Mathematics: Solving Equations and Inequalities with Properties of Real Numbers, Lecture notes of Algebra

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Beginning Algebra

M ATH 100B

Math Study Center

BYU-Idaho

Preface

This math book has been created by the BYU-Idaho Math Study Center for

the college student who needs an introduction to Algebra. This book is the product

of many years of implementation of an extremely successful Beginning Algebra

program and includes perspectives and tips from experienced instructors and

tutors.

Videos of instruction and solutions can be found at the following url:

https://youtu.be/-taqTmqALPg?list=PL_YZ8kB-SJ1v2M-oBKzQWMtKP8UprXu8q

The following individuals have assisted in authoring:

Rich Llewellyn

Daniel Baird

Diana Wilson

Kelly Wilson

Kenna Campbell

Robert Christenson

Brandon Dunn

Hannah Sherman

We hope that it will be helpful to you as you take Algebra this semester.

The BYU-Idaho Math Study Center

Chapter 3 – Lines

Section 3.1 – Graphing…………………………………………………………………………

Graphing by Pick ‘n Stick, Intercepts

Section 3.2 – Slope……………………………………………………………………………..

Section 3.3 – Graphing with slope, Slope-Intercept……………………………………………

Section 3.4 – Graphing with slope, Standard……...……………………………………………

Section 3.5 – Writing Equations………………………………………………………………..

Find the Equation of a line given a slope and a point or two points

Chapter 4 – Exponents and Polynomials

Section 4.1 – Laws of Exponents………………………………………………………………

Mutliplication, Power, Division, Zero rules for exponents; Scientific Notation Arithmetic

Section 4.2 – Intro to Polynomials……………………………………………………………..

Terminology, Addition and Subtraction of Polynomials

Section 4.3 – Multiplication of Polynomials…………………………………………………..

Monomial × Polynomials, Special Cases, Binomial Squared, Binomial × Binomial

Section 4.4 – Division of Polynomials…………………………………………………………

Division of Polynomials by Monomials

Chapter 5 – Factoring

Section 5.1 – Intro to Factoring, Methods 1 and 2……………………………………………..

Factoring by pulling out GCF, Grouping with 4 terms

Section 5.2 – Factoring Trinomials, Method 3…………………………………………………

Factoring Trinomials with lead coefficient =

Section 5.3 – Factoring Trinomials, Method 4…………………………………………………

Factoring Trinomials with lead coefficient ≠ 1, ac-method

Section 5.4 – Factoring Special Cases………………………………………………………….

Factoring Perfect Squares and Difference of Squares

Section 5.5 – Factoring With All Methods……………………………………………………..

Holistic Approach to determine which method to use

Section 5.6 – Solving Polynomial Equations…………………………………………………..

Zero Multiplication; Solving Polynomial Equations by factoring

Chapter 1 :

ARITHMETIC &

VARIABLES

OVERVIEW

1.1 LCM and Factoring

1.2 Fractions

1.3 Decimals

1.4 Exponents, Order of Operations, Rounding

1.5 Variables and Formulas

1.6 Negatives

1.7 Laws of Simplifying

ONE: If two numbers only have the number one as a common factor that means these numbers are

“relatively prime”. Example: The numbers 7 and 12 are relatively prime because the only factor they have in common 1.

FACTOR: The numbers that are multiplied together to get the product are factors.

Example: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 because these are the only natural numbers that can multiply to get a product of 42.

LCM: Least Common Multiple. You can find this when the multiples of two or more numbers meet

up at a common number. Example: The multiples of 8 are 8, 16, 24, 32, 40, 48, and so on. The multiples of 12 are 12, 24, 36, 48, 60, and so on. To find the LCM you find the smallest multiple these two numbers have in common. 8 and 12 both have 24, 48 in common but we are looking for the least common multiple, which means the smallest number. Therefore, the LCM is 24 because it is the smallest common multiple between 8 and 12.

I. Find All Factors Factoring is a useful and necessary skill when adding and subtracting fractions and will be a very helpful skill to have in algebra.

For factoring number, we simply write down all the numbers that go into it. Number to be Factored Factors

12 1 , 2 , 3 , 4 , 6 , 12

Find Factors

1. Start with 1 and move up finding numbers that are factors.
2. List the numbers you have found. These are all the factors

EXAMPLES

Find all factors of 48

Step 1: Find all factors that multiply to be the product 48, starting with the number 1 and 48 and moving up the number line.

Note: In the last box we see 8x6=48. The 8 has already been used in the factors, so we know that all of the factors have been found. To make them a little easier to see we can put them in numerical order from smallest to largest. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 3: Now we will list all the numbers we used up until we saw the repeated number and these will be our factors. 1, 48, 2, 24, 3, 16, 4, 12, 6, 8

II. Find Prime Factorization of a Number Once again, factor each number individually just like we did in the previous example.

Find the prime factorization of 60 and 72

Step 1: Find the factors of 60 until the factors are all prime..

Step 2: Repeat step one for 72

EXAMPLES

EXAMPLES

Prime Factorize

1. Find a factor, break the number up.
2. Repeat until all factors are prime.

Find the LCM of 4 and 5

Step 1: Write out the multiples for 4 and 5

Step 2: The first number that both multiples hit is 20

Step 3: The LCM of 4 and 5 is 20.

We can also find the LCM of numbers by using prime factorization

2 × 2 × 3 Prime factorization of the smallest number that both will go in to 2 × 2 × 3 = 12

Thus 12 is the LCM of 4 and 6

4 5 8 10 12 15 16 20 20 25 24 30 28 35 32 40

Prime factorization of 4 Prime factorization of 6

4 = 2 × 2 6 = 2 × 3

EXAMPLE

EXAMPLE

Find the LCM of 40 and 36.

2 × 2 × 2 × 3 × 3 × 5 Prime factorization of the smallest number that both will go in to 2 × 2 × 2 × 3 × 3 × 5 = 360

Thus 360 is the LCM of 40 and 360

Prime factorization of 40 Prime factorization of 36

40 = 2 × 2 × 2 × 5 36 = 2 × 2 × 3 × 3

Find the LCM (Prime Factorization)

1. Prime factorize
   2. Write out the smallest number that they all can go into.
   3. Multiply it out = LCM

EXAMPLE

Answers

3. 1 , 37 25. In Class.

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• 1. 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 23.
• 2. 1 , 2 , 3 , 5 , 6 , 10 , 15 , 25 , 30 , 50 , 75 , 150 24.
• 5. 1 , 3 , 5 , 15 , 25 , 4. 1 , 2 , 3 , 4 , 6 , 12 26. In Class.
• 6. 1 , 3 , 9 , 27 ,
• 7. 2 × 5 ×
• 8. 2 × 2 × 2 ×
• 9. 3 × 3 ×
• 10. 2 × 2 × 5 ×
• 11. 3 × 3 ×

NUMERATOR: The top of a fraction. This is always an integer, never a decimal.

DENOMINATOR: The bottom of the fraction. This is always an integer, never a decimal,

and never zero.

SIMPLIFY: Fractions are simplified when the numerator and denominator have no factors in

common. You can also say that the fraction is reduced****.

Example: The fraction ⡩⡳⡱⡨ can be simplified or reduced down to ⡩⡰.

ONE: Any number divided by itself is 1.

1.2 Arithmetic of Fractions

A RULES OF ARITHMETIC AND VARIABLES (Overview)

B ARITHMETIC OF FRACTIONS

DEFINITIONS & BASICS

CHAPTER ONE TOPICS

LCM AND FACTORING Find Factors Find Least CommonMultiples

FRACTIONS Addition/Subtraction Multiplication/Division

DECIMALS AND PERCENTS

Addition/Subtraction Change to decimals

Multiplication Change to fractions

Division ROUNDING, ESTIMATION, EXPONENTS, ORDER OF OPERATIONS

Nearest place value; Round and then compute

VARIABLES AND FORMULAS

Replace numbers and make formulas NEGATIVES Addition/Subtraction^ Multiplication^ Division

Laws of Simplifying

OBJECTIVES

• Know the parts of a fraction and when it is simplified fully
• Add, subtract, multiply, and divide fractions

2 3

Add

Step 1: Common denominator. If we multiply the denominator here, we’ll have some big numbers to work with. Let’s use prime factorization to find the LCD.

×

=

×

=

Prime factorization of 30: 2×3× Prime factorization of 24: 2×2× We need a number whose factors include each of these: 2 × 2 × 3 × 5 = 60

= (^61) Step 2: Now that the denominators are the same, add the numerators.

⡰⡴ ⡴⡨

Answer: ➃❸ ➃❷

Subtract ➂

➆ ㎘^

❸ ➀ ⡳ ⡷

㎘

⡩ ⡱

×⡱ ×⡱

The common denominator is 9, so change the

to a

5 9

㎘

3 9

=

2 9

Subtract the numerators.

Answer:

❹ ➆

EXAMPLES

EXAMPLES

II. Multiplication of Fractions When multiplying fractions, common denominators are not needed. This is different from addition and subtraction.

Multiply

×

×

For multiplication don’t worry about getting common denominators

⡳ ⡴ ×

=

Multiply the numerators straight across

×

=

Multiply the denominators straight across

Answer: ➂ ❸➅

III. Division of Fractions Dividing fractions is an interesting idea, because a fraction itself is a division (i.e. ½ can also be said as 1 divided by 2). Because of this, there is a special process for dividing fractions that actually simplifies it. To divide a number by a fraction, reciprocate the fraction and multiply instead. Now you’re doing a multiplication problem, one you already know how to do.

Multiplication of Fractions

1. No common denominators
2. Multiply numerators
3. Multiply denominators

EXAMPLES

Division of Fractions

1. Change any fractions into improper fractions.
2. Keep the first fraction the same, change the division sign to multiplication, and flip the second fraction’s numerator and denominator: Keep it, change it, flip it.
3. Multiply straight across.

Find Factors.

1. 16 2. 48 3. 110

Find the prime factorization.

4. 60 5. 630 6. 225 7. 210

Find the least common multiple (LCM).

8. 3 & 13 9. 8 & 22 10. 6 & 7 11. 35 & 21 12. 108 & 32 13. 1500 & 180 14. If two planets are aligned with the sun and one planet goes around the sun every 12 years and the other planet takes 22 years, how long will it be before they are in alignment again?

Add by hand.

15. ⡩ ⡴ +^

⡩ ⡱

⡱

⡶ +^

⡱ ⡳

Subtract by hand.

18. ⡳ ⡶ ㎘^

⡩ ⡰

⡵ ㎘^1

⡰ ⡱

⡩⡩ ㎘^

⡱ ⡵

Multiply by hand.

21. (^) 3 × ⡩ ⡩⡰

⡳ ×^

⡩ ⡴

⡳ ×^

⡵ ⡩⡰

Divide by hand.

24. ⡳ ⡩⡰ 㐂^

⡩ ⡱

⡷ 㐂^6

⡩⡩ 㐂^

⡳ ⡴

Solve and simplify with calculator.

27. ⡲ ⡵ +^

⡩ ⡷

⡩⡷ +^

⡰ ⡩⡵

⡳⡩ +^

⡩⡩ ⡱⡲

30. ⡶ ⡩⡱ ㎘^

⡩⡩ ⡰⡴

31.^11

12 ㎘^

1 21

⡱⡱ ㎘^

⡱ ⡩⡩

33. ⡲⡳ ⡱ ×^

⡲ ⡩⡷

⡰⡱ ×^

⡩⡩ ⡷

⡱⡩ ×^

⡶ ⡩⡵

36. ⡱⡲ ⡱⡵ 㐂^

⡰ ⡵

⡱⡱ 㐂^

⡩⡵ ⡰⡩

⡩⡱ 㐂^

⡴ ⡲⡩

Preparation. After reading some of section 1.3, find the following:

39. 21.34 + 12.01= 34.2 ㎘ 18 = 72.1 + 11.03 = 10.4 ㎘ 4.9 = 40.. 04 × .26= 48.2 㐂 1.6= 9.3 × 4.1= 4.4 㐂 2.2=

1.2 EXERCISE SET

Answers

1. 1 , 2 , 4 , 8 , 16 23. (^) ⡩⡳⡵ 2. 1 , 2 , 3 , 4 , 6 , 8 , 12 , 16 , 24 , 48 24. ⡳⡲ 3. 1 , 2 , 5 , 10 , 11 , 22 , 55 , 110 25. ⡩ ⡷ 4. 2 × 2 × 3 × 5 26. ⡲⡰ ⡳⡳ 5. 2 × 3 × 3 × 5 × 7 27. ⡲⡱ ⡴⡱ 6. 3 × 3 × 5 × 5 28. ⡰⡵⡴⡱⡰⡱ 7. 2 × 3 × 5 × 7 29. (^) ⡩⡨⡰⡶⡱ 8. 39 30. (^) ⡰⡴⡳ 9. 88 31. ⡵⡱⡶⡲ 10. 42 32. (^) ⡩⡩⡴ 11. 105 33. ⡴⡨ ⡩⡷ 12. 864 34. ⡳⡳ ⡴⡷ 13. 4500 35. (^) ⡳⡰⡵⡷⡴ 14. 132 years^ 36. ⡩⡩⡷ ⡱⡵

15. ⡩ ⡰ 37.^

⡩⡲ ⡩⡵

16. ⡰⡴ ⡱ 38. ⡲⡳⡩ ⡵⡶ 17. ⡳⡷ ⡲⡨ or 1 ⡩⡷ ⡲⡨ 39. In Class. 18. ⡩⡶ 40. In Class. 19.^2 ㄘㄗㄠ or^ ⡲⡴ ⡰⡩ 20. (^) ⡵⡵⡷ 21. ⡩⡲ 22. (^) ⡩⡳⡰

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