Name: Period: Estimated Test Date:danielsroar.weebly.com/uploads/5/3/1/4/5314494/math_1_unit_1... · Solve linear equations with rational number ... theorem to find the distance between

Name:____________________________

Period: _____

Estimated Test Date: ___________

Common Core Math 1 Unit 1 One Variable Equations and Inequalities

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Main Concepts Page #

Study Guide 2 – 3

Vocabulary 4 – 9

Order of Operations/Distributive Property 10 – 11

Writing & Simplifying Expressions 12 – 18

Solving Multi-step Equations 19 – 21

Solving Multi-step Word Problems 22 – 24

Pythagorean Theorem & Distance Formula 25 – 26

Geometry & Literal Equations 27 – 30

Solving Multi-step Inequalities 30 – 31

Unit 1 Review 32 – 35

Answers 36 – 40


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Common Core Standards

8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8.G.6 Explain a proof of the Pythagorean theorem and its converse.

8.G.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean theorem to find the distance between two points in a coordinate system.

A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.


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Learner Objectives:

Rewrite expressions involving radicals and rational exponents using the properties of exponents

Write and simplify expressions

Interpret parts of expressions such as terms, factors, constants, and coefficients

Solve multi-step equations using the Distributive Property

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solution

Determine how many solutions an equation has by successively transforming the equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Create equations with one variable and use them to solve multi-step problems.

Explain a proof of the Pythagorean theorem and its converse

Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions

Apply the Pythagorean theorem to find the distance between two points in a coordinate system.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

Solve multi-step inequalities

Solve inequality word problems

Know the difference between equations and inequalities.

Represent, analyze and solve linear equations and inequalities.

Essential Understandings:

Rational exponents and roots can be used to write and simplify expressions.

The distributive property can be used to solve multi-step equations.

There can be no solution or infinitely many solutions when solving a multi-step equation.

There are strategies to clear fractions and algebraic proportions when solving multi-step equations.

Word problems can be solved by creating and solving multi-step equations.

The Pythagorean theorem and its converse can be used to identify a right triangle.

Real-world problems with right triangles can be modeled using the Pythagorean theorem and distance formula.

Formulas can be derived to find the volume of cylinders, cones, and spheres.

Literal equations can be solved for a variable

Strategies to solve equations can also be applied to solve multi-step inequalities. Understand the differences. Essential Questions:

Why is it helpful to write numbers in different formats?

How can you determine possible solutions from an equation or inequality?

Why do we need to define a variable before writing an equation or inequality from a real-world situation?

How can algebraic concepts be applied to geometry?

Why are formulas important in math and science?


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Vocabulary: Define each word and give examples and notes that will help you remember the word/phrase.

Additive Inverse

Example and Notes to help YOU remember:

Algebraic Equation


Base of a Power


Coefficient


Constant


Cube Root



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Definition of Subtraction


Distributive Property


Equivalent Expression


Evaluate


Exponent


Exponential Form



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Expression


Hypotenuse


Identity Property of Addition

(Additive Identity Property of Zero)


Inverse Property of Addition

(Addition of Opposites)


Integer


Irrational Number



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Identity Property of Multiplication

(Multiplicative Identity Property of One)


Inverse Property of Multiplication

(Multiplication of Reciprocals)


Like Terms


Leg


Order of Operations


Perfect Cube



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Perfect Square


Power


Property of Dividing Powers

with the Same Base


Property of Raising a Power to a Power

(Multiplicative Inverse)


Property of Zero as an Exponent


Reciprocal



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Simplify


Square Root


Substitution Property of Equality


Term


Variable



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Order of Operations

P

E

M D

A S

Examples:

a) 3 + 6 ⋅ 5 ÷ 3 b) 9 −5

8−3+ 6 c) 150 ÷ (6 + 3 ∙ 8)


Examples:

d) – 7(−8 + 4𝑟) e) 𝟑(𝟕𝒏 + 𝟏) + 𝟓 f) 𝟓 − (𝒙 + 𝒚)


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Order of Operations

Evaluate each expression.

1. 𝟑𝟎 − 𝟑 ÷ 𝟑 2. (𝟐𝟏 − 𝟓) ÷ 𝟖 3. 𝟕 × 𝟕 − (𝟖 − 𝟐) 4. 𝟕 × 𝟗 − 𝟕 − 𝟑 × 𝟓

5. 𝟒(𝟒 ÷ 𝟐 + 𝟒) 6. 𝟏

𝟐(𝟒 + 𝟔) − 𝟑 7. (𝟑 − 𝟓)𝟐 + −𝟒 8. (𝟔 − 𝟒) × 𝟒𝟗 ÷ 𝟕

9. 𝟒𝟓

𝟖(𝟒−𝟔)𝟐−𝟕 10.

(𝟑−𝟏)𝟑

𝟐𝟐 − 𝟏𝟕 11. 𝟒𝟑−𝟏

𝟒+𝟐+ 𝟏𝟎 12.

𝟑

𝟒(𝟓 − 𝟏)𝟐


Simplify each expression.

13. 𝟑(𝟒 + 𝟓𝒙) 14. – 𝟐(𝟑 + 𝒚) 15. 𝟓(−𝟏 − 𝟗𝒕) 16. 𝟒(𝟐𝒙 + 𝒚 − 𝟑)

17. 𝟑𝟑 + 𝟒(𝒕 − 𝟏) 18. – (−𝟐 − 𝒏) 19. – 𝟒 + 𝟐(𝟓𝒅 + 𝟔) 20. 𝟏

𝟐(𝟑𝒏 − 𝟏𝟔)


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Writing and Simplifying Algebraic Expressions

Look at each vocabulary term. In the front of the packet, write an example and any notes that will help YOU

remember the definition.

algebraic expression, coefficient, constant, equivalent expression, integers, like terms, simplify,

substitution property of equality, term and variable

Writing Algebraic Expressions

Write the verbal phrase or word that mean the same thing as the algebraic symbol at the top of the column.

(Note we will not use x for multiplication. Why?)

+ - or / x or ^ =

Examples: Write each phrase as an algebraic expression.

1. thirteen plus v ______13 + v_______

2. six minus w _________________

3. $18 less than the sale price _________________

4. the quotient of n and 12 _________________

5. 8 less than the product of 25 and a number q _________________

6. 8 times the sum of 28 and a number g _________________

7. 10 more than a number s times 5 _________________


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Combining Like Terms and Simplifying Expressions

Like terms must have the same _____________________ to the same _______________.

Like Terms Not Like Terms

An expression is a(n) ___________________________________when it has no ________________and

no _____________________.

Examples: One Variable

Identify the coefficients and constants, then combine the like terms.

1) -13c + c 2) 2x + 3x – 2 + 4x + 5

a. Coefficients: _______________________ a. Coefficients: _______________________

b. Constants: _______________________ b. Constants: _______________________

c. Simplify: _______________________ c. Simplify: _______________________

Examples: Two or More Variables

Identify the terms in the expression, then combine the like terms.

3) 0.3a – b + 0.9a + 3b 4) 8f – 2t + 3f + t




Examples: Variables Raised to Different Exponents

Identify the terms in the expression, then combine the like terms.

5) 3x + 2x² – 2.6x + 5x² + 7 6) 3a² – a – 7 + 5a² + a + 4





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Examples: With Parentheses

When multiplying two factors, and at least one factor has multiple terms, use the

_______________________ property to simplify the product.

1. 3(b+9) +10 2. 4y – 7 + 8(y+5)

a. Distribute: _______________________ a. Distribute: _______________________

b. Simplify: _______________________ b. Simplify: _______________________

HW: Writing Expressions: Write the following expressions in algebraic form.

3. the quotient of z and 9 _______________

4. the total of n and 40 _______________

5. the sum of 8 and m __________________

6. x divided by 5 _______________________

7. the difference of h and 7 ______________

8. 23 less than p _______________________

9. the product of g and 2 ________________

10. 77 plus twice v ______________________

11. 9 more than c ______________________

12. b minus 4 ___________________

13. two times the quantity of r increased by twelve ______________________________

Simplifying Expressions: Identify the coefficient and constant(s) in expressions.

14. 8x2 + 9x – 3 15. 17a4 – 2a2 + a – 1

a) coefficient(s): ______________ a. coefficient(s): ______________________

b. constant(s): ________________ b. constant(s): _______________________

Simplify the following expressions. Clearly show your work.

16. 3(4x – 5) = __________________

17. –4(x – 2) = ____________________

18. 9 – 7(b – 10) = _________________

19. 2(b – 3) – 4(2b + 2) = _____________

20. 2p4 + 3p + 12 – 18p4 – p – 7 = ______________


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Annotating Math Word Problems - CUBES

Just like in Language Arts, we sometimes need to annotate problems to better understand them.

C – Circle important numbers and variables

U – Underline important words

B – Box what the problem is asking you to solve

E – Equation

S – Solve

Example: Amanda has 2 pencils. Jacob has 3 more than 2 times the number of pencils that Amanda has.

Write an expression for how many pencils Jacob has.

HW 2: Write an algebraic expression for each of the follow verbal expressions.

1. 8 minus the quotient of 15 and y

2. The absolute value of a number squared, less 8

3. the product of 8 and z plus the product of 6 and y

4. The sum of the square root of a number and 6

Write a verbal expression for each of the following algebraic expressions.

8. Write an algebraic expression to model the following situation: The local video store

charges a monthly membership fee of $5 and $2.25 per video rented.

9. Error Analysis: Describe and correct the error in the problem below:

A student writes “the quotient of 5 and x” to describe the algebraic expression 𝑥

5

10. Describe a real world situation for the expression below. Be sure to identify the

variable(s). b + 5

5. 174

x 6. 3x + 10 7. 𝑥3 + 7


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Rational Exponents Recall that in an exponential expression of the form 𝑏𝑛, 𝑏 is called the base, 𝑛 is called the exponent and 𝑏𝑛 is a power. You can say that 𝑏 is raised to the 𝑛th power. Because of this, the exponent is also referred to as a power. For example, 54 would be read as “five raised to the fourth power.” In expanded form, we would write 54 = 5 ∙ 5 ∙ 5 ∙ 5. So the exponential form can be used to shorten an expression with repeated multiplication. Raising five to the fourth power means multiplying 5 times itself four times.

So what about 5.12? How can we multiply 5 by itself one-half times?

In previous courses, you were taught properties for rewriting expressions involving exponents. Let’s look at some patterns and see if we can rediscover those properties. Multiplying powers with the same base

1. 23 ∙ 24 = (2 ∙ 2 ∙ 2)(2 ∙ 2 ∙ 2 ∙ 2) = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 2___ 2. 𝑥5 ∙ 𝑥3 = (𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥)( ) = 3. 102 ∙ 105 = ( )( ) =

4. 𝑥𝑚 ∙ 𝑥𝑛 = 5. When multiplying powers with the same base, you can _____________ the exponents. Dividing exponential expressions with like bases

6. 45

42=

4∙4∙4∙4∙4

4∙4=

4

4∙

4

4∙ 4 ∙ 4 ∙ 4 = 1 ∙ 1 ∙ 4 ∙ 4 ∙ 4 = 4 ∙ 4 ∙ 4 = 4

7.𝑥8

𝑥6=

= 𝑥

8. (0.94)6

(0.94)4=

= (0.94)

9. 𝑥𝑚

𝑥𝑛= 𝑥

10. When dividing powers with the same base, you can _______________ the exponents. Raising a power to a power

11. (23)4 = (23)(23)(23)(23) = 23+3+3+3 = 2_____

12. (𝑥5)2 = (𝑥5)( ) = 𝑥______+______ = 𝑥______

13. (102)6 = ( )( )( )( )( )( ) = 10

14. (𝑥𝑚)𝑛 = ___________ 15. When raising a power to a power, you can __________________ the exponents.


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Negative exponents

𝐸𝑥𝑎𝑚𝑝𝑙𝑒. 23

24=

2∙2∙2

2∙2∙2∙2=

2

2∙

2

2∙

2

2∙

1

2= 1 ∙ 1 ∙ 1 ∙

1

2=

1

21=

1

2

16. 45

47=

4∙4∙4∙4∙4

4∙4∙4∙4∙4∙4∙4= ∙ ∙ ∙ ∙ ∙ ∙ =

17.𝑥3

𝑥8=

= ∙ ∙ ∙ ∙ ∙ ∙ ∙ =

Apply the previous rule about dividing powers with the same base to rewrite each of the expressions.

23−4 = 2−1

18. 45−7 = 4 19. 𝑥3− 8 = 𝑥

Fully simplify each exponential expression and complete the table below.

Power of 3 Value

34

33

32

31

30

3−1

3−2

*Study the table and the previous examples to determine the meaning of a negative exponent. What patterns do you see as the exponent increases by 1? Decreases by 1? What is the meaning of a negative exponent?

20. 𝑥−𝑚 =


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Back to rational exponents . . . In order to understand the meaning of rational exponents, consider the following table of exponential expressions and their values:

Exponential Expression

Value

28 256

24 16

22 4

21 2

2.12

21. What is the pattern for the exponents? 8, 4, 2, 1,1

2, …

22. What is the pattern of the simplified values? 256, 16, 4, 2, …

23. If you continue the pattern, what would be the value of 2.12?

24. What is the meaning of 𝑥 .1𝑛?

25. 𝑥 .1𝑛 =

Radical expressions: √𝑏𝑎

. The √ is the radical symbol. a is the index. If there isn’t an index written, the

default value is 2. These are called square roots. If the index is 3, it is called a cube root. b is the radicand.

Rewrite the following exponential expressions as radical expressions.

26. 5.13 27. 𝑦 .

12 28. 16.

14

Rewrite the following radical expressions as exponential expressions.

29. √𝑥3

30. √6 31. √325

Simplify each expression.

32. 4 12 = ______ 33. 9−0.5 = _________ 34. 8

13 = ________


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Solving Equations:

1. When solving equations with a single variable with an exponent of 1, there are three possible solutions: a

_________ solution, ____________ _______ solutions, or _____ solution.

Vocabulary Review:

2. addition property of equality ______________________________________________________________

________________________________________________________________________________________

3. subtraction property of equality ___________________________________________________________

________________________________________________________________________________________

4. multiplication property of equality _________________________________________________________

________________________________________________________________________________________

5. division property of equality ______________________________________________________________

________________________________________________________________________________________

As long as equality is maintained, equations can be rewritten. To solve for a variable, it needs to be isolated

and have a coefficient of 1. Typically, these steps work the best when solving for a variable.

Simplify each side of the equation. To avoid sign errors, use the definition of subtraction to rewrite

subtraction as addition.

Use the addition property of equality and the inverse property of addition so that all of the terms with

the variable you are solving for are on one side of the equation and all of the other terms are on the

other side.

If the variable’s coefficient is an integer, use the division property of equality to make the variable’s

coefficient 1. If the coefficient is a fraction, use the multiplication property of equality and the inverse

property of multiplication to make the variable’s coefficient 1.

Use the substitution property of equality to check the answer.

Solve each equation for x. Clearly show neat steps. Check your solution by using the substitution property of

equality. If your solution is “no solution” carefully check each step of your work.

6. 6 – 5x = –7

7. 8 – 2(x – 3) = 9

8. 2(x – 1) – 3 = –14 – 4x

9. 5(x + 2) = 2 + 3(x + 4) – 4


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When solving equations, sometimes you don’t get a single value for the variable.

For what value(s) of x will this equation be true? 3x = 3x

For what value(s) of x will this equation be true? 2x + 1 = 2x + 1

If we zero out the variable term from one side of the first equation, we get 0 = 0. When we do it with the

second equation, we get 1 = 1.

The reflexive property of equality (a = a) states that these equations are true. An equation that is true for all

values of the variables is called an identity. There are infinitely many solutions to an identity.

Another type of equation that does not have a single value as a solution is one that has no solution.

Consider the following equation: x = x + 1 For what value(s) of x would this equation be true? If we zero

out the variable term from one side of the equation, we get 0 = 1.

10. What do you think is the best way to check your answer when you get “no solution” or “infinitely many

solutions” as your answer?

11. 8 – 2(3x – 7) = 5x – 11x – 3 + 5

12. 3(x – 5) = 3x – 18 + 3

13. 8x = 3x

14. 12 = –4(–6x – 3)

15. 24x – 22 = –4(1 – 6x)

16. 6x + 5 – 2x = 4 + 4x + 1

17. 13 – (2x + 2) = 2(x + 2) + 3x

18. 7x – 4y + 12z + 4 = 5 – 3y + 7x – y + 12z


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19. Create an equation that has no solution.

20. Create an equation that has infinitely many solutions.

When equations have coefficients and constants that are not integers, you can use the multiplication

property of equality and the least common denominator to rewrite the equation using integers. While this is

not a necessary step, it is a skill that you should master.

21. −5

3𝑥 − 2𝑥 = −

11

9

22. 9

5𝑥 +

3

2𝑥 = −

33

10

23. 3

2𝑥 +

4

3= −

85

9

24. –2.7 – x = 7.2

Declare a variable and model the situation with an equation. Solve. Check your answer with the original problem.

25. Aloysius’s cell phone plan is $29.99 per month for the first 500 minutes and $0.27 for each additional

minute. His bill last month before taxes was $35.12. For how many minutes did Aloysius use his cell phone

last month?

Declare a variable: Let m = # of minutes used over the initial 500

Model the situation with an equation: 29.99 + 0.27m = 35.12

Solve & check.

26. Mr. Wilson spends three-fifths of his cash at Thai Villa. On his way home, he spends another $11.17 to

fill up his car with gas. When he gets home, he only has $1.83 left. How much cash did he have in his wallet

when he arrived at Thai Villa?

Declare a variable: Let c = the initial amount of cash in dollars

Model the situation with an equation: 𝑐 −3

5𝑐 − 11.17 = 1.83

Solve & check.


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Solve Word Problems

When solving a word problem, carefully read the problem and write a variable or variable expression

for each unknown. Use as few variables as possible.

Use your variables to write an equation that models the problem.

Use properties of equality to solve the equation.

Check to make sure that you answered the question. Use the original problem to check your answer.

For each of the following word problems, annotate using CUBES and then

a) declare the variable(s), b) model the word problem with an equation, and c) solve the equation.

Check your solution with the original problem.

1. After Simon donated four books to the school library, he had 28 books left. How many books did Simon

have before he donated the books to the school library?

2. One day Reeva baked several dozen muffins. The next day she made 12 more muffins. If she made 20

dozen muffins in all, how many dozen did she make the first day?

3. When asked how hold he was, Jerry said, “400 reduced by 4 times my age is 188.” How old is Jerry?

4. The Cooking Club made some pies to sell during lunch to raise money for a field trip. The cafeteria helped

by donating three pies to the club. Each pie was cut into seven pieces and sold. There were a total of 91

pieces to sell. How many pies did the club make?

5. A health club charges a $50 initial fee plus $2 for each visit. Mary has spent a total of $144 at the health

club this year. Use an equation to find how many visits she has made.

6. Find two consecutive even integers such that the sum of the larger and twice the smaller is 62.

7. Find three consecutive odd integers such that the sum of the smallest and 4 times the largest is 61.


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8. The sum of two numbers is 35. Three times the larger number is equivalent to 4 times the smaller

number. Find the numbers.

9. Find three consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.

10. Find four consecutive odd integers who sum is 56.

11. The larger of two numbers is 1 less than 3 times the smaller. Their sum is 63. Find the numbers.

12. The sum of two numbers is 172. The first is 8 less than 5 times the second. Find the first number.

13. Find two numbers whose sum is 92, if the first is 4 more than 7 times the second.

14. The sum of three numbers is 61. The second number is 5 times the first, while the third is 2 less than the

first. Find the numbers.

15. The sum of three numbers is 84. The second number is twice the first, and the third is 4 more than the

second. Find the numbers.

16. An 84-meter length cable is cut so that one piece is 18 meters longer than the other. Find the length of

each piece.

17. The length of a rectangle is 2 cm less than 7 times the width. The perimeter is 60 cm. Find the width and

length.


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18. The first side of a triangle is 7 cm shorter than twice the second side. The third side is 4 cm longer than

the first side. The perimeter is 80 cm. Find the length of each side.

19. The length of a rectangle is 6 cm longer than the width. If the length is increased by 9 cm and the width

by 5 cm, the perimeter will be 160cm. Find the dimensions of the original rectangle.

20. The first side of a triangle is 8 m shorter than the second side. The third side is 4 times as long as the

first side. The perimeter is 26 m. Find the length of each side.

21. A triangular sail has a perimeter of 25 m. Side a is 2 m shorter than twice side b, and side c is 3 m longer

than side b. Find the length of each side.

22. The length of a rectangular field is 18 m longer than the width. The field is enclosed with fencing and

divided into two parts with a fence parallel to the shorter sides. If 216 m of fencing are required, what are

the dimensions of the outside rectangle?

23. Matthew is 3 times as old as Jenny. In 7 years, he will be twice as old as she will be then. How old is each

now?

24. Melissa is 24 years younger than Joyce. In 2 years, Joyce will be 3 times as old as Melissa will be then.

How old are they now?

25. In the Championship game, Julius scored 5 points fewer than Kareem, and Wilt scored 1 point more

than twice Kareem’s points. If Wilt scored 20 points more than Julius, how many points were scored by each

player?


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Pythagorean Theorem

If a triangle is a right triangle, then the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (legs).

*Review: What does the triangle inequality theorem state?

*Review: Define rational numbers and irrational numbers.

Study the two squares below. The length of the thick, bold line segment is a units. The length of the thin line segment is b units. The length of each side of each square is (a + b) units.

*How would you calculate the area of each square?

The converse of the Pythagorean theorem is also true. If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

1. Label the legs and the hypotenuse of the right triangle.

2. Which sides of a right triangle are the legs?

3. Define “hypotenuse”.

4. What does the Pythagorean theorem state?

The distance formula is used to calculate the distance between two points on a coordinate plane. It is based on the Pythagorean theorem.

Use the Pythagorean theorem to calculate the missing side of the right triangle. Sketch each triangle. If the

square root can be simplified, write it as a rational number.

5. Leg = 8 cm ; leg = 1 cm 6. Leg = 1.5 inches ; hypotenuse = 2 inches

Use the converse of the Pythagorean theorem to determine whether or not the given lengths could form a

right triangle. Sketch each triangle. The triangles do not need to be drawn to full-scale.

7. 12 cm, 8 cm, 10 cm 8. 3 in, 4 ft, 5 in 9. 8 m , 17 m, 15 m


26 | P a g e

Calculate the length of the dashed line.

Sketch the two points and draw a right triangle. Label the coordinates of the third point. Use Pythagorean

theorem to calculate the distance between the two given points.

14. Label the coordinates of the third point and draw a right triangle.

15. Write a variable expression for the length of the vertical leg.

16. Write a variable expression for the length of the horizontal leg.

17. Use your expressions and the Pythagorean theorem to solve for the distance from (x1, y1) to (x2, y2).

18. Name one point that is √2 units away from (–1, 5).

Bonus: Name as many points that are √2 units away from (–1, 5) as you can. Explain your logic.

y

x

y

x

10. 11.

12. (0, –2), (–5, –1) 13. (–8, –10), (–6, 7)

(x1, y1)

(x2, y2)


27 | P a g e

Geometry Formulas

Area of a Circle


Area of a Triangle


Circumference of a Circle


Volume of a Cylinder or Right Prism


Volume of a Sphere


Volume of a Cone or Pyramid



28 | P a g e

Calculate the area of the following:

1. 2.

3. 4.

Calculate the perimeter or circumference:

5. 6. Calculate the value of x using the given information.

7. Perimeter = 41 8. Perimeter = 24

9. Area = 15 10. Perimeter = 18

Calculate the volumes:

11. 12.

13. 14.

2x 3x – 7

5x – 2

x – 7 3x

4x – 1

6

2x + 1

3x + 2

x – 1


29 | P a g e

Sometimes you have a formula, such as something from geometry, and you need to solve for some variable

other than the "standard" one. For instance, the formula for the perimeter P of a square with sides of length s

is P = 4s. You might need to solve this equation for s, so you can substitute in a value for the perimeter and

figure out the side length.

Equations with several variables are called literal equations.

* Previously, we have dealt with one-variable equations. 2 – 5(x + 1) = 12 ; Solve for x

* What does “solve for x” mean?

1 – 6 : Identify each formula and solve for the given variable.

1. 𝑉 =1

3𝜋𝑟2ℎ Solve for h.

3. 𝑉 =4

3𝜋𝑟3 Solve for r3.

5. Solve #4 for b2.

7. 𝑟 =𝑎

𝑑2𝑟 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑎.

8. x + 3y = 6

a) Solve for x. b) Solve for y.

9. 3x – 2y = 4

a) Solve for x. b) Solve for y.

10. Annie has a cylindrical container, but she does not know its radius or height. She does know that the

radius and the height are the same and that the volume of the container is 512π cubic inches. Find the

radius of Annie’s container.

2. 𝑉 =1

3𝐵ℎ Solve for h.

4. 𝐴 =1

2(𝑏1 + 𝑏2)ℎ Solve for h

6. S = 2r2 + 2rh Solve for h

http://www.purplemath.com/modules/geoform.htm


30 | P a g e

11. A cone with a radius of 6 centimeters and a height of 12 centimeters is filled to capacity with liquid. Find

the minimum height of a cylinder with a 4 centimeter radius that will hold the same amount of liquid.

12. The volume of a cylinder is 980π in. . The height of the cylinder is 20 in. What is the radius of the

cylinder?

Bonus. w(h – y) = w+hy Solve for y. Why? Because it’s fun!

Inequalities

The steps and properties for solving inequalities are similar to those involving equations.

*Review: Why does the inequality symbol get reversed when both sides of the inequality are

multiplied or divided by a negative quantity?

Review: Solve, graph, and do a full check before you check your work and answers with the answer key.

1. 5 – x < 3 2.

3. 9 > 4x – 7 4.

5. –2x + 3 > 7 6. 0.1x + 6 2

7. Sally earns $58 per week plus a 4% commission on the seashells she sells. What must her total sales be for

the week if she wants to earn at least $102?

a) Declare a variable. Let x = the value (in $) of the seashells Sally sells by the seashore in a week

b) Write an inequality to model the situation. 0.04x + 58 ≥ 102

c) Solve & check.

2

3𝑥 + 7 ≤ 9

5𝑥 − 3 ≥ 7


31 | P a g e

Multi-step inequalities. Solve for x.

8. 3x – 2(6x – 4) > 4 – (x + 6) 9. 6x – 4 < 2x + 12

10. 2x + 17 < –2(8 – x) 11. 2(x + 2) > –4 + 2x

Practice the steps for solving word problems. Use CUBES to annotate.

12. Carlos goes to the fair where it costs $5 to get in and $0.80 per ride. He has $28. How many

times can he go on rides at the fair?

a) Declare a variable. b) Write an inequality.

c) Solve & check.

13. Wednesday morning, Kristina bought suckers for her children. She gave Linus half of the ones she

bought because he had cleaned his room and made his bed in the morning. She gave Nicholas one third of

the ones she bought because he made his bed, but he had not yet cleaned his room. Later in the day, she

bought 3 extra suckers and gave them to Nicholas due to good behavior. However, Linus made fun of

Nicholas, so she took 5 suckers away from Linus. By the end of the day, Linus had at least as many suckers as

Nicholas. What is the minimum number of suckers purchased by Kristina?

a) Declare a variable. b) Write an inequality.

c) Solve & check.

Bonus: Because this is an inequality, we don’t know for sure how many suckers were purchased. List 3

possible amounts of suckers that Kristina could have purchased on Wednesday.


32 | P a g e

Unit 1 Review # 1

1. Calculate the value of x. The right triangle is not drawn to scale.

2. Calculate the value of x. The right triangle is not drawn to scale.

3. Use Pythagorean theorem to determine whether the given sides can be the sides of a right triangle? (yes or no) Show your work to support your answer. 3 m, 4 m, 5 cm

4. Calculate the distance from (1, 7) to (2, 3).

5. Simplify 49.12 6. Simplify 64.

13

Solve for x.

7. 3(4x – 9) = 6(2x – 5) + 2

8. 5 – 2(3x – 4) = 7 – 5x

9. 2x – 7 = –19

10. 3(x + 8) = 4(x + 6)

11. 5x > 20

12. 2x – 7 > 11

13. x – 8 < 12

14. 5 + 2(4 – x) + x < –7

Write each radical expression as an exponential expression with a rational exponent.

15. √24

16. √93

3 cm 7 cm

x

2 cm

7 cm

x


33 | P a g e

Simplify.

17. 3 – 5n + 2n – 7 – 6

18. 5 – 7(x – 3) – 2x + 4x – 3 + 4

How many terms are there? Identify any coefficients. Identify any constants.

19. 5x – y – 3z – 8

Calculate the volume of each figure. Figure 20 is a cylinder with a smaller cylinder removed.

20. 21. 22.

23. The volume of a cylinder is 980π cubic inches. The height of the cylinder is 20 inches. What is the radius

of the cylinder?


34 | P a g e

Unit 1 Review #2. Solve each equation below.

1. 2

3+

3𝑥

4=

7

12

2. 18𝑥 − 5 = 3(6𝑥 − 2)

3. 9 + 5𝑥 = 2𝑥 + 9

4. −8𝑥 + 14 = −2(4𝑥 − 7)

5. 2(6 + 4𝑥) > 12 − 8𝑥

6. 2 = −3𝑥−(−4)

8

7. 𝑥 − 1

3=

4

5

8. 𝑥−2

𝑥+5=

3

8

Solve each equation for x.

9. 3y + 2 = 9x - 4

10. 𝑥+𝑦

𝑧= 𝑎

11. 𝑎

𝑥=

𝑏

𝑐

12. 2xy + z = 5x


35 | P a g e

Write each radical expression as an exponential expression. Simplify the exponential expression if it is

possible.

13. √213

14. √𝑥22

15. √𝑥16

16. √646

Write an expression.

17. 4 more than twice a number. Let x = the number

18. 12 times the quantity of x minus 8

19. The quotient of the sum of x and 3 and 5.

Declare a variable, write an equation or inequality and calculate the solution.

20. There are 4 more boys than girls in Spanish class. The class has 38 members. How many boys and girls

are there separately?

21. Find three consecutive odd integers with a sum of 45.

22. Ten less than two times a number is equivalent to the number increased by 6.

23. The low temperatures for the previous two days were 62o and 58o. What would the temperature need to

be for the third day such that the average daily temperature is at least 64o?

24. Manuel is taking a job translating English instruction manuals to Spanish. He will receive $15 per page

plus $100 per month. He’d like to work for 3 months during the summer and make at least $1,500. Write an

inequality and find the minimum number of pages Manuel must translate in order to reach his goal.


36 | P a g e

Answers

Pages 4 – 9

Vocabulary Word Definition

Additive Inverse the opposite of a quantity

Algebraic Equation an equality of two algebraic expressions

Base of a Power a number that is raised to a power

Coefficient a number that is multiplied by a variable

Constant a term containing no variables

Cube Root One of the three equal factors of a number

Definition of Subtraction subtracting a quantity is equivalent to adding its opposite

Distributive Property For every real number a, b, and c: a(b + c) = ab + ac and a(b - c) = ab - ac.

Equivalent Expression expressions that have the same value for all the values of the variables

Evaluate to calculate the value of a numerical or algebraic expression

Exponent the quantity to which a base is raised

Exponential Form a number is written as a base with an exponent.

Expression (Algebraic Expression) A mathematical phrase that can include numbers, variables, and operation symbols

Hypotenuse longest side of a right triangle, opposite the right angle

Identity Property of Addition the sum of zero and any quantity is that quantity

Inverse Property of Addition zero is the sum of a quantity and its opposite

Integer a whole number or the opposite of a whole number

Irrational Number a number that cannot be expressed as a ratio of two integers; a non-terminating, non-repeating decimal

Identity Property of Multiplication the product of one and any quantity is that quantity

Inverse Property of Multiplication one is the product of a quantity and its reciprocal

Like Terms Like terms have identical variables; that is, they have the same variable to the same power. Constants are classified as like terms as well.

Leg either of the two sides of a right triangle that make up the right angle

Order of Operations a rule used for evaluating expressions which establishes the order in which operations should be done

Perfect Cube the cube of a rational number

Perfect Square the square of a rational number

Power a base with an exponent ; the exponent can also be called a power

Property of Dividing Powers with the Same Base

For every nonzero number a and integers m and n, 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛 =1

𝑎𝑛−𝑚

Property of Raising a Power to a Power

For every nonzero number a and integers m and n, (am)n=amn

Property of Zero as an Exponent For every non-zero number a, a0 = 1.

Reciprocal the multiplicative inverse of a quantity

Simplify to write an expression in simplest form

Substitution Property of Equality When two quantities are equivalent, one can be substituted for the other.

Term part of an expression that adds with other terms ; algebraic terms are

numbers, variables, or products of numbers and variables

Variable a symbol representing an unknown quantity or a quantity that can have a

variety of values


37 | P a g e

Page 10.

a) 13 b) 14 c) 50 d) –28r + 56 e) 21n + 8 f) –x – y + 5

Page 11.

1. 29 2. 2 3. 43 4. 41 5. 24 6. 2 7. 0 8. 14 9. 9

5= 1

4

5= 1.8 10. –15 11. 17 12. 12

13. 15x + 12 14. –2y – 6 15. –45t – 5 16. 8x + 4y – 12 17. 4t + 23 18. n + 2 19. 10d + 8

20. 3

2𝑛 − 8

Page 12.

2. 6 – w 3. x – 18 4. 𝑛

12 5. 25q – 8 6. 8(28 + g) 7. 5s + 10

Page 13.

1. a. –13, 1 2. a. 2, 3, 4 3. a. 0.3, –1, 0.9, 3 b. none b. –2, 5 b. none c. –12c c. 9x + 3 c. 1.2a + 2b

4. a. 8, –2, 3, 1 5. a. 3, 2 –2.6, 5 6. a. 3, –1, 5, 1 b. none b. 7 b. –7, 4 c. 11f – t c. 7x2 + 0.4x + 7 c. 8a2 – 3 Page 14. 1. a. 3b + 27 + 10 ; b. 3b + 37 2. a. 4y – 7 + 8y + 40 ; b. 12y + 33 3. z/9 4. n + 40 5. 8 + m 6. x/5 7. h – 7 8. p – 23 9. 2g 10. 77 + 2v 11. c + 9 12. b – 4 13. 2(r + 12) 14. a. 8, 9 ; b. –3 15. a. 17, –2, 1 ; b. –1 16. 12x – 15 17. –4x + 8 18. –7b + 79 19. –6b – 14 20. –16p4 + 2p + 5

Page 15.

1. 8 −15

𝑦 2. |x2| – 8 3. 8z + 6y 4. √𝑥 + 6 5 – 7 Answers will vary. 5. 17 less than the quotient of a number and 4.

6. 10 more than the product of 3 times x. 7. 7 more than the square of x. 8. 5 + 2.25v 9. Answers will vary. There is no commutative property of division. The order of the divisor and dividend matter. What the student wrote would be written as 5/x. 10. Answers will vary. Billy has some comic books. I have 5 more comic books than Billy.

Pages 16 – 18.

1. 7 2. (x ∙ x ∙ x) = x8 3. (10 ∙ 10)(10 ∙ 10 ∙ 10 ∙ 10 ∙ 10) = 107 4. xm+n 5. add 6. 3

7.x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x

x ∙ x ∙ x ∙ x ∙ x ∙ x, 2 8.

0.94 ∙ 0.94 ∙ 0.94 ∙ 0.94 ∙ 0.94 ∙ 0.94

0.94 ∙ 0.94 ∙ 0.94 ∙ 0.94 , 2 9. m − n 10. subtract 11. 12

12. 5, 5, 10 13 (102)(102)(102)(102)(102)(102) , 12 14. Xmn 15. Multiply 16. 4

4∙

4

4∙

4

4∙

4

4∙

4

4∙

1

4∙

1

4=

1

42

17.x ∙ x ∙ x

x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x=

x

x∙

x

x∙

x

x∙

1

x∙

1

x∙

1

x∙

1

x∙

1

x=

1

x5 18. − 2 19. − 5 20.

1

xm

21. divide the previous number by 2 22. square root of the previous term 23. √2 24. nth root of x 25. √𝑥𝑛

26. √53

27. √𝑦 28. √164

29. x

13

30. 6

12

31. 32

15

32. 2 33. 1

3 34. 2


38 | P a g e

Pages 19 – 21. The key only has the answers. If you are not sure if your work is acceptable and correct, ask for clarification. Proper work is more important than the correct answer.

1. single, infinitely many, no 2. If a = b, then a + c = b + c 3. If a = b, then a – c = b – c 4. If a = b and c ≠ 0, then ac = bc 5. If a = b and c ≠ 0, then a/c = b/c 6. x = 13/5 7. x = 5/2 8. x = –3/2 9. x = 0 10. Answers will vary. 11. No solution 12. Infinitely many solutions 13. x = 0 14. x = 0 15. No solution 16. Infinitely many solutions 17. x = 1 18. No solution 19. & 20: Answers will vary. 21. x = 1/3 22. x = –1 23. x = –194/27 24. x = –9.9 25. m = 19 ; 519 minutes (Did you remember to answer the question?) 26. c = 32.5 ; $32.50

Pages 22 – 25. Variables and equations may vary. If you are not sure if your work is acceptable and correct, ask for clarification. Proper work is more important than the correct answer. Numerical answers without proper units are incorrect.

1. 32 books 2. 19 dozen muffins 3. 53 years old 4. 10 pies 5. 48 visits 6. 20 and 22 7. 9, 11, and 13 8. 15 and 20 9. 24, 25, and 26 10. 11, 13, 15, and 17 11. 16 and 47 12. 142 (You were only asked for the first number.) 13. 11 and 81 14. 7, 9, and 45 15. 32 and 36 16. 33 meters and 51 meters 17. The width is 4 cm. The length is 26 cm. 18. 23 cm, 27 cm, and 30 cm 19. 30 cm by 36 cm. 20. 3 m, 11 m, and 12 m 21. 6 m, 9 m, and 10 m 22. The width is 36 m. The length is 54 m. 23. Jenny is 7 years old. Matthew is 21 years old. 24. Joyce is 34 years old. Melissa is 10 years old. 25. Kareem scored 14 points, Julius scored 9 points, and Wilt scored 29 points

Pages 25 & 26.

1. We will go over the diagram in class. 2. The legs are the two sides that form the right angle. 3. The hypotenuse is the longest side of a right triangle. It is opposite the right angle. 4. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. 5 – 9 & 12 – 13: The sketches are not included. If you are not sure about your work, ask for clarification.

5. √65 𝑐𝑚 6. √1.75 𝑖𝑛𝑐ℎ𝑒𝑠 7. No 8. No. If you didn’t carefully watch units, analyze the error and check for it while you are working on problems in the future. Unit errors are common, but usually easy to catch and fix. 9. Yes 10. 5 units 11. 10 units

12. √26 𝑢𝑛𝑖𝑡𝑠 13. √293 𝑢𝑛𝑖𝑡𝑠 14. (x1, y2) 15. y2 – y1 16. x2 – x1 17. √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Why don’t the squares

and the square root simplify? 18. Answers will vary. 19. How many points were you able to find? How did you find them?

Page 28. Remember to include the correct units.

1. 196π in2 2. 3 in2 3. 228.01π cm2 4. 23.1 yd2 5. 26 mm 6. 20π yd 7. x = 5 8. x = 4 9. x = 2

10. x = 2 11. 100π in3 12. 288π mi3 13. 484 cm3 14.28

3π mi3

Pages 29 – 30.

1. h =3V

πr2 , volume of a cone 2. h =3V

B ; volume of a pyramid 3. r3 =

3V

4π ; volume of a sphere

4.2A

b1 + b2 ; area of a trapezoid 5. b2 =

2A

h− b1 6. h =

S − 2πr2

2πr ; surface area of a cylinder 7. a = r2d2

8. a. x = 6 − 3y or x = −3y + 6 8 b. y =6 − x

3 or y = −

1

3x + 2 9. a. x =

2y + 4

3 or x x =

2

3y +

4

3

9b. y =3x − 4

2 or y =

3

2x − 2 10. 8 inches 11. 9 cm 12. 7 inches


39 | P a g e

Pages 30 – 31

1. 2.

3. 9 > 4x – 7 4.

5. –2x + 3 – 3 > 7 – 3 6. 0.1x + 6 – 6 2 – 6

(3) (2

3𝑥 + 7 ) ≤ (3)(9)

2x + 21 – 21 ≤ 27 – 21 ; 2x ≤ 48

(½)(2x) ≤ (½)(48) ; x ≤ 24

CHECK #1: 2

3(24) + 7 = 9

16 + 7 = 9 ; 9 = 9

CHECK #2: 30≤24

2

3( − 30) + 7 ≤ −9; 20 +7≤ 9

13 ≤ 9

5x – 3 +3 ≥ 7 + 3

5x ≥ 10

5𝑥

5≥

10

5; 𝑥 ≥ 2

CHECK #1: 5(2) – 3 = 7 ; 10 – 3 = 7

7 = 7

CHECK #2: 3>2 ; 5(3) – 3 ≥ 7

15 – 3 ≥ 7 ; 12 ≥ 7

0 2

5 + (5) +(–x) < 3 + (5)

x < 2 ; −𝑥

−1<

−2

−1

x > 2

CHECK #1: 5 – (2) = 3 ; 3 = 3

CHECK #2 3 > 2 ; 5 – (3) < 3 ; 2 < 3 24 0

9+7 > 4x – 7 + 7

16 > 4x ; 16

−4>

−4𝑥

−4

x > 4

CHECK #1: 9 = (4)(4) – 7 ; 9 = 16 – 7

9 = 9

CHECK #2 0 > 4 ; 9> 4(0) – 7 ; 9 > 7

0 4 2

0

2x > 4

(−1

2) (−2𝑥) > (−

1

2) (4)

x < 2

CHECK #1: 2(2) + 3 = 7 ; 4 + 3 = 7

7 = 7

CHECK #2: 3 < 2 ; 2(3) + 3 > 7

6 + 3 > 7 ; 10 > 7

0 2 0.1x ≤ 4

(0.1𝑥)(10) ≤ (4)(10)

x ≤ 40

CHECK #1: 0.1(40) + 6 = 2

4 + 6 = 2 ; 2 = 2

CHECK #2: 50 < 40

0.1(50) + 6 ≤ 2 ; 5 + 6 ≤ 2 ; 1 ≤ 2

0 40


40 | P a g e

Pages 30 – 31

7. 0.04𝑥 + 58 − 58 ≥ 102 − 58

0.04𝑥 ≥ 44

0.04𝑥

0.04≥

44

0.04 ; 𝑥 ≥ 1100 ; Sally must sell at least $1100 worth of seashells.

8 – 14: Only the answers are given. If you are not sure about how your work, graph, or solution, ask for clarification.

8. 𝑥 <5

4 9. 𝑥 ≤ 4 10. No solution 11. Infinitely many solutions 12. He can go on as many as 28 rides. If you only solved

the equation, but didn’t answer the questions, carefully analyze your error. Carefully look for this error on similar problems in the future. Sometimes you need to answer a question. Solving an equation or inequality is only part of the work. You must interpret the solution you get in order to answer the questions being asked. Attend to precision. 13. 48 suckers.

Pages 32 – 35: You should have mastered how to solve and check problems like the ones in the review. Ask for help if you have not yet fully mastered the skills.

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