chapter 1 set: a collection of objects called elements important sets of numbers: natural numbers...

Chapter 1Chapter 1

Set: a collection of objects called elements

Important sets of numbers:

Natural numbers

Whole numbers

Integers

Use the ‘roster method’ to write a set

Inequality symbols

Additive inverses

Absolute value

Addition of integers

• SAME SIGNS

1. Add the absolute values of the numbers ( ignore the signs and add)

2. Attach the common sign

• DIFFERENT SIGNS 1. Find the difference of the absolute values (ignore the

signs and subtract)

2. Attach the sign of the number with the larger absolute value.

Subtraction of integers

1. Rewrite the “—” as “+ the opposite of the

number”

2. Follow the rules for addition.

Subtraction of integers

• We might say this subtraction process as “Change to the opposite then add”

Multiplication of integers

• The product of two numbers with the same sign is positive.

positive · positive = +

negative · negative = +

Multiplication of integers

• The product of two numbers with different signs is negative.

positive · negative = negative

negative · positive = negative

When multiplying or dividing:

ODD # of negative signs makes the answer negative.

EVEN # of negative signs makes the answer positive.

Division of integers (The rules are like those for multiplication)

• The quotient of two numbers with the same sign is positive.

positive ÷ positive = +

negative ÷ negative = +

• The quotient of two numbers with different signs is negative

positive ÷ negative = negative

negative ÷ positive = negative

• If a fraction is negative, the ‘–’ sign can be place in any of three different positions, and all are considered equivalent.

• 0 ÷ any number = 0

• Division by 0 is not defined!

• To compute the arithmetic mean or average:

Operations with rational numbers

• Rational numbers are fractions and decimals which repeat or stop.

• Reduce fractions to simplest form

• Change fractions into decimals which repeat or stop

• Add, subtract, multiply and divide fractions

• Add, subtract, multiply and divide decimals

• Don’t forget to use our rules for + and - signs!

Working with %

• Change % to a decimal

• Change a decimal to a %

• Change a fraction to a %

• Change a % to a fraction

Order of OperationsOrder of Operations

1. Do operations within grouping symbols first.

2. Exponents3. Multiply and divide from

left side of problem toward right side.

4. Add and subtract last!

Working with Exponents

24 = (2)(2)(2)(2)

( –2)4 is not the same

as – 24

A good way to remember the order is

PEMDAS

Remember the “parens” can also be brackets, absolute value symbols, or a fraction bar.

Chapter 2

• Variable: a letter used to stand for a quantity

• A variable expression is made up of terms

Types of terms:variable termsconstant terms

A variable term has a coefficient

When evaluating variable expressions, remember PEMDAS to get the right order of operations.

Also, watch

your signs!

Like terms – terms which have the exact same variable part

Constant terms are also “like terms”

Only ‘like’ terms can be added or subtracted!!!

Be able to use the distributive property

Translating Verbal Expressions

See p. 67 for a list of common phrases…..

When you see the phrase “in terms of”… let the part which follows be x

Chapter 3

Solve an equation means to find the value which makes the equation true.

We want to end with :

Variable = constant

• To do this, perform ‘opposite operations’ to both sides of the equation.

• On your assignments, please show the process!

• To check whether a value is really a solution to an equation, put it in place of the variable and see if it makes a true statement.

% Problems

1. Change % to a decimal

2. Of multiply

3. Is =

4. Write an equation

5. Solve the equation

Solving Equations

1. First simplify each side of the equation. Distribute to get rid of parens and combine like terms

2. Add or subtract terms to move all constants to one side and all variable terms to the other side of the =

3. Divide or multiply to get rid of the coefficient of the variable.

End with : variable = constant

3.4 Translating Sentences into Equations

These words or phrases are replaced with an = sign:

equals is

is equal to amounts to

represents

Steps:

1. Assign a variable to the unknown quantity.

2. Translate the words into math symbols. Write the equation.

3. Solve the equation.

4. Check your answer.

Recall the integers are the positive and negative whole numbers:

{… -4, -3, -2, -1, 0, 1, 2, 3, 4 …}

An even integer is an integer that is divisible by 2 like 12, -4, 28, 0

An odd integer is not divisible by 2 like 33, -27, and 5

Consecutive Integers (none in exercises)

5, 6, 7 or -11, -10, -9 or n, n + 1, n + 2

Consecutive Even Integers-12, -10, -8 or 4, 6, 8 or n, n + 2, n + 4

Consecutive Odd Integers5, 7, 9 or -13, -11, -9 or n, n + 2, n + 4

Find three consecutive even integers such that three times the second is four more than the sum of the first and the third.

Five times the first of two consecutive odd integers equals three times the second integer. Find the integers.

Translate “three more than twice a number is the number plus six” into an equation.

Four less than one-third of a number equals five minus two-thirds of the number. Find the number

The sum of two numbers is sixteen. The difference between four times the smaller number and two is two more than twice the larger number. Find the two numbers.

The sum of two numbers is twelve. The total of three times the smaller number and six amounts to seven less than the product of four and the larger number. Find the two numbers.

The difference between a number and twelve is twenty. Find the number.

A board 20 ft long is cut into two pieces. Five times the length of the smaller piece is 2 ft more than twice the length of the longer piece. Find the length of each piece.

A company makes 140 televisions per day. Three times the number of black and white TV’s made equals 20 less than the number of color TV’s made. Find the number of color TV’s made each day.

Translating Sentences into Equations

1. Assign a variable or an expression to the unknown quantity or quantities.

2. Translate the verbal expressions into math symbols. We want two expressions equal to each other.

3. Solve the equation.

4. Check your answer.

Chapter 4 • Monomial….a number, a

variable, or the product (mult.) of numbers and variables

• Polynomial…2 or more monomials added or subtracted (The monomials are then called terms.)

Special polynomials

• Binomial: 2 terms

• Trinomial: 3 terms

• To add or subtract polynomials, you can use a vertical or a horizontal format.

• When subtracting, remember to distribute the ‘–’ to all terms inside the parens.

Basic Rules for exponents

1. axn +bxn = (a+b)xn

In ‘like’ terms, add coefficients but not the exponents.

2. xn • xp = xn +p

3. (axn) • (bxp) = (ab)xn +p

(Mult. coef. and add expon.)

4. (xn)p = xnp

5. (axn)p = ap xnp

(Do coef. to power, but mult. expon.)

6. (xnyb)p = xnp ybp

(Multiply exponents)

Using the rules…..

1. monomial • monomial

2. monomial • polynomial

3. binomial • binomial

4. binomial • trinomial

5. Special products

Rules for exponents

1. axn +bxn = (a+b)xn

In ‘like’ terms, add coefficients but not the exponents.

2. (axn) • (bxp) = (ab)x n +p

(Mult. coef. and add expon.)

3. (axn)p = ap xnp

(Do coef. to power, but mult. expon.)

4. (xnyb)p = xnp ybp

(Multiply exponents)

5. Rule for division:

Divide coefficients.

Do top exponent minus bottom exponent to get new exponent….or cancel equal amount from top and bottom.

6. Fraction to a power:

Apply exponent to top and bottom

7. a0 = 1 if a 0

8. x -p = 1 / xp and

ax -p = a / xp

A negative exponent makes that part move but does not make it negative!

9. a / x -p = axp

10. (fraction) -p =

(reciprocal of fraction ) p

Negative exponent inverts fraction…but does not make it negative!

Simplest form is written with no negative exponents!

Scientific notationUsed to write really large and

very small numbers in compact form

1. 2.4 10 -3

2. 1.7 10 4

• Know what your instructor requires!– Read your syllabus; – keep it for future reference.

• Don't fall behind! Math skills must be learned immediately and reviewed often. Keep up-to date with all assignments.

• Most instructors advise students to spend two hours outside of class studying for every hour spent in the classroom. Do not cheat yourself of the practice you need to develop the skills taught in this course!

• Take the time to find places that promote good study habits. Find a place where you are comfortable and can concentrate. (library, quiet lounge area, study lab)

• Survey each chapter ahead of time.

– Read the chapter title, section headings and the objectives listed to get an idea of the goals and direction for the chapter.

• Take careful notes and write down examples.

• The book provides material to read and examples for each objective studied. It also has answers to the odd-numbered exercises in the back of the book so that you can check your answers on assignments.

• Be sure to read the Chapter summary and use the Chapter Review and Chapter Test exercises to prepare for each Chapter exam. (All answers are in the back for these)

• Spaced practice is generally superior to massed practice. You will learn more in 4 half-hour study periods than in one 2 hour session.

• Review material often because repetition is essential for learning. You remember best what you review most. Much of what we learn is soon forgotten unless we review it.

• Attending class is vital if you are to succeed in any math course.

• Be sure to arrive on time…. and stay the entire class period!

• You are responsible for everything that happens in class, even if you are absent.

If you must be absent :• 1. Deliver due assignments to

instructor as soon as possible (even ahead of time if you know in advance).

• 2. Copy notes taken by a classmate while you were absent.

• 3. Ask about announcements, assignments or test changes made in your absence.

• If you have trouble in this course – seek help!–1. Instructor–2. Tutors–3. Video Tapes–4. Computer Tutoring

Study Tips: Preparing for Tests

• Try the Chapter Test at the end of each chapter before the actual exam. Do these exercises in a quiet place and pretend you are in class taking the exam.

Study Tips: Preparing for Tests

• If you missed questions on the practice test, review the material, practice more problems of the same type, get help as needed.

Try these strategies of successful test takers:

• 1. Skim over the entire test before you start to solve any problems.

• 2. Jot down any rules, formulas or reminders you might need.

• 3. Read directions carefully.

• 4. Do the problems that are easiest for you first.

• 5. Check your work to be sure you haven't made any careless errors.

Test today... or tomorrow!!!!

Are you ready???

Test today... or tomorrow!!!!

Don’t forget!!!!

Today is the last day for the Chapter

Test….Don’t forget!!!!

The Chapter 4 Test opens today…..

Are you ready????

The Chapter 6 Test opens tomorrow…..Are you ready????

The Chapter 6 Test opens tomorrow…..Are you ready????

The Chapter 7 Test opens tomorrow…..Are you ready????

Finding and Factoring out Common Factors

Factor 15 Factor 30 To factor a number into prime

numbers, we break it into the prime numbers which would multiply to equal the original number

• Factor a polynomial, means the same thing….break it into monomials, binomials, polynomials, etc. that are prime and would multiply together to equal the original polynomial.

Always look for a common factor first thing!

Sometimes the common factor is a monomial…….

• Sometimes the common factor

is a binomial…………..

• And sometimes you have to factor out a -1 in order to have a common binomial.

If you have a polynomial with 4 terms:

1. Make 2 groups of 2 terms each

2. Factor the groups

3. Find the common binomial factor and write it in parens.

4. Write the second set of parens.

Factoring Trinomials like x2 + bx +c

1. Factor out all common factors and write them in front of the parens.

2. If the coefficient of x2 is -1, factor out the -1; put the terms in descending order.

3. If the constant is + , make 2 binomials with the same sign as the middle term.

4. If the constant is - , one binomial will have a + , the

other will have a -.

5. Outside product + inside

product must = middle term.

6. Check your binomial factors by using the FOIL process.

5.3 Factoring trinomials like ax2 + bx + c

1. Factor out any common factors

2. Make sets of 2 binomials for each combination of factors of the coefficient of x2.

3. Use the signs of the constant and middle term to determine the signs within the binomials:

constant is +

signs are both + if middle term is + ; signs are both - if middle term is -

constant is -

one binomial has +, the other has -

4. Remember from FOIL that the outside product + the inside product must = the middle term.

5. An important fact is that if the terms of the trinomial do not have a common factor, then you cannot have a common factor within either binomial.

We will not do factoring by grouping….ignore it in the book!

5.4 Special factoring situations

a2 + b2 is nonfactorable over the integers

a2 – b2 is called the difference of perfect squares and it is factorable! a2 – b2 = (a + b)(a - b)

Some trinomials in this section will be perfect squares:

x2 + 4x + 4 x2– 6x + 9

4x2– 20x + 25

9y2 + 30y + 25

16a2 – 8a + 1

This section will also have some polynomials with four terms like we had earlier :

x3 – 3x2 – 4x + 12

a2b2 – 49a2 – b2 + 49

Good checklist for the factoring process on p. 229:

1. Is there a common factor?

2. Only 2 terms?

Try (a + b)(a – b)

3. Trinomial ? Make 2 binomials! Check by FOIL

4. 4 terms ? Make 2 groups of 2 terms then factor each

group watching for a common factor to pull out in front.

5. Are all factors prime or can they be factored more?

Using Factoring to Solve Equations

If ab = 0, then a = 0 or b = 0

This is called the “principle of zero products”.

STEPS

1. Make the equation = 0

2. Then factor

3. Then set each factor = 0 and solve.

1. (x – 3)(x – 5) = 0

2. 3x(x + 2) = 0

3. (2x + 3)(3x – 1) = 0

4. 2x2 + x – 6 = 0

5. y2 – 8y = –15

6. 2x2 – 50 = 0

7. (x + 2)(x – 7) = 52

Application problems:

1. The sum of the squares of two consecutive positive integers is 61. Find the two integers.

2. A rectangle has a width of x inches. The length is 4 inches longer than twice the width. The area of

the rectangle is 96 in2. Find the width and the length of the rectangle.

3. The sum of two numbers is six. The sum of the squares of the two numbers is twenty. Find the two numbers.

4. Sometimes a formula is given for the

problem….see page 241

Rational expression:

a fraction with a polynomial in the numerator and / or the denominator

To simplify:

1. Factor the top and bottom

2. Divide out common factors from the top and the bottom

To Multiply:

1. Factor the top and bottom

2. Divide out common factors from the top and the bottom

3. Multiply straight across, leaving the top and bottom in simplified factored form

To Divide:

1. Flip the second fraction and multiply by that reciprocal

2. Factor and cancel as in multiplication

To add or subtract rational expressions:

1.Factor all denominators

2. Change all fractions to a common denominator by multiplying the top and bottom of each fraction by the factors it needs to match the common denominator.

3. Add / subtract the tops and put that answer over the common denominator

4. Simplify by factoring the top and canceling common

factors from the numerator and the denominator.

Complex Fractions have at least one fraction within a fraction.

To simplify:

1. Find the smallest common denominator (LCD) of all the denominators in the top and bottom of the fraction

2. Multiply the whole numerator and the whole denominator by this LCD. This should clear the fractions from the top and the bottom of the “main” fraction.

3. Simplify by factoring the new top and bottom and canceling common factors

To solve an equation containing fractions:

1. Factor all denominators and find the common denom

2. Multiply each side of the equation by the LCD. This should clear all the denominators!

3. Solve this “fraction free” equation

4. Check your answer into the original equation and reject any answer that makes a denom = 0

Proportion: an equation stating that 2 fractions are equal

To solve a proportion,

1. Multiply both sides of the = by the common denom

OR

2. Cross multiply and set the cross products =

Application problems

Write the fraction’s pattern in words first!

Application problems

1. Sixteen tiles are needed to tile an area that is 9 ft2. At this rate, how many square feet can be tiled using 256 tiles?

2. A monthly loan payment is $29.75 for each $1000 borrowed. At this rate, find the monthly payment for a $9800 car loan.

3. Package directions say that 3.5 ounces of a certain

medication are required for a 150 lb. adult. At this rate, how many additional ounces are needed for a 210 lb. adult?

Literal equations have more than one variable.

You will have to rewrite the given equation so that a specified variable is isolated.

To do this:

1. Clear fractions by multiplying both sides of the equation by the LCD and get rid of parens.

2. Use opposite operations to move all terms with the

specified variable to one side of the = and all terms without that variable to the other side of the =

3. Isolate the specified variable (usually by division…but if 2 or more terms have the specified variable, you will have to factor first, then divide)

Important concepts and formulas for sections 7.1, 7.2

1. Rectangular coordinate system

2. Ordered pairs

3. Graph ordered pairs

4. Ordered pair solutions to equations

5. Make relation from given data

6. Is a relation a function?

7. Function notation…f(x)

8. Evaluate functions

Graphing lines using x|y roster ★ Find 3 solution points

★You can choose any x-value, then put it into the equation to get the y-value

Special Points

★ x-intercept and y-intercept

★ Find them

★ Use them to draw the graph

Let x = 0 to get the y-intercept.

Y-intercept is on the y-axis!

Let y = 0 to get the x-intercept.

X-intercept is on the x-axis!

Slope

Slope is a measure of the steepness of a line. There are several ways to find the slope:

Ways to find slope:

1. From the graph

2. From 2 points

3. From the equation

To graph the slope,

Top # go up if + , down if –

Bottom # go to right

Draw graphs

of lines

using slope and y- intercept

Set….a collection of objects

Naming sets:

1. Roster method

2. Set-builder notation

Inequalities

• Symbols

• Graph on a number line

Operations on sets

Union (A B)

Set of all elements of A together with all elements of B. They are in A or in B… all are included!

☺☺☺☺☺☺☺☺☺☺

Operations on sets

Intersection (A B)

Set of all elements that are common to A and B. They are in both A and in B Some are left out!

☹☹☹☹☹☹☹☹☹☹

Solving Inequalities

Do operations exactly like solving equations, but if you divide both sides by a negative number or multiply both sides by a negative #, you must reverse the inequality symbol.

constant

• Put answer as

Variable < constant

OR

Variable constant

Chapter 10

Operations with Square Roots

= the number whose square is x

Some simplify to whole numbers because the radicands are perfect squares

= a

x

2a

Other radicands are not perfect squares, but can be simplified

= ab a b

Adding and Subtracting

1. You must have the same number or expression under the to + or –

2. Add the coef. of like radicals

Multiplying Square Roots

1. Multiply coefficients

2. Multiply expressions under the radicals and put under one √

3. Simplify

Dividing Square Roots

1. Divide out common factors to reduce

2. Simplify

3. Rationalize if denominator still has

You cannot leave a radical in the denominator

Two types:

1. Monomial in denom

2. Binomial in denom

Use a number between 1 and 10 times a power of 10.

There will be one digit then the decimal point !

Dividing Polynomials

1. Polynomial ÷ monomial

2. Polynomial ÷ binomial