STANDARDS OF LEARNING

CONTENT REVIEW NOTES

ALGEBRA I

1st Nine Weeks, 2016-2017

2

OVERVIEW

Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

for students and parents. Each nine weeks’ Standards of Learning (SOLs) have been identified and a detailed

explanation of the specific SOL is provided. Specific notes have also been included in this document to assist

students in understanding the concepts. Sample problems allow the students to see step-by-step models for

solving various types of problems. A “ ” section has also been developed to provide students with the

opportunity to solve similar problems and check their answers. The answers to the “ ” problems are found

at the end of the document.

The document is a compilation of information found in the Virginia Department of Education (VDOE)

Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE

information, Prentice Hall textbook series and resources have been used. Finally, information from various

websites is included. The websites are listed with the information as it appears in the document.

Supplemental online information can be accessed by scanning QR codes throughout the document. These will

take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the

document to allow students to check their readiness for the nine-weeks test.

To access the database of online resources scan this QR code,

or visit http://spsmath.weebly.com

The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of

questions per reporting category, and the corresponding SOLs.

Algebra I Blueprint Summary Table

Reporting Categories No. of Items SOL

Expressions & Operations 12 A.1

A.2a – c

A.3

Equations & Inequalities 18 A.4a – f

A.5a – d

A.6a – b

Functions & Statistics 20 A.7a – f

A.8

A.9

A.10

A.11

Total Number of Operational Items 50

Field-Test Items* 10

Total Number of Items 60

* These field-test items will not be used to compute the students’ scores on the test.

It is the Mathematics Instructors’ desire that students and parents will use this document as a tool toward the

students’ success on the end-of-year assessment.

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4

Expressions and Order of Operations A.1 The student will represent verbal quantitative situations algebraically and evaluate these

expressions for given replacement values of the variables.

Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables. An expression can be represented algebraically… Example 1: 6x + 5 Example 2: a – 9b or in written form. Example 1: The sum of a number and eleven Example 2: One half of a number squared minus four Some common words are used to indicate each operation. Many of these are shown in the table below, but there are others.

Add Subtract Multiply Divide Equals

Plus

Sum

More than

Increased by

Total

All together

Add to

And

Difference

Minus

Less than

Decreased by

Take away

How many left

Remaining

Subtracted by

Less

Times

Product

Multiplied By

Doubled (x2)

Tripled (x3)

By

Squared (a·a)

Cubed (a·a·a)

Part

Quotient

Divided by

Each

Half ( ÷ 2)

Split (÷ 2)

Is

Are

Is Equal To

Is equivalent to

Equals

Expressions and Order of Operations Translate the written expressions to algebraic expressions, and algebraic expressions to written

expressions.

1. the difference of eleven and x

2. three times the sum of a number and ten

3. four times the difference of n squared and five

4. 12g ÷ 4

5. a² - b⁴

5

Expressions are simplified using the order of operations and the properties for operations with real numbers. The order of operations is as follows: First: Complete all operations within grouping symbols. If there are grouping symbols within other grouping symbols, do the innermost operation first. Grouping symbols

include parentheses (a), brackets [a], radical symbols √𝒂, absolute value bars |𝒂|,

and the fraction bar 𝒂

𝒃.

Second: Evaluate all exponents. Third: Multiply and/or divide from left to right. Fourth: Add and/or subtract from left of right. To evaluate an algebraic expression substitute in the replacement values of the variables and then evaluate using the order of operations.

Example 1: 15 − 9 ÷ (−3)2

Step 1: 15 − 9 ÷ (−3)2 Step 2: 15 − 9 ÷ 9 Step 3: 15 − 1

Step 4: The answer is 14

Example 2: (𝑝 − 3)2 + 2𝑝 − 4, 𝑝 = −7

Step 1: (−7 − 3)2 + 2(−7) − 4

Step 2: (−10)2 + 2(−7) − 4 Step 3: 100 + 2(−7) − 4 Step 4: 100 + (−14) − 4 Step 5: 86 − 4

Scan this QR code to go to an

order of operations video tutorial!

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Step 6: The answer is 82

Example 3: 20 − √3 ∙ 10 − 14 + |7 − 5 ∙ 4|

Step 1: 20 − √30 − 14 + |7 − 20|

Step 2: 20 − √16 + |−13| Step 3: 20 − 4 + 13 Step 4: The answer is 29

Expressions and Order of Operations Evaluate each expression. a=2, b=5, x= - 4, and n=10.

6. [𝑎 + 8(𝑏 − 2)]2 ÷ 4

7. (2𝑥)2 + 𝑎𝑛 − 5𝑏

8. 𝑛2 + 3(𝑎 + 4) 9. 𝑏𝑥 − 𝑎𝑥

Evaluate each expression using the order of operations.

10. 63 − √10 ÷ 2 + 2 ∙ 22

11. √75 + (9 + 3)2 − 33

+ 19

12. (7∙3−18)3

√63−14+21

13. |6 − √33 + 22| − 8

Scan this QR code to go to a video for more complicated order

of operations help.

7

Properties of Real Numbers A.4 The student will solve multistep linear and quadratic equations in two variables,

including b) justifying steps used in simplifying expressions and solving equations, using field

properties and axioms of equality that are valid for the set of real numbers and its subsets;

A.5 The student will solve multistep linear inequalities in two variables, including

b) justifying steps used in solving inequalities, using axioms of inequality and properties

Property Definition Examples

Multiplicative Property

of Zero

Any number multiplied by zero

always equals zero.

𝑎 (0) = 0

0 ∙ (−14) = 0

Additive Identity Any number plus zero is equal

to the original number.

𝑎 + 0 = 𝑎

126 + 0 = 126

Multiplicative Identity Any number times one is the

original number.

𝑎 ∙ 1 = 𝑎

1 ∙ 78 = 78

Additive Inverse A number plus its opposite

always equals zero.

𝑎 + (−𝑎) = 0

−21 + 21 = 0

Multiplicative Inverse

A number times its inverse

(reciprocal) is always equal to

one.

𝑎 ∙ 1

𝑎= 1

5

2 ∙

2

5= 1

Associative Property

When adding or multiplying

numbers, the way that they are

grouped does not affect the

outcome.

(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)

5 + (3 + 8) = (5 + 3) + 8

(𝑎𝑏)𝑐 = 𝑎(𝑏𝑐)

6(3𝑎) = (6 ∙ 3) 𝑎

Commutative Property

The order that you add or

multiply numbers does not

change the outcome.

𝑎 + 𝑏 = 𝑏 + 𝑎

14 + 6 = 6 + 14

𝑎𝑏 = 𝑏𝑎

8 ∙ 3 = 3 ∙ 8

Distributive Property

For any numbers a, b, and c:

a(b + c) = ab + ac

5 ( 3 − 2) = 5 ∙ 3 − 5 ∙ 2

−3 (𝑎 + 𝑏) = −3𝑎 + (−3𝑏)

𝑜𝑟 − 3 (𝑎 + 𝑏) = −3𝑎 − 3𝑏

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Substitution property of

equality

If a = b, then b can replace a.

A quantity may be substituted

for its equal in any expression.

𝐼𝑓 5 + 2 = 7, 𝑡ℎ𝑒𝑛 (5 + 2) ∙ 4

= 7 ∙ 4

𝐼𝑓 𝑎 = 5, 𝑡ℎ𝑒𝑛 11𝑎 = 11 ∙ 5

Reflexive Property of

Equality Any quantity is equal to itself.

𝑎 = 𝑎

5

3=

5

3

Transitive Property of

Equality

If one quantity equals a second

quantity and the second

quantity equals a third, then the

first equals the third.

𝐼𝑓 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.

𝐼𝑓 2 + 4 = 6, 𝑎𝑛𝑑 2(3) = 6,

𝑡ℎ𝑒𝑛 2 + 4 = 2(3)

Symmetric Property of

Equality

If one quantity equals a second

quantity, then the second

quantity equals the first.

𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎

𝐼𝑓 25 = 13𝑎 − 1, 𝑡ℎ𝑒𝑛 13𝑎 − 1 = 25

Properties of Real Numbers Match the example on the left to the appropriate property on the right.

1. (𝑥 + 3) + 𝑦 = 𝑥 + (3 + 𝑦)

2. 1 = 1

2𝑥∙ 2𝑥

3. 3𝑥 + 6 = 6 + 3𝑥

4. (5 – 𝑥)2 = 10 – 2𝑥

5. 𝑏5 + 0 = 𝑏5

6. (𝑥 + 3) + 𝑦 = 𝑦 + (𝑥 + 3)

7. 0 ∙ 17𝑛 = 0

8. 5𝑥 + (−5𝑥) = 0

9. 𝑥𝑦𝑧 = 𝑥𝑦𝑧

10. If one dollar is the same as four quarters,

and four quarters is the same as ten dimes,

then ten dimes is the same as one dollar.

A. Multiplicative Property of Zero

B. Additive Identity

C. Multiplicative Identity

D. Additive Inverse

E. Multiplicative Inverse

F. Associative Property

G. Commutative Property

H. Distributive Property

I. Substitution Property of Equality

J. Reflexive Property of Equality

K. Transitive Property of Equality

L. Symmetric Property of Equality

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Solving Equations

A.4 The student will solve multistep linear and quadratic equations in two variables,

including d) solving multistep linear equations algebraically and graphically; f) solving real-world problems involving equations and systems of equations.

You will solve an equation to find all of the possible values for the variable. In order to solve an equation, you will need to isolate the variable by performing inverse operations (or ‘undoing’ what is done to the variable). Any operation that you perform on one side of the equal sign MUST be performed on the other side as well. Drawing an arrow down from the equal sign may help remind you to do this.

Example 1: 𝑚 − 9 = −3

+9 + 9

𝑚 = 6

Check your work by plugging your answer back in to the original problem.

6 − 9 = −3

Example 2: 𝑥+4

5 = −12

∙ 5 ∙ 5

𝑥 + 4 = −60

−4 − 4

𝑥 = −64

Check your work by plugging your answer back in to the original problem.

−64 +4

5 =

−60

5= −12

Scan this QR code to go to a video tutorial on two-step

equations.

10

You may have to distribute a constant and combine like terms before solving an equation. Example 3: −4(𝑔 − 7) + 2𝑔 = −10

−4𝑔 + 28 + 2𝑔 = −10

−2𝑔 + 28 = −10

−28 − 28

−2𝑔 = −38

÷ (−2) ÷ (−2)

𝑔 = 19

Check your work by plugging your answer back in to the original problem.

−4 (19 − 7) + 2(19) = −10

−4(12) + 2(19) = −10

−48 + 38 = −10

If there are variables on both sides of the equation, you will need to move them all to the same side in the same way that you move numbers. Example 4: 3𝑝 − 5 = 7(𝑝 − 3)

3𝑝 − 5 = 7𝑝 − 21

−3𝑝 − 3𝑝

−5 = 4𝑝 − 21

+21 + 21

16 = 4𝑝

÷ 4 ÷ 4

𝑝 = 4

Check your work by plugging your answer back in to the original problem.

3(4) − 5 = 7(4) − 21 12 − 5 = 28 − 21

7 = 7

Scan this QR code to go to a video tutorial on multi-step

equations.

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Example 5: 𝑥+10

5𝑥=

−1

5 You can begin this problem by cross multiplying!

5(𝑥 + 10) = −1(5𝑥)

5𝑥 + 50 = −5𝑥

+5𝑥 + 5𝑥

10𝑥 + 50 = 0

−50 − 50

10𝑥 = −50

÷ 10 ÷ 10

𝑥 = −5

Check your work by plugging your answer back in to the original problem.

−5+10

5(−5)=

−1

5

5

−25 =

−1

5

− 1

5 = −

1

5

Solving Equations

Solve each equation 1. 𝑘 + 11 = −8

2. 9 − 3𝑥 = 54

3. −17 = 𝑦−6

2

4. 5 (2𝑛 + 6) + 8 = 33

5. 3 − (4𝑘 + 2) = −15

6. 5𝑔 + 4 = −9𝑔 − 10

7. −2(−4𝑚 − 1) + 3𝑚 = 4𝑚 − 8 + 𝑚

8. 𝑥−4

2=

−2(3𝑥−3)

6

Scan this QR code to go to a video tutorial on equations with

variables on both sides.

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Transforming Formulas A.4 The student will solve multistep linear and quadratic equations in two variables,

including a) solving literal equations (formulas) for a given variable;

Transforming Formulas is done the same way as solving equations. Treat the variables the same way that you treat numbers, being sure to combine like terms when you can. Remember that in order to be like terms, both terms need to have the same variables, and those variables have to have the same exponent.

Example 1: 𝑆𝑜𝑙𝑣𝑒 𝑃 = 2𝑙 + 2𝑤 𝑓𝑜𝑟 𝑤

𝑃 = 2𝑙 + 2𝑤

− 2𝑙 − 2𝑙

𝑃 − 2𝑙 = 2𝑤

÷ 2 ÷ 2

𝑃−2𝑙

2= 𝑤

Example 2: 𝑆𝑜𝑙𝑣𝑒 12𝑥 − 3𝑦 = 18 𝑓𝑜𝑟 𝑦

12𝑥 − 3𝑦 = 18

−12𝑥 − 12𝑥

−3𝑦 = 18 − 12𝑥

÷ (−3) ÷ (−3)

𝑦 = −6 + 4𝑥

Example 3: 𝑆𝑜𝑙𝑣𝑒 2𝑎𝑥 + 6𝑎 = 9𝑏 + 3 𝑓𝑜𝑟 𝑎

We will have to “un-distribute” the a from each term on the left.

2𝑎𝑥 + 6𝑎 = 9𝑏 + 3

𝑎 (2𝑥 + 6) = 9𝑏 + 3

÷ (2𝑥 + 6) ÷ (2𝑥 + 6)

𝑎 = 9𝑏+3

2𝑥+6

Scan this QR code to go to a video tutorial on transforming

formulas.

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Example 4: 𝑆𝑜𝑙𝑣𝑒 3𝑏

2𝑦=

6

4𝑥 𝑓𝑜𝑟 𝑥

You can cross multiply to rewrite this problem without fractions.

3𝑏

2𝑦=

6

4𝑥

3𝑏 (4𝑥) = 6 (2𝑦)

12𝑏𝑥 = 12𝑦

÷ 12𝑏 ÷ 12𝑏

𝑥 = 12𝑦

12𝑏 Don’t forget to simplify your fractions!

𝑥 = 𝑦

𝑏

Example 5: 𝑆𝑜𝑙𝑣𝑒 𝐴 =1

2𝑏ℎ 𝑓𝑜𝑟 𝑏

To divide by 1

2 , you can multiply by the reciprocal, which is

2

1, or just 2.

𝐴 = 1

2𝑏ℎ

∙ 2 ∙ 2

2𝐴 = 𝑏ℎ

÷ ℎ ÷ ℎ

2𝐴

ℎ = 𝑏

Transforming Formulas Solve each equation for the stated variable.

1. 3𝑥 + 2𝑦 = 8 𝑓𝑜𝑟 𝑦

2. 𝐶 = 2𝜋𝑟 𝑓𝑜𝑟 𝑟

3. 𝑦 = 2𝑧+12

5 𝑓𝑜𝑟 𝑧

4. 𝑉 = 1

3𝜋𝑟2ℎ 𝑓𝑜𝑟 ℎ

5. 15 = 𝑥 + 𝑎𝑥 𝑓𝑜𝑟 𝑥

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Inequalities A.5 The student will solve multistep linear inequalities in two variables, including

a) solving multistep linear inequalities algebraically and graphically; c) solving real-world problems involving inequalities

An inequality is solved the same way as an equation. The only important thing to remember is that if you multiply or divide by a negative number, you need to switch the direction of the inequality sign. A proof of this is included in the online video tutorials or on the top of page 179 in your text book. You will also need to know how to graph inequalities on the number line. If the inequality has a greater than or equal to (≥) or less than or equal to (≤) sign, then you will use a closed point to mark the spot on the number line. This closed point indicates that the number that the point is on IS included in the solution. For a greater than (>) or less than (<) sign, you will use an open point on the number line. This open point indicates that the number that the point is on is NOT included in the solution. Example 1: Solve and graph the following inequality.

𝑦 + 3 < −4

−3 − 3

𝑦 < −7

Graph:

Example 2: Solve and graph the following inequality.

−4 ≥ −3𝑤 + 8

−8 − 8

−12 < −3𝑤

÷ (−3) ÷ (−3) Don’t forget to switch the sign direction!

4 ≤ 𝑤

Graph:

0 3 6 9 120–3–6–9

0 3 6 90–3–6–9

Scan this QR code to go to a video tutorial on solving and

graphing inequalities.

15

Example 3: 7𝑚 + 4 < 3𝑚 − 16

−3𝑚 − 3𝑚

4𝑚 + 4 < −16

−4 − 4

4𝑚 < −20

÷ 4 ÷ 4

𝑚 < −5

Example 4: Dan’s math quiz scores are 88, 91, 87, and 85. What is the minimum score he would need on his 5th quiz to have a quiz average of at least 90?

88+91+87+85+𝑥

5≥ 90 The average of his 5 quiz scores must be greater than or equal to 90.

351+𝑥

5≥ 90

∙ 5 ∙ 5

351 + 𝑥 ≥ 450

−351 − 351

𝑥 ≥ 99

Dan needs to score a 99 or better on his final quiz to have a 90% quiz average.

Inequalities 1. Solve and graph: 5𝑥 − 3 > −18 2. Solve and graph: 9𝑤 + 5 − 11𝑤 ≥ −11 3. Solve: −3(𝑑 + 16) ≥ 5𝑑 4. Solve: 2 − (6𝑔 − 8) < 4 (𝑔 − 5) 5. A salesman earns $410 per week plus 10% commission on sales. How many dollars in sales will the salesman need in order to make more than $600 for the week?

Scan this QR code to get help on setting up and solving

inequalities word problems.

16

Justifying Steps using Properties A.4 The student will solve multistep linear and quadratic equations in two variables, including

b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets;

You are using the properties of real numbers to solve equations and inequalities, and to simplify expressions. You will need to be able to identify the property that you are using in each step of the simplification or solution. When you solve an equation or inequality and perform the same operation on both sides of the equal sign this is a special property of equality.

Property Equation Example Inequality Example

Addition Property of

Equality and

Inequality

𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐

𝑥 − 6 = 4

+6 + 6

𝑥 = 10

𝐼𝑓 𝑎 > 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 > 𝑏 + 𝑐

𝐼𝑓 𝑎 < 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 < 𝑏 + 𝑐

𝑤 − 1 ≥ 4

+1 + 1

𝑤 ≥ 5

Subtraction Property

of Equality and

Inequality

𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 = 𝑏 − 𝑐

𝑑 + 4 = −3

−4 − 4

𝑑 = −7

𝐼𝑓 𝑎 > 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 > 𝑏 − 𝑐

𝐼𝑓 𝑎 < 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 < 𝑏 − 𝑐

𝑔 + 1 < 3

−1 − 1

𝑔 < 2

Multiplication

Property of Equality

and Inequality

𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 = 𝑏𝑐 𝑛

5= −1

∙ 5 ∙ 5

𝑛 = −5

𝐼𝑓 𝑎 > 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 > 𝑏𝑐

𝐼𝑓 𝑎 < 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 < 𝑏𝑐 *Don’t forget to switch the sign if you multiply or divide

by a negative!

−𝑐

2≤ 4

∙ (−2) ∙ (−2)

𝑐 ≥ −8

Division Property of

Equality and

Inequality

𝐼𝑓 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎

𝑐=

𝑏

𝑐

−2ℎ = 12

÷ (−2) ÷ (−2)

ℎ = −6

𝐼𝑓 𝑎 > 𝑏, 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎

𝑐>

𝑏

𝑐

𝐼𝑓 𝑎 < 𝑏, 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎

𝑐<

𝑏

𝑐

*Don’t forget to switch the sign if you multiply or divide

by a negative!

9𝑓 < −81

÷ 9 ÷ 9

𝑓 < −9

17

Example 1: 2𝑥 + 3 (5 – 6𝑥) − 4

2𝑥 + 15 − 18𝑥 − 4 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦

2𝑥 − 18𝑥 + 15 − 4 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛

−16𝑥 + 15 − 4 𝐴𝑑𝑑(𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

−16𝑥 + 11 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

Example 2: 7𝑥 + (5 − 3𝑥) = 21

7𝑥 + (−3𝑥 + 5) = 21 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛

(7𝑥 − 3𝑥) + 5 = 21 𝐴𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛

4𝑥 + 5 = 21 𝐴𝑑𝑑 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

−5 − 5 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐸𝑞𝑢𝑎𝑙𝑖𝑡𝑦

4𝑥 = 16 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦

÷ 4 ÷ 4 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐸𝑞𝑢𝑎𝑙𝑖𝑡𝑦

𝑥 = 4 𝐷𝑖𝑣𝑖𝑑𝑒 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

Example 3: 12 < 𝑥 + 6 − 4𝑥

12 < 𝑥 − 4𝑥 + 6 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛

12 < −3𝑥 + 6 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

−6 − 6 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐼𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦

6 < −3𝑥 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

÷ (−3) ÷ (−3) 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐼𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦

−2 > 𝑥 𝐷𝑖𝑣𝑖𝑑𝑒 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

x < −2 Symmetric Property of Inequality

Example 4: 𝑆𝑜𝑙𝑣𝑒 3 (2𝑥 – 𝑦) = 12 𝑓𝑜𝑟 𝑦

6𝑥 − 3𝑦 = 12 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦

−6𝑥 − 6𝑥 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐸𝑞𝑢𝑎𝑙𝑖𝑡𝑦

−3𝑦 = −6𝑥 + 12 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛

÷ (−3) ÷ (−3) 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐸𝑞𝑢𝑎𝑙𝑖𝑡𝑦

𝑦 = 2𝑥 − 4 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)

18

Justifying Steps using Properties List the properties used to justify each step in the problems below. 1. 8 + 4𝑥 − 3(2 + 𝑥)

8 + 4𝑥 − 6 − 3𝑥

8 − 6 + 4𝑥 − 3𝑥

2 + 4𝑥 − 3𝑥

2 + 𝑥

2. 𝑆𝑜𝑙𝑣𝑒 (5 + 3𝑥) + 𝑥 ≥ 4𝑦 − 8 𝑓𝑜𝑟 𝑥

5 + (3𝑥 + 𝑥) ≥ 4𝑦 − 8

5 + 4𝑥 ≥ 4𝑦 − 8

−5 − 5

4𝑥 ≥ 4𝑦 − 13

÷ 4 ÷ 4

𝑥 ≥ 4𝑦−13

4

19

Answers to the problems: Expressions and Order of Operations 1. 11 − 𝑥

2. 3 (𝑛 + 10)

3. 4 (𝑛² − 5)

4. The product of twelve and a number

divided by four

5. a squared minus b to the fourth power

6. 169

7. 59

8. 118

9. −12

10. 209

11. 25

12. 3

13. −7

Properties of Real Numbers 1. F - Associative 2. E - Multiplicative Inverse 3. G - Commutative 4. H - Distributive 5. B - Additive Identity 6. G - Commutative 7. A - Multiplicative Property of Zero 8. D - Additive Inverse 9. J - Reflexive Property of Equality 10. K - Transitive Property of Equality

Solving Equations 1. 𝑘 = −19

2. 𝑥 = −15 3. 𝑦 = −28

4. 𝑛 = −1

2

5. 𝑘 = 4 6. 𝑔 = −1

7. 𝑚 = −5

3

8. 𝑥 = 2

Transforming Formulas

1. 𝑦 = 8−3𝑥

2 or 𝑦 =

−3𝑥+8

2

2. 𝑟 = 𝐶

2𝜋

3. 𝑧 = 5𝑦−12

2

4. ℎ = 3𝑉

𝜋𝑟2

5. 𝑥 = 15

1+𝑎 or 𝑥 =

15

𝑎+1

Inequalities 1. 𝑥 > −3

2. 𝑤 ≤ 8

3. −6 ≥ 𝑑 𝑜𝑟 𝑑 ≤ −6 4. 3 < 𝑔 𝑜𝑟 𝑔 > 3 5. 𝑠 > $1900 Justifying Steps using Properties 1. Distributive Commutative Subtraction (Substitution) Subtraction (Substitution) 2. Associative Addition (Substitution) Subtraction Property of Inequality Subtraction (Substitution) Division Property of Inequality Divide (Substitution)

0 3 6 9 120–3–6–9

0 3 6 90–3–6–9