WEEK 5 Day 1

Three traveling salesman stop at a hotel for the night, they ask how much is a room. The manager says the room is $30. Each man puts a $10 dollar bill on the counter, they get the key and go to their room. The manager notices that he made a mistake, the room is only $25 not $30. He gives five $1 dollar bills to his assistant to return to the gentleman. The assistant walks to the room thinking that he can't give $5 dollars to 3 people. He gets to the room and gives each man one dollar back and keeps two for himself. Each man(3) spent $9=$27 The assistant kept $2 That's a total of $29! What happen to the last dollar?

Actually, there is no "last dollar." If you're going to deal with the amount spent by the guests and the amount stolen by the assistant, you should subtract the two dollars, not add it. The room costs $25, the guests spent a total of $27, and the assistant stole the $2 difference. The manager originally took in $30, he gave $5 to the assistant, of which $3 went to the guests, and $2 went to the assistant. $30 minus $3 = $27. Each guest spent $9, for a total of $27. The room really costs $25. The assistant stole the $2 difference. It all checks out.

Quiz on Thursday February 4 th.

Chapter 4 Equations and Their Graphs Functions and their equations can be graphed. Page 145

5 = 2x + 1x = 2 y = 2x + 1 x, y 2, 5 3, 7 4, 9

Y = f (x) X Y 2, 5 3, 7 4, 9

4.2 GRAPHING EQUATIONS page 152 Plotting points from order pairs. Plotting is fundamental to correct graphs.

From (ordered) pairs to plotting points to graphing.

4.2 GRAPHING EQUATIONS PAGE 150 Doug’s tips for graphing a function. For X use -1, 0, 1, 2 The pair will be near the origin. The pair will allow for possible negative and positive outcomes. The numbers are mathematically easy to work with.

Page 152 A linear equation with two unknowns is an equation of degree one in the form with a and b not both 0. Degree one means no exponents.

4.2 GRAPHING EQUATIONS page 152

Y-intercept

Page 154 Solving for x = 0 This graphically means finding the point or points, if any, where the graph crosses the y axis. x y (0, 2)

4.3 THE STRAIGHT LINE Page 162 Y intercept may be solved mathematically. (section 4.3)

Slope

4.3 THE STRAIGHT LINE page 159 The slope of a line.

Any 2 ordered pair can be used.

X, Y 1, 2 6, 4

X, Y 1, 2 6, 4

If the line has zero slope, then the line is horizontal (“flat”). y 2 – y 1 = 0

If the line is vertical, then the line has undefined slope because of 0.

4.3 THE STRAIGHT LINE page 161 Point-Slope form is a simple manipulation of the slope formula.

4.3 THE STRAIGHT LINE page 161 This allows us to find the equation for a line given the slope of the line and a point (ordered pair).

Find the equation for a line with point (-1, 2) and a slope of 3. Substitution Multiply by 3 Add 2 to both sides

4.5 THE DISTANCE AND MIDPOINT FORMULAS page 170

4.5 THE DISTANCE AND MIDPOINT FORMULAS page 171

Chapter 5 Factoring and Algebraic Fractions 5.1 Special Products 5.2 Factoring Algebraic Expressions 5.3 Other Forms of Factoring 5.4 Equivalent Fractions

Chapter 5 Factoring and Algebraic Fractions 5.5 Multiplication Division of Algebraic Expression 5.6 Addition and Subtractions of Algebraic Expressions 5.7 Complex Fractions 5.8 Equations with Fractions

5.1 Special Products

5.1 SPECIAL PRODUCTS page 180 In Chapter 1 we introduced certain fundamental algebraic concepts and operations. page 2 The following are some properties of real numbers:

5.1 SPECIAL PRODUCTS page 180 “We must be able to do the multiplications quickly and mentally.”

5.1 SPECIAL PRODUCTS page 181 There are two general forms of the square of a binomial. A binomial is an algebraic expression containing exactly two terms.

5.1 SPECIAL PRODUCTS page 181 Why is there no yx or xy ?

5.1 SPECIAL PRODUCTS page 182 There are two general forms of the cube of a binomial. Not for us

5-2 Factoring Algebraic Expressions

A product is the result obtained by multiplying two or more quantities together. Factoring is finding the numbers or expressions that multiply together to make a given number or equation.

5.2 FACTORING ALGEBRAIC EXPRESSIONS page 183 Slide number 6.

5.2 FACTORING ALGEBRAIC EXPRESSIONS page 183 Greatest or largest.

5.2 FACTORING ALGEBRAIC EXPRESSIONS Greatest or largest common factor. 15ab – 6ac = 3a (5b – 2c)

5.2 FACTORING ALGEBRAIC EXPRESSIONS page 184

page 185 A summary about the signs in trinomials. If the trinomial to be factored is one of the following forms, use the corresponding sign patterns.

page 185

page 186

From section 5.2

page 186

page 188 When the factors of a trinomial are the same two binomial factors, the trinomial is called a perfect square trinomial.

Page 188 And we have come back to the beginning.

5.3 OTHER FORMS OF FACTORING page 189 Some algebraic expressions may be factored by grouping their terms so that they are of the types we have already studied. Move on to section 5.4

5.4 Equivalent Fractions

5.4 EQUIVALENT FRACTIONS page 189 Two fractions are equivalent when both the numerator and the denominator of one fraction can be multiplied or divided by the same nonzero number in order to change one fraction to the other. a/a = 1

5.4 EQUIVALENT FRACTIONS page 189 A fraction has three signs associated with it: 1. The sign of the fraction. 2. The sign of the numerator. 3. The sign of the denominator.

5.4 EQUIVALENT FRACTIONS page 192 Any two of these three signs may be changed without changing the value of the fraction. Not seen much.

5.4 EQUIVALENT FRACTIONS page 192 A negative sign of an algebraic fraction may be removed by placing a negative sign before the numerator or the denominator in parentheses.

5.4 EQUIVALENT FRACTIONS page 192 A fraction is in lowest terms when its numerator and denominator have no common factors except 1.

5.4 EQUIVALENT FRACTIONS page 194 If you can do this You can do this

5.4 EQUIVALENT FRACTIONS page 194 Then this:

5.5 Multiplication Division of Algebraic Expressions

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195 This is good.

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195 Factor each of the terms in the numerator and denominator. Divide by common factors. Then multiply the numerators and denominators.

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195

Cancellation marks are useful for showing each completed division.

3 goes in to 3 once. 3 goes into 6 twice.

5 goes into 20, 4 times. 5 goes in to 5 once.

2 goes into 4, 2 times. 2 goes in to 2 once.

Doug’s technique.

Factor each of the terms in the numerator and denominator. Divide by common factors. Then multiply the numerators and denominators.

Reorganize like terms.

Factor each of the terms in the numerator and denominator. Divide by common factors. 60 a y 30 b x

60 a y 30 b x 2ay 1bx 2ay bx

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195 You can never be too good at this.

Page 196

5.6 Addition and Subtractions of Algebraic Expressions

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Fractions may be added or subtracted if they have a common denominator. That is why a Least Common Denominator (LCD) must be determined.

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Denominators get factored.

Factoring is finding the numbers or expressions that multiply together to make a given number or equation.

A prime number is a positive that is evenly divisible only by itself and one. The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Why 2 not one (its meaningless changes nothing it remains 1 and the number.

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Why 3 because 2 doesn’t work with 27.

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Notice 540 is greater (larger) then any denominator and we are looking for Least Common Denominator.

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 199

Example 3 page 199 Your home work is not this complex.

Other means that work and work in a broad context.

5.6 ADDITION AND SUBTRACTION OF ALGEBRAIC FRACTIONS page 198 Fractions may be added or subtracted if they have a common denominator. That is why a Least Common Denominator (LCD) MUST be determined. That is why a Least Common Denominator (LCD) SHOULD be determined. That is why a Common Denominator (CD) MUST be determined.

A common denominator by multiplying the denominators.

A common denominator