1.1 variables and expressions how can a verbal … variables and expressions how can a verbal...

1.1 Variables and Expressions

How can a verbal expression be translated to an algebraic

expression? Recall:

Variable:

Algebraic Expression:

Examples of Algebraic Expressions:

Different ways to show multiplication:

Product Factors

Powers

Power:

Base:

Exponent:

Symbols Words Meaning 31 3 to the first power 32 3 to the second power 33 3 ∙ 3 ∙ 3

34 3 ∙ 3 ∙ 3 ∙ 3

2𝑏6 𝑥𝑛 x to the nth power

Evaluate:

Examples:

1. Evaluate 26

2. Evaluate 43

You Try:

3. Evaluate 35

4. Evaluate 24

Translating Expressions:

You will often need to translate between verbal expressions and algebraic expressions. What is

the difference between a verbal expression and an algebraic expression?

Writing Algebraic Expressions

**The phrase that always gives the most problems is “less than”

Which expression means three less than seven?

7 – 3 or 3 - 7

This often gets confused with three less seven, which means…

Examples:

Write an algebraic expression for each verbal expression.

1. Eight more than a number

2. 7 less the product of 4 and a number x

3. One third of the original area a

4. The product of 7 and m to the fifth power

You Try:


5. 5 less than a number

6. 9 plus the product of 2 and the number d

7. Two thirds of the original volume V

8. The product of 3 and a to the seventh power

Writing Verbal Expressions

**There are many ways to write verbal expressions that all mean the same thing

Examples:

Write a verbal expression for each algebraic expression.

1. 4𝑚3

2. 𝑐2 + 21𝑑

3. 𝑥4 −𝑦

9

You Try:


4. 7𝑥3 + 1

5. 𝑦5 − 16𝑦

6. 8𝑥2

5

Practice Problems Evaluate each expression

1. 92 2. 44 3. 82 4. 106 5. 35 6. 153


7. The sum of a number and 14

8. 6 less than a number t

9. 24 less than 3 times a number

10. 1 minus the quotient of r and 7

11. Two-fifths of a number j squared

12. n cubed increased by 5

13. X more than 7

14. A number less 35

15. One-third of a number

16. F divided by 10

17. The quotient of 45 and r

18. 49 increased by twice a number

19. 18 decreased by 3 times d

20. K squared minus 11

21. 20 divided by t to the fifth power

22. The area of a circle is the number 𝜋 times the square of the radius. Write an

expression that represents the area of a circle with radius r.


23. 1

8𝑦

24. 𝑤 − 24

25. 3𝑥2

26. 𝑟4

9

27. 2𝑎 + 6

28. 𝑛3 ∙ 𝑝5

29. 1

2𝑛3

30. 𝑎2 − 18𝑏

31. 17 − 4𝑚5

32. 12𝑧2

5

1.2 Order of Operations

How do you evaluate numerical and algebraic expressions using the

Order of Operations? Order of Operations

Step 1:

Step 2:

Step 3:

Step 4:

Evaluate Numerical Expressions

Evaluate each expression using the order of operations.

Examples:

1. 15 ÷ 3 ∙ 6 − 42

2. 8 − 6 ∙ 4 ÷ 3

You Try:

3. 32 + 72 − 5 ∙ 2

4. 48 ÷ 23 ∙ 3 + 5

Evaluate Numerical Expressions with Grouping Symbols

Examples of Grouping Symbols:

Examples:

Evaluate each expression.

1. 2(5) + 3(4 + 3)

2. 2[5 + (30 ÷ 6)2]

3. 6+4

32∙4

4. (8 − 3) ∙ 3(3 + 2)

You Try:

Evaluate each expression.

5. (15 − 9) + 3 ∙ 6

6. 45 + [(1 + 1)3 ÷ 4]

7. 62−8

4(3+7)

8. 4[12 ÷ (6 − 2)]2

9. 25−6∙2

33−5∙3−2

Evaluate Algebraic Expressions

To evaluate an algebraic expression, replace the variables with their values. Then use the order

of operations.

Examples:

1. Evaluate 𝑎2 − (𝑏3 − 4𝑐) if 𝑎 = 7, 𝑏 = 3, 𝑎𝑛𝑑 𝑐 = 5

2. Evaluate 𝑥(𝑦3 + 8) ÷ 12 if 𝑥 = 3 𝑎𝑛𝑑 𝑦 = 4

You Try:

3. Evaluate 2(𝑥2 − 𝑦) + 𝑧2 if 𝑥 = 4, 𝑦 = 3, 𝑎𝑛𝑑 𝑧 = 2

Real World Practice:

1. The Pyramid Arena in Memphis is the third largest pyramid in the world. The area of its

base is 360,000 square feet, and it is 321 feet high. The volume of a pyramid is one-

third of the product of the area of the base B and its height h.

a. Write an expression that represents the volume of a pyramid.

b. Find the volume of the Pyramid Arena in Memphis

2. According to market research, the average consumer spends $78 per trip to the mall on

weekends and only $67 per trip during the week.

a. Write an algebraic expression to represent how much the average consumer spends

at the mall in x weekend trips and y weekday trips.

b. Evaluate the expression to find what the average consumer spends after going to

the mall twice during the week and 5 times on the weekends.

Practice Problems: Evaluate each expression

1. 22 + 3 ∙ 7

2. 18 ÷ 9 + 2 ∙ 6

3. 10 + 83 ÷ 16

4. 12 ÷ 3 ∙ 5 − 42

5. (11 ∙ 7) − 9 ∙ 8

6. (12 − 6) ∙ 52

7. 35 − (1 + 102)

8. 108 ÷ [3(9 + 32)]

9. [(63 − 9) ÷ 23]4

10. 8+33

12−7

11. (1+6)9

52−4

12. 3+23

52(6)

13. 2∙82−22∙8

2∙8

Evaluate each expression if 𝑎 = 4, 𝑏 = 6, 𝑎𝑛𝑑 𝑐 = 8

14. 8𝑏 − 𝑎

15. 2𝑎 + (𝑏2 ÷ 3)

16. 𝑏(9−𝑐)

𝑎2

Evaluate each expression if 𝑟 = 2, 𝑠 = 3, 𝑎𝑛𝑑 𝑡 = 11

17. 𝑟 + 6𝑡

18. 7 − 𝑟𝑠

19. (2𝑡 + 3𝑟) ÷ 4

20. 𝑠2 + (𝑟3 − 8)5

21. 𝑡2 + 8𝑠𝑡 + 𝑟2

22. 3𝑟(𝑟 + 𝑠)2 − 1

23. A sales representative receives an annual salary s, an average commission each month c,

and a bonus b for each sales goal that she reaches.

a. Write an algebraic expression to represent her total earnings in one year if she

receives four equal bonuses.

b. Suppose her annual salary is $52,000 and her average commission is $1225 per

month. If each bonus is $1150, how much does she earn in a year?

1.3 Open Sentences

How do you solve open sentence equations and inequalities? Activator: Find the truth values of the following sentences:

a. There are 12 inches in a foot. __________

b. October is the 11th month of the year. ____________

c. 30 – 14 ÷ 2 = 5 ∗ 5 – 1 ∗ 2 ____________

d. Broccoli tastes good. ___________

Open Sentence:

Variable:

If the open sentence has = then it is called an _______________________

If the open sentence has <, >, ≤, ≥ then it is called an ______________________

Examples:

a. It’s the best movie of the year.

b. She likes her job.

c. x + 5 = 25

d. Which statement in the activator is an open sentence? Why?

Replacement Set:

Element:

Solution Set:

Examples:

1. 2𝑎 + 5 = 21 If the replacement set is {6, 7, 8, 9}, find the solution set.

a 2𝑎 + 5 = 21 True or False?

So the solution set is:

2. 24 − 2𝑦 ≥ 13 If the replacement set is {3, 4, 5, 6}, find the solution set.

y 24 − 2𝑦 ≥ 13 True or False?


You Try:

3. 6𝑛 + 7 = 37 If the replacement set is {3, 4, 5, 6, 7}, find the solution set.

n 6𝑛 + 7 = 37 True or False?


4. 19 > 2𝑦 − 5 If the replacement set is {5, 6, 7, 8}, find the solution set.

y 19 > 2𝑦 − 5 True or False?


Practice Problems: Find the solution of each equation is the replacement sets are

a = {0, 3, 5, 8, 10} and b = {12, 17, 18, 21, 25}

1. 𝑏 − 12 = 9

2. 22 = 34 − 𝑏

3. 15

𝑎= 3

4. 5(𝑎 − 1) = 10

5. 40

𝑎− 4 = 0

6. 27 = 𝑎2 + 2

Find the solution set for each inequality using the given replacement set

7. 𝑠 − 2 < 6 {6, 7, 8, 9, 10, 11}

8. 5𝑎 + 7 > 22 {3, 4, 5, 6, 7}

9. 3 ≥25

𝑚 {1, 3, 5, 7, 9, 11}

10. 2𝑎

4≤ 8 {12, 14, 16, 18, 20, 22}

11. 2.7(𝑥 + 5) ≥ 17.28 {1.2, 1.3, 1.4, 1.5}

12. 𝑥 +2

5≤ 1

3

20 {

1

4,

1

2,

3

4, 1, 4, 11 }

1.4, 1.5, 1.6 Properties of Algebra

What are the properties of Algebra, and how can you recognize

these properties?

Use your flip book to help you recall the properties below.

Examples:

Name the property shown by each statement.

1. 3 + 7 + 9 = 7 + 3 + 9

2. (𝑎 ∙ 6) ∙ 5 = 𝑎 ∙ (6 ∙ 5)

3. 0 ∙ 12 = 12

You Try:

Name the property shown by each statement.

4. 3 ∙ 10 ∙ 2 = 3 ∙ 2 ∙ 10

5. (2 + 5) + 𝑚 = 2 + (5 + 𝑚)

6. 17 ∙ 1 = 17

Find the value of n in each equation. Then name the property that is used.

Examples:

1. 42 ∙ 𝑛 = 42

2. 28𝑛 = 0

3. 5 + 𝑦 = 5

You Try:

4. 𝑛 ∙ 9 = 1

5. 7 + 𝑥 = 0

6. 𝑛 ∙1

5= 1

Additive Identity Symmetric Property

Multiplicative Identity transitive Property

Additive Inverse commutative Property (addition)

Multiplicative Inverse Commutative property (multiply)

Substitution associative property (addition)

Reflexive Property Associative property (multiply)

Multiplicative Property of Zero

Simplifying Expressions

Term:

Coefficient:

Like Terms:

Equivalent Expressions:

Simplest Form:

Examples:

Combine Like Terms to Simplify. If already in simplest form, write simplified.

1. 15𝑥 + 18𝑥

2. 3𝑛2 + 10𝑛 + 9𝑛2

3. 7𝑝2 − 8𝑝 − 2𝑝2

You Try:

Combine Like Terms to Simplify. If already in simplest form, write simplified.

4. 17𝑎 + 21𝑎

5. 12𝑥2 − 8𝑥2 + 6𝑥

6. 𝑏2 + 13𝑏 + 13

Distributive Property

What does the word distribute mean? Give an example.

In math, distribute means to multiply a value by all other quantities in parentheses.

The Distributive Property:

This can be used for numerical and algebraic expressions.

Examples:

Rewrite each product using the Distributive Property, then simplify.

1. 5(𝑔 − 9)

2. 3(𝑥2 + 𝑥 − 1)

3. −6(𝑟 − 𝑠 − 𝑡)

You Try:


4. 12(𝑦 + 3)

5. (8 + 𝑛)2

6. 4(𝑥2 + 8𝑥 + 2)

Evaluate Expressions Using Multiple Properties

Examples:

Simplify the expression and name the property used in each step.

1. 6(𝑥 − 2𝑦) + 4(−3𝑥 + 𝑦)

2. Three times the sum of 3x and 2y added to five times the sum of x and 4y

You Try:


3. 4(𝑎 + 𝑏) + 2(𝑎 + 2𝑏)

4. 5 times the difference of q squared and r, plus eight times the sum of 3q and 2r

Practice Problems Find the value of n in each equation. Then name the property that is used

1. 1 = 2𝑛

2. 6 = 6 + 𝑛

3. 𝑛 ∙ 1 = 5

4. 4 − 𝑛 = 0

5. 4 ∙1

4= 𝑛


6. 2(𝑥 + 4)

7. (5 + 𝑛)3

8. 8(4 − 3𝑚)

9. −3(𝑥 − 6)

10. 27(1

3− 2𝑏)

11. 4(𝑝 + 𝑞 − 𝑟)

12. −6(2 − 𝑥2 + 𝑥)

13. 5(6𝑚3 + 4𝑛 − 3𝑛)

Simplify each expression. If not possible, write simplified.

14. 4𝑥 + 5𝑦 + 6𝑥

15. 5𝑎 + 3𝑏 + 2𝑎 + 7𝑏

16. 3(4𝑥 + 2) + 2𝑥

17. 7𝑚 + 6𝑛 − 8

18. 7(𝑎𝑐 + 2𝑏) + 2𝑎𝑐

19. 3(4𝑚 + 𝑛) + 2𝑚(4 + 𝑛)

20. 6(0.4𝑓 + 0.2𝑔) + 0.5𝑓

21. 𝑥2 + 3𝑥 + 2𝑥 + 5𝑥2

22. (𝑑 + 5)8 + 2𝑓

23. 14𝑎2 + 13𝑏2 + 27


24. 1

5(5) ∙ 17

25. 2(3 ∙ 2 − 5) + 3 ∙1

3

26. Twice the sum of s and t, decreased by 2

27. 5 times the product of x and y increased by 3xy

28. 6 times the sum of x and y squared, minus 3 times the sum of x and y squared

1.8 Number Systems

What are the relationships among the various number sets in the

real number system? How do you order real numbers?

Sets of Numbers Definition Examples Symbol

Natural Numbers N

Whole Numbers W

Integers Z

Rational Numbers Q

Irrational

Numbers

I

Real Numbers R

Classify Real Numbers

Examples:

Name ALL sets of numbers to which each real number belongs.

1. 5

22

2. √81

3. √56

4. 6

11

You Try:

Name ALL sets of numbers to which each real number belongs.

5. √17

6. 1

6

7. √169

8. −√9.16

Closure Property

If something is closed, what does that mean?

Sets of numbers can also be closed, under certain operations.

This means that if you perform that operation on numbers from the set, that the answer will

also be from the set.

Examples:

1. Is the set of whole numbers closed under multiplication?

2. Is the set of integers closed under division?

3. Is the set of real numbers closed under subtraction?

You Try:

4. Is the set of whole numbers closed under subtraction?

5. Is the set of integers closed under addition?

6. Is the set of natural numbers closed under division?

Comparing Real Numbers

There are three signs we use to compare numbers. < > =

Examples:

Replace each ◊ with <, >, or = to make the sentence true.

1. √19 ◊ 3. 8̅

2. 22

3 ◊ √5

You Try:


3. 7. 2̅ ◊ √52

4. 0. 8̅ ◊ 8

9

Ordering Real Numbers

We can take this comparison a step further and compare multiple numbers at once. This is

called ordering.

Examples:

1. Order from least to greatest. 2. 63̅̅̅̅ , −√7,8

3,

53

−20

2. Order from greatest to least. −1. 46̅̅̅̅ , 0.2, √2, −1

6

You Try:

3. Order from least to greatest. 12

5, √6, 2. 4̅,

61

25

4. Order from greatest to least. √0.42, 0. 63̅̅̅̅ , √4

9

Practice Problems Name ALL sets of numbers to which each real number belongs.

1. −√64

2. 8

3

3. √28

4. 56

7

5. √10.24

6. −54

19

7. −√22

8. √72

2

Determine whether each set of numbers is closed under the indicated operation.

9. Whole numbers, Division

10. Rational Numbers, Addition

11. Rational Numbers, Division

12. Natural Numbers, Subtraction

13. Irrational Numbers, Addition

14. Natural Numbers, Addition

15. Whole Numbers, Multiplication

16. Integers, Subtraction

17. Integers, Multiplication


18. √17 ◊ 41

10

19. 2

9 ◊ 0. 2̅

20. 1

6 ◊ √6

21. 5. 72̅̅̅̅ ◊ √5

22. √2

3 ◊

2

3

Order each set of numbers from least to greatest

23. 1

8, √

1

4, 0. 15̅̅̅̅ , −15

24. √30, 54

9, 13, √

1

30

25. 0.6, √16

49,

5

9

26. √0.06, 0. 24̅̅̅̅ , √9

144

1.1 variables and expressions how can a verbal … variables and expressions how can a verbal...

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