over eight cube root of 8 square root of 25

'; var real_number= 'Each of the following is a real number.

' +'

square root of two square root of five the irrational number pi cube root of seven cube root of four square toot of five divided by square root of 2 square root of four sevens

' +'

0 1 2 3 4 5 6 7 8 10 568749 -4 -5 -3 -29 -98762 three over four thirteen over two seven over eight cube root of 8 square root of 25

'; var example_distributive_property= '

The distributive property tells us (without calculations) that ' +'3(2 + 9) = (3)(2) + (3)(9). This is easily verified by ' +'observing that the left side of the equation is the product of 3 and 11 which is 33 ' +'and the right side of the equation is the sum of 6 and 29 which is 33.

' +'

The distributive property occasionally makes mental arithmetic easier. Suppose ' +'you want to determine the product of 5 and 27. Think of 27 as 20 + 7, so that (5)(27) ' +'becomes(5)(20) + (5)(7) which is easily seen to be 100 + 35 so the product of 5 and 27 is 135.

' +'

Keep in mind that equality works both ways. That is\; if x equals y then y equals x. ' +'Applying that fact to the Distributive Property sometimes makes it possible to write ' +'a sum as a product. That process is called factoring.
For example observe that ' +'35 + 49 = 7(5 + 7) which is easily seen to be (7)(12) or 84. In this case the ' +'Distributive Property again simplified some computations.

' +'

The Distributive Property is frequently used to factor expressions like ' +'3x² +15x + 18 as 3(x² +5x + 6)

' +'

The Distributive Property is also the basis for multiplication of a term times a ' +'polynomial. For example ' +'3x(4x⁴ + 5x² + 5) = 12x⁵ + 15x³ + 15x.' +'

'; var example_transitive_property= '

In simple words, if two things are equal to a third thing, then they are equal to one ' +'another.

' +'

As an example, if we know that 3x - 5 = 2y + 3 and that 2y + 3 = 5z - 1 ' +'then we can conclude that 3x - 5 = 5z - 1.

' +'

Here is a simple illustration. Suppose I know that the weight of my math book ' +'is the same as the weight of my history book, and that the weight of my history book ' +'is the same as the weight of two English books. Then the transitive property tells me ' +'that the weight of my math book is the same as the weight of two English books.

' +'

The Transitive Property is fundamental to the process of creating a mathematical model for ' +'so called word problems or real life applications of mathematics. The following are ' +'examples of that very important application of the Transitive Property. They are plucked directly from some word problems.

' +'

The distance D between two cars is given by D = 67.5t - 54.8t.

' +'The distance D between two cars is known to be D = 7.
' +'The Transitive Property gives the equation 67.5t - 54.8t = 7 as the mathematical model.

' +'

The volume of water in the tank is V = (12)(3)(h) cu.ft.
' +'The volume of water in the tank is V = 70 gal. = 9.3576 cu.ft.
' +'The Transitive Property gives the equation (12)(3)(h) = 9.3576 as the mathematical model.

' +'

The total amount of concentrate is x + (.4)(55 – x).
' +'The total amount of concentrate is (.75)(55).
' +'The Transitive Property gives the equation ' +'x + (.4)(55 – x) = (.75)(55) as the mathematical model.

' +'

The amount of copper in the final alloy will be (0.5)(x + 62).
' +'The amount of copper in the final alloy will be 0.75x +22.75.
' +'The Transitive Property gives the equation (0.5)(x + 62) = 0.75x +22.75 as the mathematical model.

' +'

'; var example_zero_factor_property= '

The Zero Factor Property is an obvious truth about the real numbers. It states that ' +'if the product of two numbers is zero, then at least one of the factors is zero. ' +'The fact that it is such an obvious fact may, in fact, be a reason for not thinking ' +'of it as a useful tool. However, it is an extremely powerful tool frequently used ' +'during the process of solving equations.

' +'

The following examples illustrate that ' +'application of The Zero Factor Property. If they are not familiar at this time, ' +'don\'t fret, you will get lots of practice with this type of exercise.

' +'

If (x - 3)(x + 5) = 0 then

by The Zero Factor ' +'Property
x - 3 = 0 OR x + 5 = 0

' +'

If (2x - 3)(x + 7)(x - 5) = 0 then
by The Zero Factor ' +'Property
2x - 3 = 0 OR x + 7 = 0 OR ' +'x + 5 = 0

'; var example_law_of_trichotomy= '

Just like The Zero Factor Property, the Law of Trichotomy (when applied strictly to real ' +'numbers) is pretty obvious and may easily be overlooked when its application is ' +'appropriate.

' +'

Just as with The Zero Factor Property, the really significant applications of The Law ' +'of Trichotomy come into play in Chapter 2 and later in the course. When The Law of ' +'Trichotomy is applied to things like equations and inequalities we obtain some very ' +'helpful ideas as hinted in the following examples.

' +'

When we consider an equation like 3x + 5 = 7, the two ' +'corresponding inequalities 3x + 5 < 7 and ' +'3x + 5 > 7 are lurking in the background just begging for ' +'attention. Moreover, their solutions sets are nicely related via The Law Of Trichotomy.

' +'

When we consider an equation like ' +'x³ + 5x² - 7x = 9 , the two ' +'corresponding inequalities x³ + 5x² - 7x < 9 ' +'and x³ + 5x² - 7x > 9 ' +'are lurking in the background just begging for attention. ' +'Moreover, their solution sets are nicely related via The Law Of Trichotomy.

' +'

When we consider any equation or inequality the other corresponding equation and/or ' +'inequalities are also in consideration. And their solution sets are nicely related via The Law Of Trichotomy

' +'

When we consider an equation like y = 3x + 5 , the two ' +'corresponding inequalities y < 3x + 5 and ' +'y > 3x + 5 are lurking in the background just begging for ' +'attention. Moreover, their solutions sets are closely related via The Law Of Trichotomy.

' +'

'; // MISCELLANEOUS var forgotten_terms= 'Here are some important but forgotten terms from earlier math classes. They are ' +'used frequently to explain algebraic concepts and operations.

' +' sum and addend

' +'

' +'

' +'

'; var example_opposite= '

The opposite of 4 is -4.

The opposite of -4 is 4.
4 and -4 are opposites of each other.

' +'

The opposite of is -.
The opposite of - is .
and - are opposites of each other.

' +'

The opposite of - is .

The opposite of is -.
and - are opposites of each other.

' +'
The opposite of 0 is 0.
0 is it own opposite.
0 is the only real number which is its own opposite.'; var example_variable= '

Each of the following are examples of variables. Notice that sometimes we use subscripts on variables.

' +'

a A b B c C d D P z x t ζ θ φ Φ ω Σ σ ' +'λ ε μ ξ α β γ a₁ a₃₄ a₅ ' +'a_i a_j a_k b₂ b₇ b_i x₃ x₆ x_j M_k

' +'

The following are examples of things which are NOT variables.

' +'

1 4 -8 43 1/2 3x+2=0 5x xy 2x-7y -x -b -3w x³ ' +'a^-3 3y⁷ 9λ⁵

' +'

Here are a few, more involved, examples, of mathematical things which are not variables.

' +'

' +'

The greek letter pi is in fact a letter and therefore, according to this ' +'definition of variable, could be included in the list of examples of variables. ' +'However, this greek letter has become synonomous with the irrational number obtained ' +'when the circumference of a circle is divided by its diameter. Therefore it is unwise ' +'to think of greek letter pi ' +'as a variable but rather ' +'should be considered to be a constant.

' +'

'; var example_algebraic_expression= '

Each of the following are examples of algebraic expressions. Notice each of the ' +'examples in the first list are variables. The definition of algebraic expression states ' +'that variables are algebraic expressions. Therefore each thing in the first list is an ' +'algebraic expression.

' +'

a A b B c C d D P z x t ζ θ φ Φ ω Σ σ ' +'λ ε μ ξ α β γ a₁ a₃₄ a₅ ' +'a_i a_j a_k b₂ b₇ b_i x₃ x₆ x_j M_k

' +'

The following are examples of things which are algebraic expressions. Notice that none of these algebraic expressions are variables.

' +'

1 4 -8 43 1/2 5x xy 2x-7y -x -b -3w x³ ' +'a^-3 3y⁷ 9λ⁵

' +'

Here are a few, more involved, examples, of mathematical things which are algebraic expressions.

' +'

' +'

You should now realize that every number is an algebraic expression but not all algebraic expressions are numbers.

' +'You should also realize that every variable is an algebraic expression but not all algebraic expressions are variables.
' +'You should know that no variable is a number and no number is a variable.

' +'

In summary, ' +'

The set of numbers is a proper subset of the set of rational expressions.
The set of variables is a proper subset of the set of rational expressions.
The intersection of the set of numbers and the set of variables is the empty set.

' +'

' +'

'; //SETS var set_and_element= '

The collection of all ' +' natural numbers' +' is a set and each natural number is an element of that set.
The collection of all dogs in Arnold, MO. is a set and each dog in Arnold is an element of that set.
The collection of letters a, d, c, m, and p is a set and each of those letters is an element of that set.
The collection of all ' +' real numbers' +' less than 14 and greater than or equal to 12 is a set and each of the real numbers which satisfies the stated condition, is an element of the set.
The collection of students at STLCC is a set and each student at STLCC is an element of that set.

' +'A set is something like a container with elements inside it. Suppose you have a clear ' +'plastic bag containing an apple, an orange, and a lemon. That bag with the three fruits ' +'inside is like a set. The apple is an element of the set. The lemon is an element of ' +'the set. The orange is an element of the set. Observe that if the bag contains no fruit, ' +'we say the bag is empty. In the same way a set may contain no elements and it is then ' +'called the empty set.' +' end of example

'; var roster_method= 'Each of the following is an example of the roster method for specifying (or defining) ' +'a set.' +'

{2, 4, 6, 8, 10} is a set whose elements are the numbers between the braces.
{a, h, k, z, p, 7, , } is a set whose elements are some letters, two numbers, and an icon.
{Tom, Steve, Mike} is a set whose elements are three names.
{, , , 43, -81} is a set whose elements are the things between the braces.
{3x=4, y=9, 12x²+3x - 5 = 11, 14 = 8, , } is a set whose elements are a collection of equations.

' +'It is common to use the ellipsis … to extend the roster method to specify certain infinite sets.' +'

{1, 2, 3, 4, …} is the set of ' +' natural numbers.
{…, -3, -2, -1, 0, 1, 2, 3, …} is the set of ' +' integers.
{1, 3, 5, 7, 9, 11, …} is the set of odd natural numbers.
{a, b, c, …} is the set whose elements are the letters of the alphabet.
{3, 6, 9, 12, …} is the set of natural numbers which are divisible by 3.

' +'

'; var set_builder_method= 'Each of the following is an example of the set builder method for specifying (or defining) ' +'a set.' +'

{x|x is a natural number greater than 5}
{x|x is a prime number}
{x|x is a letter of the alphabet which precedes m}
{x|x is a dog and x is white and x is crosseyed}
{x|x is a car with Iowa license plates}
{x|x is a voter registered to vote in the state of Missouri}
{x|x Q and x < 1}
{x|x < 0 or x > 1}

' +'

'; var isin_and_isnotin= 'The following illustrate how to use the symbols is an element of

and

' +'

7 {3, 8, 19, 7, 5}
4 {3, 8, 19, 7, 5}
13 {x|x is a prime number}
15 {x|x is a prime number}
d {x|x is a letter of the alphabet which precedes m}
p {x|x is a letter of the alphabet which precedes m}
{x|x Q and x < 1}
- {x|x Q and x < 1}
-9 {x|x Q and x < 1}
{x|x Q and x < 1}
{x|x < 0 or x > 1}
-4 {x|x < 0 or x > 1}
8 {x|x < 0 or x > 1}

' +'

'; var subset_examples= '

{8, 9, 11} is a subset of {x|x is a natural number greater than 5} because every number in the first set is a ' +' natural numbers' +' greater than 5 and is therefore an element of the second set.
{17, 29, 31, 43} is a subset of {x|x is a prime number}
{3, 2, 5} is not a subset of {3, 2, 8, 9, 10}
{4} is a subset of {4}
{4, 9} is not a subset of {4}
{x|x is a prime number} is a subset of {x|x is a natural number}

' +'

'; var subset_symbol= '

{8, 9, 11} {x|x is a natural number greater than 5} because every number in the first set is a ' +' natural numbers' +' greater than 5 and is therefore an element of the second set.
{17, 29, 31, 43} {x|x is a prime number}
{3, 2, 5} {3, 2, 8, 9, 10}
{4} {4}
{4, 9} {4}
{x|x is a prime number} {x|x is a natural number}

' +'

'; var set_equality= '

{8, 9, 11} = {9,11, 8} because every number in the first set is an element of the second set and every element of the second set is an element of the first set.
{17, 29, 31, 43} = {29, 43, 17, 31} because {17, 29, 31, 43} {29, 43, 17, 31} and {29, 43, 17, 31} {17, 29, 31, 43}
{3, 2, 5} is not equal to {3, 2, 8, 9, 10} because 8 is in the second set but not in the first set
{1, 2, 3, 4, …} = {x|x is a natural number}
{1} = {x|x N and x is not prime and x is not composite}
{x|x is a prime number} = {x|x is a natural number greater than 1 which is not composite}

' +'

'; var intervals_and_rays= '

intervals
' +' rays

'; var example_set_union= 'Example 1: Suppose A = {1, 2, 3, 4} and ' +'B = {3, 4, 5, 6}, then A ∪ B = {1, 2, 3, 4, 5, 6}.

' +'Example 2: Consider the two intervals (-3, 4)' +'and (1, 6], then (-3, 4) ∪ (1, 6] = (-3, 6].

' +'Example 3: {x|x < 3} ∪ {x|x > 3} = {x|x ≠ 3}.

' +'Example 4: {x|x < 3} ∪ {x|x > 0} = R.' +'

' +' end of example

'; var example_set_intersection= 'Example 1: Suppose A = {1, 2, 3, 4} and ' +'B = {3, 4, 5, 6}, then A ∩ B = {3, 4}.

' +'Example 2: Consider the two intervals (-3, 4)' +'and (1, 6], then (-3, 4) ∩ (1, 6] = (1, 4).

' +'Example 3: {x|x < 3} ∩ {x|x > 3} = ∅.

' +'Example 4: {x|x < 3} ∩ {x|x > 0} = (0, 3).' +'

' +' end of example

'; //FORMULAS var square_area= '

' +'

three unit square Question: What is the area of a square whose side is 3 units long?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula A = x² with ' +'x = 3 to obtain ' +'A = 3² = 9 sq. units.

' +'

five and two thirds unit square Question: What is the area of a square whose side is ' +' units long?

' +'

Comment: Begin by sketching the figure with dimensions. Measurements are frequently expressed as mixed numbers but mixed ' +'numbers are ill suited for algebra. Therefore before beginning any algebraic work we ' +'should convert this mixed number to an improper fraction. The question then becomes:

' +' seventeen thirds unit square What is the area of a square whose side is units long?' +'

Disscussion: Redraw or simply change the dimensions on the figure with dimensions expressed as improper fractions and then use the formula A = x² with ' +'x = seventeen thirds to obtain

' +' area calculations

' +'

Comment: In mathematics we are interested in EXACT values. Therefore, it is ' +'appropriate in a math class to express this area as the improper fraction ' +' 289 over 9 . ' +'The application of this result will determine what units are to be used, ' +'whether the improper fraction should be converted to a mixed number or a decimal, ' +'and will determine the accuracy desired.' +'
end of example ' +'

' +'

'; var parallelogram_area= '

' +'

5 by 4 parallelogram Question: What is the area of a parallelogram whose length is 5 units and whose height is 4 units?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula A = xy with ' +'x = 5 and y = 4to obtain ' +'A = (5)(4) = 20 sq. units.

' +'

' +'

' +'

'; var rectangle_area= '

' +'

3 by 7 rectangle Question: What is the area of a rectangle which is 7 units long and 3 units wide?

' +'

Comment::In any discussion of a rectangle, the long side is the length of the rectangle and the short side is the width of the rectangle.' +'

Disscussion: Sketch the figure with dimensions and then use the formula A = xy with ' +'x = 7 and y = 3 to obtain ' +'A = (7)(3) = 21 sq. units.

' +'

' +'

' +'

'; var triangle_area= '

' +'

Question: What is the area of a triangle with base 11 units long and 5 units high?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula with ' +'b = 11 and h = 5 to obtain ' +' computations of area of a triangle sq. units.

' +'

Comment: A ' +' right triangle' +'with base b and height h is clearly one-half a rectangle with length b and width h whose area is bh. ' +' triangle as half a rectangle

' +'It follows that the area of the right triangle is .
' +'Although this derivation used a right triangle rather than a general triangle, it is easily remembered that the same formula is true for all triangles.' +'
end of example ' +'

' +'

'; var trapezoid_area= '

' +'

Question: What is the area of a trapezoid with bases 8 units and 11 units and height 5 units?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula with ' +'B = 11, b = 8, and h = 5 to obtain ' +' computations of area of a trapezoid sq. units.

' +'

' +'

' +'

'; var discussion_trapezoid_area= '

' +'

trapezoid

' +'A diagonal of a trapezoid partitions the trapezoid into two triangles as shown by the ' +'red and green triangles in Figure 1. Because the two bases of a trapezoid are parallel, ' +'the heights of the two triangles are equal. Clearly the area of the trapezoid is equal to ' +'the sum of the areas of the red triangle and the green triangle. This yields the formula for the area of a trapezoid.' +' derivation of area of trapezoid

' +'

' +'

' +'

'; var circle_area= '

' +'

circle with radius 6. Question: What is the area of a circle whose radius is 6 units?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula Area of a circle with ' +'r = 6 to obtain calculations of area of a circle sq. units.' +'
end of example ' +'

' +'

'; var circle_circumference= '

' +'

circle with radius 6. Question: What is the circumference of a circle whose radius is 6 units?

' +'

Disscussion: Sketch the figure with dimensions and then use the formula Circumference of a circle with ' +'r = 6 to obtain calculations of circumference of a circle ' +'
end of example ' +'

' +'

'; //SEQUENCES var sequence_example1= 'Consider the function b whose rule is b(n) = n² and whose domain is N. ' +'Because the domain of b is N, b is a sequence. The first few terms of b are:' +'

b(1) = 1
b(2) = 4
b(3) = 9
b(4) = 16
b(5) = 25

' +'A portion of the graph of b is:

' +' graph of squaring sequence

' +'

It is important to recognize that the graph of this function is a collection ' +'of dots each directly above a natural number.'; var sequence_example2= 'Consider the function b whose rule is b(n) = 2n - 1 and whose domain is N. ' +'Because the domain of b is N, b is a sequence. The first few terms of b are:' +'

b(1) = 1
b(2) = 3
b(3) = 5
b(4) = 7
b(5) = 9

' +'A portion of the graph of b is:

' +' graph of squaring sequence

' +'

It is important to recognize that the graph of this function is a collection ' +'of dots each directly above a natural number.'; var sequence_example3= 'Consider the function b whose rule is b(n) = (-1)ⁿ2n and whose domain is N. ' +'Because the domain of b is N, b is a sequence. The first few terms of b are:' +'

b(1) = -2
b(2) = 4
b(3) = -6
b(4) = 8
b(5) = -10

' +'A portion of the graph of b is:

' +' graph of squaring sequence

' +'

It is important to recognize that the graph of this function is a collection ' +'of dots each directly above or below a natural number. Note the effect of raising -1 to a power.'; var sequence_tau= 'The function whose name is the Greek letter τ (pronounced tau) is a function whose ' +'domain is the Natural Numbers N. So τ is a sequence. The rule for τ is not ' +'given here by a formula. The rule for τ is: τ(n) is the number of positive ' +'divisors of n. To compute the range value associated with a particular domain ' +'element n, it is necessary to count all positive divisors of n. ' +'It is convenient to think of τ as a function which counts the number of positive ' +'divisors of domain elements.

' +' τ(1) = 1 ' +' τ(2) = 2 ' +' τ(3) = 2 ' +' τ(4) = 3 ' +' τ(5) = 2 ' +' τ(6) = 4 ' +' τ(7) = 2 ' +' τ(8) = 4 ' +' τ(9) = 3 ' +' τ(10) = 4 ' +' τ(11) = 2 ' +' τ(12) = 6

' +'The graph of the first 12 terms of Tau consists of the points:

' +' (1, 1) (2, 2) ' +' (3, 2) (4, 3) ' +' (5, 2) (6, 4) ' +' (7, 2) (8, 4) ' +' (9, 3) (10, 4) ' +' (11, 2) (12, 6)

' +'A portion of the graph of tau is:

' +' graph of tau sequence

' +'

' +'Notice that if p is a prime number, then τ(p) = 2 and if k is a composite number, then ' +'τ(k) > 2. In fact sometimes τ is used to define prime numbers in the following way.

' +'Definition: A natural number p is prime if and only if τ(p) = 2.

' +'Notice how neatly this definition prohibits classifying 1 as a prime number.'; //ABSOLUTE VALUE var example_absolute_value= '' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +' ' +'

\|4\| = 4	\|-4\| = 4	\|-17\|= 17	\|17\| = 17
\|\| =	\|-\| =	\|-\|=	\|\| =
\|\| =	\|-\| =	\|-43\|= 43	\|0\| = 0

'; var discussion_absolute_value= 'The algebraic definition of absolute value warrants some explanation because it is ' +'probably the first definition you have encountered in mathematics which is presented ' +'in two parts each of which is dependent on some condition.

' +'

You should read -x as the opposite of x rather than negative x.
The letter x inside the absolute value symbol | | is a variable which means it represents any algebraic expression.
The absolute value of an expression is either the expression (top line) or the opposite of the expression (bottom line).
The absolute value of an expression depends on whether that expression is negative or not.
If the expression is non-negative (the if part of the top line) then the absolute value of that expression is that expression.
If the expression is negative (the if part of the bottom line) then the absolute value of that expression is the opposite of that expression.

'; var example_graph_equation_one_variable= 'ALERT: In these examples, do not be concerned about how the equations ' +'are solved. We will address the solution process shortly. Pay attention to the relation between the following three items:' +'

The number of variables
The nature of a soluton and the solution set
The environment of the graph

' +'Example 1: Consider the equation 3x - 5 = 7.
' +'The only variable is x
' +'Therefore this is an equation in one variable.
' +'Therefore solutions of the equation are real numbers.
' +'Therefore the solution set of the equation is a subset of the set of real numbers.
' +'Therefore the graph of the equation is on the number line.
' +'The solution set of the equation is{4}.
' +'Therefore the graph of the equation is the single dot shown on the number line below.
' +' 3x-5=7

' +'

' +'Example 2: Consider the inequality 3x - 5 < 7.
' +'The only variable is x
' +'Therefore this is an inequality in one variable.
' +'Therefore solutions of the inequality are real numbers.
' +'Therefore the solution set of the inequality is a subset of the set of real numbers.
' +'Therefore the graph of the inequality is on the number line.
' +'The solution set of the inequality is{x|x < 4}.
' +'Therefore the graph of the inequality is the ray shown on the number line below.
' +' 3x-5<7

' +'

' +'Example 3: Consider the inequality 3x - 5 > 7.
' +'The only variable is x
' +'Therefore this is an inequality in one variable.
' +'Therefore solutions of the inequality are real numbers.
' +'Therefore the solution set of the inequality is a subset of the set of real numbers.
' +'Therefore the graph of the inequality is on the number line.
' +'The solution set of the inequality is{x|x > 4}.
' +'Therefore the graph of the inequality is the ray shown on the number line below.
' +' 3x-5>7

' +'

' +'Summary of Examples 1, 2, and 3:
' +'It is instructive to graph the equation and the corresponding two inequalities on the ' +'same number line.
' +' 3x-5>7

' +'Observe that the entire number line is "used" and that there is no overlap between ' +'the three graphs.
' +'Most importantly note that the graph of the equation forms a boundary between the ' +'graphs of the two inequalities.' +'

' +'Example 4: Consider the equation x² - 3x - 10 = 0.
' +'The only variable is x
' +'Therefore this is an equation in one variable.
' +'Therefore solutions of the equation are real numbers.
' +'Therefore the solution set of the equation is a subset of the set of real numbers.
' +'Therefore the graph of the equation is on the number line.
' +'The solution set of the equation is{-5, 2}.
' +'Therefore the graph of the equation is the two dots shown on the number line below.
' +' quadratic equation

' +'

' +'Example 5: Consider the inequality x² - 3x - 10 < 0.
' +'The only variable is x
' +'Therefore this is an inequality in one variable.
' +'Therefore solutions of the inequality are real numbers.
' +'Therefore the solution set of the inequality is a subset of the set of real numbers.
' +'Therefore the graph of the inequality is on the number line.
' +'The solution set of the inequality is{x|-5 < x < 2}.
' +'Therefore the graph of the inequality is the interval shown on the number line below.
' +' quadratic inequality

' +'

' +'Example 6: Consider the inequality x² - 3x - 10 > 0.
' +'The only variable is x
' +'Therefore this is an inequality in one variable.
' +'Therefore solutions of the inequality are real numbers.
' +'Therefore the solution set of the inequality is a subset of the set of real numbers.
' +'Therefore the graph of the inequality is on the number line.
' +'The solution set of the inequality is{x|-5 < x}∪{x|x < 2}.
' +'Therefore the graph of the inequality is the two rays shown on the number line below.
' +' quadratic

' +'

' +'Summary of Examples 4, 5, and 6:
' +'It is instructive to graph the equation and the corresponding two inequalities on the ' +'same number line.
' +' quadratic

' +'Observe that the entire number line is "used" and that there is no overlap between ' +'the three graphs.
' +'Most importantly note that the graph of the equation forms a boundary between the ' +'graphs of the two inequalities.' +'

' +' end of example

'; var example_graph_equation_two_variables= 'ALERT: In these examples, do not be concerned about how the equations ' +'are solved. We will address the solution process shortly. Pay attention to the relation between the following three items:' +'

The number of variables
The nature of a soluton and the solution set
The environment of the graph

' +'Example 7: Consider the equation y = 3x - 5.
' +'There are two variables x and y.
' +'Therefore this is an equation in two variables.
' +'Therefore solutions of the equation are ordered pairs of real numbers.
' +'Therefore the graph of the equation is in the two dimensional rectangular coordinate system.
' +'The graph of the equation is the line shown in the coordinate system below.
' +' two variable equation

' +'

' +'Example 8: Consider the inequality y < 3x - 5.
' +'There are two variables x and y.
' +'Therefore this is an equation in two variables.
' +'Therefore solutions of the equation are ordered pairs of real numbers.
' +'Therefore the graph of the equation is in the two dimensional rectangular coordinate system.
' +'The graph of the equation is the line shown in the coordinate system below.
' +' two variable equation

' +'

' +'Example 9: Consider the inequality y > 3x - 5 .
' +'There are two variables x and y.
' +'Therefore this is an equation in two variables.
' +'Therefore solutions of the equation are ordered pairs of real numbers.
' +'Therefore the graph of the equation is in the two dimensional rectangular coordinate system.
' +'The graph of the equation is the line shown in the coordinate system below.
' +' two variable equation

' +'

' +'Summary of Examples 7, 8, and 9:
' +'It is instructive to graph the equation and the corresponding two inequalities on the ' +'coordinate system.
' +' 3x-5>7

' +'Observe that the entire plane is "used" and that there is no overlap between ' +'the three graphs.
' +'Most importantly note that the graph of the equation forms a boundary between the ' +'graphs of the two inequalities.' +'

' +'Example 10: Consider y < x² - 2x - 3, y = x² - 2x - 3 and y > x² - 2x - 3.

' +' two variable quadratics

' +'

' +'Example 11: Consider y < x³ - 2x² - 3, y = x³ - 2x² - 3 and y > x³ - 2x² - 3.

' +' two variable cubics

' +'

' +' end of example

'; var example_graph_equation_three_variables= 'EXAMPLE COMING SOON. three variable equations.

' +' end of example

'; var example_graph_boundary_equation= 'EXAMPLE COMING SOON. graph of boundary equation.

' +' end of example

'; var example_graph_linear_equation_inequalities= 'EXAMPLE COMING SOON. graph of linear equation & inequalities.

' +' end of example

'; var example_graph_test_point= 'EXAMPLE COMING SOON. testing one point.

' +' end of example

'; var example_equation_no_solution= 'EXAMPLE COMING SOON. equation is a contradiction.

' +' end of example

'; var example_equation_all_reals_solution= 'EXAMPLE COMING SOON. equation is an identity.

' +' end of example

'; var example_inequality_compound= 'Each of the following pairs of inequalities joined by the words AND or OR ' +'are compound inequalitites.

' +'3x - 7 < 5 AND 2x + 4 > 0

' +'6x + 2 > 9 AND 5x + 1 > -6

' +'3x - 7 < 5 OR 2x + 4 > 0

' +'6x + 2 > 9 OR 5x + 1 > -6

' +' end of example

'; var example_compound_inequality_solution_intersection= 'Example 1: The solution set for the compound inequality 3x < 15 AND 2x > 8 is the INTERSECTION ' +'of the solution sets for the individual inequalities.

' +'The solution set for 3x < 15 is (- ∞ , 5)
' +'The solution set for 2x > 8 is (4, ∞)
' +'Therefore the solution set for 3x < 15 AND 2x > 8 is ' +'(- ∞ , 5) ∩ (4, ∞) = (4, 5).
' +'Observe the conjunction AND \"translates\" to intersection.
' +'Do not lose sight of the fact that this means every number in the interval (4, 5) ' +'makes both the inequalities 3x < 15 and 2x > 8 true.

' +'Example 2: The solution set for the compound inequality 5x - 2 > 8 AND 2x + 1 < 15 is the INTERSECTION ' +'of the solution sets for the individual inequalities.
' +'The solution set for 5x - 2 > 8 is (2, ∞)
' +'The solution set for 2x + 1 < 15 is (- ∞, 7)
' +'Therefore the solution set for 5x - 2 > 8 AND 2x + 1 < 15 is ' +'(2, ∞) ∩ (- ∞, 7) = (2, 7).
' +'Observe the conjunction AND \"translates\" to intersection.' +'

' +' end of example

'; var example_compound_inequality_solution_union= 'Example 1: The solution set for the compound inequality ' +'5x - 2 > 8 OR 2x + 1 > 15 ' +'is the UNION of the solution sets for the individual inequalities.
' +'The solution set for 5x - 2 > 8 is (2, ∞)
' +'The solution set for 2x + 1 > 15 is (7, ∞)
' +'Therefore the solution set for 5x - 2 > 8 OR 2x + 1 > 15 is ' +'(2, ∞) ∪ (7, ∞) = (2, ∞).
' +'Observe the conjunction OR \"translates\" to union.

' +'Example 2: The solution set for the compound inequality ' +'3x + 2 < 14 OR 7x > 21 ' +'is the UNION of the solution sets for the individual inequalities.
' +'The solution set for 3x + 2 < 14 is (-∞, 4)
' +'The solution set for 7x > 21 is (3, ∞)
' +'Therefore the solution set for 3x + 2 < 14 OR 7x > 21 is ' +'(-∞, 4) ∪ (3, ∞) = R.
' +'Observe the conjunction OR \"translates\" to union.

' +'Example 3: The solution set for the compound inequality ' +'3x + 2 < 14 OR 3x > 18 ' +'is the UNION of the solution sets for the individual inequalities.
' +'The solution set for 3x + 2 < 14 is (-∞, 4)
' +'The solution set for 3x > 18 is (6, ∞)
' +'Therefore the solution set for 3x + 2 < 14 OR 3x > 18 is ' +'(-∞, 4) ∪ (3, ∞).
' +'Observe the conjunction OR \"translates\" to union.

' +' end of example

'; var example_compact_compound_inequality= 'The compound inequality 3 < x AND x < 7 can be written in ' +'compact form as 3 < x < 7.

' +'The compound inequality -5 < 3x + 2 AND 3x + 2 < 8 can be written in ' +'compact form as -5 < 3x + 2 < 8.

' +'The compound inequality 15 < 3x + 2 AND 3x + 2 < 8 CANNOT be written in ' +'compact form as 15 < 3x + 2 < 8. This would imply that 15 < 8 which is not true.

' +'Compound inequalities containing the conjunction OR CANNOT be written in compact ' +'form.

' +' end of example

'; var law_trichotomy_reminder= 'Do you see the Law of Trichotomy at play here?'; var transitive_property_reminder= 'Do you see the Transitive Property at play here?'; var midpoint_coordinates_are_averages= 'Have you observed that the first coordinate of the midpoint is the average of the two ' +'first coordinates and the second coordinate of the midpoint is the average of the two ' +'second coordinates?'; var coordinate_system_with_graph= '

' +' informational coordinate system with graph ' +'The graph of an equation in two variables has been superimposed on the informational ' +'coordinate system at the right. Don\'t be concerned with how that graph was ' +'constructed -- it was computer generated. Don\'t be concerned about the detail of the ' +'equation -- just realize that it is of the form ' +'y = (an expression in x).' +'

' +'

' +'The graph of an equation should visually convey information about the equation. In ' +'particular it should show where the graph is above the x-axis, ' +'where it intersects the x-axis, and where it is below the ' +'x-axis.' +' ' +' small owl ' +'

' +'

' +'To answer these question for specific equations, it is necessary to state the questions ' +'in algebraic form. The picture at the right will help us make that translation. ' +'Note that the graph is above the x-axis if and only if the graph ' +'is in the green (Quadrant II) or the blue (Quadrant I). In these two quadrants, and not ' +'in the other two quadrants, second coordinates (y-coordinates) are positive. This allows ' +'the following translation:

' +'

Where is the graph above the x-axis? ' +'

is equivalent to
Which points on the graph of the equation satisfy the inequality ' +'y > 0?
' +'

' +'

' +'Similar reasoning related to the graph being below the x-axis ' +'leads to the following:

' +'

Where is the graph below the x-axis? ' +'
is equivalent to
Which points on the graph of the equation satisfy the inequality ' +'y < 0?
' +'

' +'

' +'Likewise the question about x-intercepts ' +'leads to the following:

' +'

Where does the graph intersect the x-axis? ' +'
is equivalent to
Which points on the graph of the equation satisfy the inequality ' +'y = 0?
' +' ' +' small owl ' +'

' +'

' +'For the remainder of this course and the next (College Algebra) most of the time will be ' +'devoted to investigating these questions.' +'

'; var example_parallel_lines= 'The graphs of the equations y = 3x - 8, ' +'y = 3x + 2, and y = 3x + 43 ' +'are three parallel lines because the equations are linear equations in two variables ' +'with the same slope and different y-intercepts.'; var example_perpendicular_lines= '

' +'The graphs of the equations y = x - 8, ' +'and y = x + 5 are perpendicular lines because ' +'the equations are linear equations in two variables whose slopes are negative reciprocals ' +'of each other.' +'

' +'

' +'The graphs of the equations y = x - 8, ' +'and y = -x + 5 are perpendicular lines because the equations are linear equations in ' +'two variables whose slopes are negative reciprocals of each other.' +'

'; var example_computing_slope= 'To compute the slope of the line through the points (2, 5) and (7, 8) we use the formula

' +' slope formula

to obtain ' +'

.'; var example_vertical_line_no_slope= 'In an attempt to calculate the slope of a vertical line use the formula

' +' slope formula

' +'with any two points (x₁, y₁) and (x₂, y₂).
' +'Because these points are on the same vertical line x₁ = x₂ ' +'so the points can be represented as (x₁, y₁) and (x₁, y₂).
' +'Now when we use the formula for slope we obtain

' +'

.
' +'Because division by zero is undefined the slope of a vertical line is undefined.
' +'The claim that a vertical line has no slope is not an arbitraty proclamation. It is an ' +'absolute necessity. The claim is consistent with the formula for slope and the ' +'definition of division.'; var example_horizontal_line_zero_slope= 'In an attempt to calculate the slope of a horizontal line use the formula

' +' slope formula

' +'with any two points (x₁, y₁) and (x₂, y₂).
' +'Because these points are on the same horizontal line y₁ = y₂ ' +'so the points can be represented as (x₁, y₁) and (x₂, y₁).
' +'Now when we use the formula for slope we obtain

' +'

.' +' ' +' small owl

'; var denominator_is_nonzero= 'It is important to notice that the denominator is not zero so the quotient is defined.'; var example_linear_equation_two_variables_graphing= 'To sketch the graph of the equation y = 3x + 6 we begin by observing ' +'this is a linear equation in two variables.

That observation is important because ' +'we therefore know the graph is a line in the Cartesian coordinate system.
Any ' +'two points will determine the line, but it is prudent to use the intercepts.
' +'Without any calculation we know from the slope-intercept form ' +'of the equation that the y-intercept is ' +'(0, 6).
Plot that point and label it with its coordinates.
' +'To determine the x-intercept, let y = 0 ' +'and solve for x. In this case we obtain 0 = 3x + 6 whose only ' +'solution is -2. The x-intercept for this graph is the point ' +'(-2, 0)
Plot that point and label it with its coordinates.' +'
Finally sketch a line through the two points.

' +'To sketch the graph of 3x - 4y = 12 we begin by observing ' +'this is a linear equation in two variables.
Therefore the graph is a line in ' +'the Cartesian coordinate system.
Any two points will determine the line, but it is ' +'prudent to use the intercepts.
' +'To determine the y-intercept, let x = 0 ' +'and solve for y. In this case we obtain -4y = 12 whose only ' +'solution is -3. The y-intercept for this graph is the point ' +'(0, -3)
Plot that point and label it with its coordinates.
' +'To determine the x-intercept, let y = 0 ' +'and solve for x. In this case we obtain 3x = 12 whose only ' +'solution is 4. The x-intercept for this graph is the point ' +'(4, 0)
Plot that point and label it with its coordinates.' +'
Finally sketch a line through the two points.

' +'The above procedure will not work for equations which are not linear in two variables.' +'That is one reason to recognize the kind of equation you are dealing with.'; var example_linear_inequality_two_variables_graphing= 'To sketch the graph of the equation y < 3x + 6 we begin by ' +'observing this is a linear inequality in two variables.

That observation is ' +'important because we therefore know the graph is a half-plane ' +'in the Cartesian coordinate system and we know it is bounded by the graph of the boundary ' +'equation y = 3x + 6.
Sketch the graph of the boundary equation ' +'as described in the example to the left.
Because y < 3x + 6' +'is a strict inequality, the boundary line is not part of the solution set. It is ' +'conventional to therefore sketch the boundary as a dotted or dashed line.

' +'Pick any point, not on the boundary line, and test it in the inequality. In this case ' +'the boundary line does not contain the origin so it is convenient to use ' +'(0,0) as a test point. Substitute the coordinates of the test ' +'point into y < 3x + 6 to obtain 0 < 0 + 6 ' +'which is TRUE.
Therefore the half-plane containing the ' +'origin and bounded by the line y = 3x + 6 is the graph of ' +'y < 3x + 6.It is customary to shade the solution set for an ' +'inequality.' +' ' +' small owl

'; +'

'; var graph_and_solutions = 'It is important to realize that the pair of coordinates of every point in that ' +'half-plane is a solution of the linear inequality ' +'y < 3x + 6.'; var example_quadratic_equation_one_variable= ' ' +' small owl

' +'The equation 3x² - 5x + 4 = 0 is a quadratic equation ' +'in one variable because it matches ax² + bx + c = 0 ' +'with a = 3, b = -5, and' +' c = 4.' +'

' +'The equation 7x² + 2x + 9 = 2x² - 8x + 1 ' +'is a quadratic equation in one variable because we can add the opposite of ' +'2x² - 8x + 1 to both sides to obtain ' +'5x² + 10x + 8 = 0 which matches ' +'ax² + bx + c = 0 ' +'with a = 5, b = 10, and' +' c = 8.' +'

' +'The equation 3x⁵ - 5x³ + 4 = 0 is not ' +'a quadratic equation in one variable because it cannot be converted to the form ' +'ax² + bx + c = 0 using the two fundamental properties of equations.' +'

' +'

'; var example_quadratic_inequality_one_variable= ' ' +' small owl

' +'The inequality 3x² - 5x + 4 < 0 is a quadratic inequality ' +'in one variable because it matches ax² + bx + c < 0 ' +'with a = 3, b = -5, and' +' c = 4.' +'

' +'The inequality 7x² + 2x + 9 < 2x² - 8x + 1 ' +'is a quadratic inequality in one variable because we can add the opposite of ' +'2x² - 8x + 1 to both sides to obtain ' +'5x² + 10x + 8 < 0 which matches ' +'ax² + bx + c < 0 ' +'with a = 5, b = 10, and' +' c = 8.' +'

' +'The inequality 3x² - 5x + 4 > 0 is a quadratic inequality ' +'in one variable because if we multiply both sides by -1 and reverse the sense of the ' +'inequality we obtain -3x² + 5x - 4 < 0 which matches ' +'ax² + bx + c < 0 ' +'with a = -3, b = 5, and' +' c = -4.' +'

' +'The inequality 3x⁵ - 5x³ + 4 < 0 is not ' +'a quadratic inequality in one variable because it cannot be converted to the form ' +'ax² + bx + c < 0 using the three fundamental ' +'properties of inequalities.' +'

' +'

'; var example_solving_factoring_zero_factor_property= 'EXAMPLE COMING SOON Examples of solving equations with factoring and the zero factor property' +' end of example

'; var example_solving_with_quadratic_formula= 'EXAMPLE COMING SOON Examples of solving equations with the quadratic formula' +' end of example

'; var example_quadratic_equation_two_variables= 'EXAMPLE COMING SOON Examples of quadratic equations in two variables' +' end of example

'; var example_quadratic_equation_two_variables_intercepts= 'EXAMPLE COMING SOON Examples of intercepts of quadratic equations in two variables' +' end of example

'; var determine_if_one_variable_equation_is_quadratic= 'To determine if a given equation is a quadratic equation in one variable, the given ' +'equation must be compared to the definitional equation ' +'ax² + bx + c = 0.

The only permissible ' +'procedures for rewritting the given equation are the two basic properties of equations.

' +'If any expression is added to both sides of an equation the resulting equation ' +'is equivalent to the original equation.

' +'If both sides of an equation are multiplied by the same non-zero real number, the resulting equation ' +'is equivalent to the original equation.'; var determine_if_one_variable_inequality_is_quadratic= 'To determine if a given inequality is a quadratic inequality in one variable, the given ' +'inequality must be compared to the definitional inequality ' +'ax² + bx + c < 0.

The only permissible ' +'procedures for rewritting the given equation are the three basic properties of inequalities.

' +'If any expression is added to both sides of an inequality the resulting inequality ' +'is equivalent to the original inequality.

' +'If both sides of an inequality are multiplied by the same positive real number, the resulting inequality ' +'is equivalent to the original inequality.

' +'If both sides of an inequality are multiplied by the same negative real number and the inequality symbol is reversed, ' +'the resulting inequality is equivalent to the original inequality.';