CLASSIFYING NUMBERS RATIONAL VS IRRATIONAL NOTES

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TABLE OF CONTENT

Rational Vs Irrational Numbers

Evaluate Square And Cube Roots And Approximating Irrational Numbers Estimating Values Of Non-Perfect Squares

Scientific Notation

Operations With Scientific Notation

Solve Complex Linear Equations

No, One, Or Infinite Solutions

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CLASSIFYING NUMBERS RATIONAL VS IRRATIONAL NOTES

https://www.youtube.com/watch?v=XMronO6wgds

Real Numbers

EX:

Rational Numbers

EX:

Irrational Numbers

EX:

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Stop at the ten-thousandth place value for values that do not terminate.

Fraction Decimal Equivalent

1/3 1/8 7/9

2/3 2/8 8/9

1/6 3/8 1/11

2/6 5/8 2/11

4/6 6/8 3/11

5/6 7/8 4/11

1/7 1/9 5/11

2/7 2/9 6/11

3/7 3/9 7/11

4/7 4/9 8/11

5/7 5/9 9/11

6/7 6/9 10/11

What observations did you make about fractions with a 3 as the denominator?

What observations did you make about fractions with a 6 as the denominator?

What observations did you make about fractions with a 9 as the denominator?

What observations did you make about fractions with a 11 as the denominator?

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NAME DATE PERIOD

EXIT TICKET

1. A rational number can always be written in what form?

A. decimal B. percent

C. fraction D. square root

2. Which is a rational number?

3. Which number is irrational? 4. Which statement is NOT correct?

A. A rational can be written as a ratio.

B. An irrational number can be written as a decimal.

C. An irrational can be written as a fraction.

D. An irrational number has endless non-repeating digits to the right

5. Which of these is a rational number?

A. √8 B. 0.13133133313…

C. 0.33333...

D. 2.718281804...

6. Which is a rational number?

A. √4/9 B. 2.123456...

C. cube root of 48 D. square root of 27

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Square and Cube Roots And Approximating Irrational Numbers On The Number Line Notes

You’ve already learned about inverse or OPPOSITE operations.

Radicals denote that the root number, or BASE is what we’re looking for. It’s the OPPOSITE or inverse of a base raised to an exponential value

The solutions to √36 can ALSO be -6 ^{because a} (Neg.) x (Neg.) = (Pos.)!

You try it!

Find the value of the radical, then plot the point on the number line.

√64 √100 √49 √81

6

²

means 6 x 6 which equals 36.

What does 5

³

mean?

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How do we find where the √14 is located on a number line since it’s not a perfect square?

First let’s write down all perfect squares until we pass the √14.

Which two perfect squares does it fall between?

Next , find the square root of those perfect squares. Place the perfect squares ABOVE those values on the number line.

Finally , which value is the √14 closer to? Place it CLOSER to that number.

Now YOU try it! Plot the radicals on the number line.

√29 √82 √102 √6

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Plotting And Comparing Rational And Irrational Numbers Practice

Part I – Perfect Square and Cubed Numbers and Their Roots

Let’s start with some background information. Fill in the chart below with the missing information.

Square What does it mean? Solution Square Root Solution

11²

12² 12 x 12 144 √144 12

13²

14² 14

15² 225 √225

16²

17² 17 x 17

18² 19² 20²

21² 21

22² 22 x 22

23² 24²

25² 625 25

Cube What does it mean? Solution Cube Root Solution

1³

2³ 8 ³√8

3³ 4³

5³ 5 x 5 x 5 125 ³√125 5

6³ 7³ 8³ 9³

10³ 10 x 10 x 10 1000 10

Part II - Estimating Radicals

Without using a calculator, determine which two integers each of the following radicals below falls between.

1. 50 is between and 2. is between and

9. is between and 10. is between and

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Practice Number Line

Use your new estimation skills to place each of the following radicals onto the number line below.

6) 3 7) 10 8) 26 9)  5 10)  14

Name the point on the number line associated with each irrational number.

Part IV – Comparing Radical Values Compare the following numbers using > or <.

16. 17. 18.

19. 20.

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SQUARE, CUBE AND THEIR ROOTS NUMBERS EXIT TICKET Solve the equation 𝑥³ = 27

A. 3 B. 9 C. 27 D. 81

Which is the ³√64?

A. 4 B. 8 C. 16 D. 21

Between which of the following pairs of numbers does the √200 lie?

A. Between 10 and 11 B. Between 196 and 225 C. Between 14 and 15 D. Between 20 and 21

The expression 8 ∙ 8 ∙ 8 can also be written as?

A. 8³ B. 3⁸ C. 24³ D. 512³

Which of the following expressions is equivalent to 8²y⁴?

A. 8y × 8y × 8y × 8y B. 16 × y × y × y × y C. 8 × 8 × y × y D. 64 × y × y × y × y

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ESTIMATING VALUES OF NON-PERFECT SQUARE ROOTS

We can find the square root of non-perfect square roots using a few easy steps.

• 1^st Find the nearest perfect square, without going over your value.

• 2^nd Divide your value by the square root.

• 3^rd Find the average of the solution from step 2 and the root.

We know how to approximate using a number line, now let’s learn to estimate using a mathematical process.

Find the √12

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

Find the √28

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

Find the√120

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

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Estimating Values Of Non-Perfect Square Roots Practice

Find the value to the nearest tenth.

F

^(Find)

D

^(Divide)

A

(Average)

√162

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

√95

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

√74

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

√60

1

^st

Find the nearest perfect square, without going over your value.

2

^nd

Divide your value by the square root.

3

^rd

Find the average of the solution from step 2 and the root.

Your Solution

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USING SQUARE AND CUBED ROOTS Looking at squares to understand square roots.

If I told you that the area of a square is 225 in², what is the measure of the side lengths?

What is the perimeter?

Solve the following.

1. James wants to buy a new rug for his living room. In a department store he finds a square rug that has an area of 9 m². How long is each side of the rug?

a. How many of those rugs are needed to cover an area of 36 square meters?

2. Jessica has a square picture with an area of 900 square inches. How long is each side of the picture?

a. What is the perimeter of the picture?

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3. The Smiths want to fence their square garden, which has an area of 256 square meters. How long is each side of the garden?

a. If one side of the garden borders the house, therefore doesn’t need a fence, how many meters of fence are necessary?

4. Tanico’s farm is 300 square miles. If the farm is a square, approximately how many miles of fencing is need to completely enclose the land? Round to the nearest tenth.

5. The painting my father purchased is 20 square feet. Approximately how many feet of trim is needed to frame it?

6. Dave is crafting a small vegetable garden. He has 52 feet of fencing to put around the square plot. How much space does he have to plant his vegetables?

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https://www.youtube.com/watch?v=VQsQj1Q_CMQ

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NEGATI VE EXPO NEN T S a

-n

= 1 1 = a

n

a

n

a

-n

POWE R O F A PRODUCT O R Q U O TI ENT

(ab)

m

= a

m

b

m

- - - - - - - - - - DIV IDING EXPO NEN T S

a

m

= a

m - n

a

n

EXPO NEN T OF ZER O

a

0

= 1 RAISI N G A POWE R TO A POWE R

(a

m

)

n

= a

m•n

MULTIP LY ING EXPO NEN T S

a

m

• a

n

= a

m + n

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A non ze ro qu antity rai sed to a z er o exponent i s __________. To rai se a pow er to a pow er , the exponents . When multiplyin g exponents with the _ bas e, _ the bas e and ______ the exponents .

a

0

b

3

=

(25c

3

d

7

)

0

= (d

5

)

3

=

(- 3x

2

y

4

)

3

= b

2

• b

5

=

(5x

2

)(9x

3

) =

4a

-3

b

6

= 16a

2

b

-2

x

4

y

0

= x

-2

(xy z)

3

=

(5bc)

3

= - - - - - - - - - - -

* O n ce th e t o p a n d b o tto m a re ra is ed , th en fo llo w q u o tie n t o f p o we rs r u les ! a

10

b

9

= a

2

b

4

12x

7

y

8

= 6x

6

y

3

A non ze ro bas e rai sed to a nega tive exponent i s e qu al to the of the bas e rai sed to a exponent T o fin d the po w er o f a p ro d u ct, a p p ly t h e ex p o n en t t o e a ch t er m. - - - - - - - - - - - - - - T o fin d the po w er o f a q u o ti en t, a p p ly t h e p o we r to t h e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ an d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . When dividing exponents with the _____ bas e, _ the bas e and ___________ the exponents .

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RULES OF EXPONENTS PRACTICE

Find the value of each expression.

1) 5 ⁵ 2) 2 ¹¹ 3) 6 ³ 4) 9 ³

Simplify each product.

5) 10¹²10³⁵  6) a⁷a¹²  7)

 

²^x² ⁴^{x y}³ ²



^

8)



^³^{a b}²



⁶^{ab c}⁴



^ ⁹⁾



^{9x z}¹⁰ ²



^^{x y}⁵ ³



^ ¹⁰⁾

 

¹¹^c⁸ ^¹⁰^{c d}⁴



^

Simplify each expression.

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 

^x² ³ ^ ¹²⁾

 

^a⁷ ⁵ ^ ¹³⁾

 

^y¹³ ⁴ ^

14)

 

^w^²¹ ^¹⁵^ ¹⁵⁾

 

^^y⁵ ⁴ ^ ¹⁶⁾



^^3h⁹



³

Simplify each quotient.

17)

210 207

9

9  18)

6 3

2 r

r  19)

6 3

40 20

s s

  20)

18 5 11 3

21 7

d e

d e  21)

5 5 5 2 3 4

a b c a b c 



22) x 6

y

  

   23)

2 2

5c d

  

 

  24)

3 3 5

4d c

 

  

 

25)

4 6

3w g

 

  

  26)

6 3 3 5

4s t r

  

 

  27)

11 6 2 18

2d f c

  

 

 

28) 29) 30) 31)

Simplify the expression. Your solution should contain NO NEGATIVE exponents.

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32) 33) 34)

35) 36) 37)

Evaluate each quotient if x =2, y = -2, and z = 10.

38) x3

x  39)

y4

y  40)

3 3

x y xy 

41)

4 2 2

z x y

zxy  42)

 

^yz ²

z  43) ³

 

²

3

3 9 y zx

x 

44)

1 x

x

z z

  45) ₃

x x y

z z



  46)

xz 3

y

 

  

 

47) What is the area of a square with side lengths equaling 3a⁵? What is the square’s perimeter?

48) What is the area of the rectangle with the width of 6x² and the length of 12x³? What is the square’s perimeter?

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EXPONENT RULES EXIT TICKET

1. Simplify the expression using a positive exponent. ^𝑥

−7 𝑥⁴. A. 𝑥⁻³ B. ¹

𝑥³

C. ¹

𝑥³

D. ¹

𝑥³

2. Which expression is equivalent to 10⁸? A. (10²)⁴

B. 10² 10⁶ C. 10⁴ × 10² D. (-10)^-8

3. Which value of N would make the statement true?

(5^-N)³ = 5¹⁵ A. N = -5 B. N = -12 C. N = -18 D. N = 18

4. Which expression is equivalent to 5^-4?

5. Which expression is equivalent to the following?

(x²y)(x³y⁴) A. x⁶y⁴

B. x⁵y⁵ C. x⁸y⁴ D. x^-1y^-3

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Rational, Irrational, Square, Cube, And Roots Quick Check 1. Solve the equation 𝑥³ = 64

E. 2 F. 4 G. 8 H. 32

2. The expression (2y⁴z³)² can also be written as?

E. 4y⁶z⁵ F. 4y⁸z⁶ G. 2y²z H. 2yz⁹

3. Between which of the following pairs of integers does the √78 lie?

E. Between 4 and 5 F. Between 7 and 8 G. Between 8 and 9 H. Between 64 and 81

4. Complete the statement.

All rational and irrational numbers are ________.

A. Positive numbers B. Whole numbers C. Real Numbers D. Repeating decimals

5. Which of the following expressions is equivalent to 2⁴p²?

E. 8p² F. 16p

G. 8 × p × p × p × p H. 16 × p × p

6. Which is the equivalent value to ⁹/11? A. 0.090909…

B.

C. 99%

D. ¹⁶/22

7.

Which of the following set of numbers is not rational?

A. 3.1111…, 6 ½ , √64, 0 B. 3.14, √0, 3 ¼ , 0.7896543

C. ¹/11, ³√8, 9.87654321, 6.66666…

D. 1.15, 9², ¹³/4, 3.1718192…

8. What is the √96?

A. 8.5 B. 9.2 C. 8.9 D. 9.7

9. Which is the equivalent of √289?

E. 9

F. 13

G. 17

H. 23

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Which properly depict the values placed on the number line?

4² 2³ √56 8 ¹/9

A.

B.

C.

D.

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Understanding Scientific Notation

https://www.youtube.com/watch?v=DXTuYjPDjqQ

Glue down your Understanding Scientific Notation Foldable

Here

Example of Standard Notation:

Example of Scientific Notation:

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Scientific Notation Exit Ticket

Convert from Scientific to Standard Notation Convert from Standard to Scientific Notation

1. 3.2 X 10^-2 6. .00001

2. 1.7 X 10³ 7. 23,000

3. 1 X 10⁵ 8. 720,000

4. 4.0 X 10^-6

9. .0000000054

5. Which is the 454,300,000 properly written in scientific notation form?

A. 45.43 × 10⁷ B. 4.543 × 10⁷

C. 4 × 10⁸ + 5 × 10⁷ + 4 × 10⁶ + 3 × 10⁵ D. 4.543 × 10⁸

10. Which of the following shows the numbers in order from least to greatest?

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ADDING & SUBTRACTING WITH SCIENTIFIC NOTATION NOTES https://www.youtube.com/watch?v=p0zVNTko7z4

ADDITION - When adding scientific notations with the SAME exponents

3.6 x 10³ + 4.8 x 10³

1^st Add the multipliers 3.6 + 4.8 = 8.4

ADDITION - When adding scientific notation with DIFFERENT exponents

4.2 x 10² + 2.9 x 10⁵

1^st Convert to STANDARD form

4.2 x 10²= 420 + 2.9 x 10⁵ = 290000

2^nd ADD your solutions 290420

Convert back to SCIENTIFIC NOTATION IF required 2.9042 x 10⁵

SUBTRACTION - When subtracting scientific notations with the SAME exponents

4.6 x 10³ - 3.8 x 10³

1^st Subtract the multipliers 4.6 - 3.8 = .8

2^nd Keep the SAME power IF and only IF the multiplier is between 1 and 10

.8 x 10³

The solution is NOT between 1 and 10 3^rd IF the multiplier is NOT between 1 and 10, THEN convert to

STANDARD form and BACK to SCI. NOT. IF required 800 =

8 x 10²

SUBTRACTION - When subtracting scientific notation with DIFFERENT exponents

4.2 x 10⁵ - 2.9 x 10²

1^st Convert to STANDARD form 4.2 x 10⁵= 420000

- 2.9 x 10² = 290

2^nd SUBTRACT your solutions 419710

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Convert back to SCIENTIFIC NOTATION IF required 4.1971 x 10⁵ You TRY it!! Convert each to standard form.

1. (3.45 x 10³) + (6.11 x 10³) 4. (8.96 x 10⁷) - (3.41 x 10⁷)

2. (9.09 x 10⁻²) + (2.07 x 10⁻²) 5. (4.23 x 10³) - (9.56 x 10²)

3. (4.12 x 10⁶) + (3.94 x 10⁴) 6. (9.7 x 10⁸) - (6.28 x 10⁴)

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ADDITION AND SUBTRACTION WITH SCIENTIFIC NOTATION

Write the answer in both scientific and standard form.

1. 7.4 x 10⁶ + 2.735 x 10⁶ 2. 2 x 10³ – 1.9 x 10²

3. 5.2 x 10⁷ + 3.01 x 10⁴ 4. 2.005 x 10² – 8.664 x 10²

5. 6.2 x 10⁵ + 9.7 x 10¹ 6. 7.32 x 10⁶ – 4.01 x 10⁸

7. ( 5.32 × 10⁸) – ( 4.6 × 10⁶) 8. ( 9.67 × 10⁶) + ( 3.45 × 10⁶)

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9. ( 2.82 × 10⁹) + ( 6.3 × 10⁷) 10. ( 3.64 × 10⁶) – ( 2.18 × 10⁴)

11. ( 9.8 × 10³) – (6.7 × 10³) 12. (6.98 × 10⁵) + (1.65 × 10⁷)

13. A factory builds a new warehouse that is approximately 1.28 ×10⁵square feet. Later, they add on 1.13 ×10³ more square feet for offices. Use scientific notation to write the total size of the new building.

14. Rochester, NY has an average of 28.2 inches of snow fall in January, while Atlanta, GA has an average of 1.3 inches of snow fall in January.

a. Rewrite the snowfall averages in scientific notation.

b. How much more snow does Rochester, NY receive in January than Atlanta, GA, on average?

Calculate this using scientific notation. Write your final answer in standard notation.

c. Buffalo, NY has an average of 25.3 inches of snow fall in January. What is the total average of snow fall of Buffalo and Rochester, NY in the month of January? Calculate this using scientific notation.

Write your final answer in standard notation.

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MULTIPLYING AND DIVIDING WITH SCIENTIFIC NOTATION PRACTICE

https://www.youtube.com/watch?v=UADVIDjdaVg

Glue down your Multiplying & Dividing Scientific Notation Foldable

Here

1. (4.3 x 10⁵) (2 x 10⁷) 4. 3.66 x 10⁻⁵ 2.0 x 10⁻³

2. (5.2 x 10³) (1.7 x 10¹⁴) 5. (5.1 x 10⁴) (2.5 x 10³)

3. 6.2 x 10⁶ 3.1 x 10³

6. 3.5 x 10⁻⁶ 5 x 10⁻²

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7. How much larger is 8 10 ⁶compared to 4 10 ²? A. 2000 times

B. 4000 times C. 20,000 times D. 40,000 times

8. The thinnest commercial glass is 9.84 x 10^-4 inches thick. The glass of an aquarium is 1,000 times as thick.

How thick is the aquarium glass written in both scientific and standard form?

9. In 2008, the total trade between the US and Japan was $2.04 x 1011. The total trade between the US and Australia was $3.28 x 1010.

A. Which was the greater trade?

B. How many times greater was it?

10. Each shrimp weighs approximately 0.000 27 grams and a shrimp company can bring in over 3,100,000,000 shrimp per year. Approximately the weight of that many shrimp. Write your solution in scientific and standard form.

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OPERATIONS WITH SCIENTIFIC NOTATION EXIT TICKET

Write your solution in scientific AND standard form.

1. The US spends on average $10,200 on each student per year. There are about 77,000,000 students in the United States. On average, how much is spent on students yearly?

2. In 2008, the number of tourists visiting France was 7.94 x 10⁷. The number of tourists visiting Italy was 4.27 x 10⁷. Who had more tourists? How many more tourists visited that country?

3. The Earth has a mass of about 1 × 10²⁵ kg. Neptune has a mass of 1.8 × 10²⁷ kg. How many times bigger is Neptune than Earth?

4. The distance from the Earth to the sun is approximately 9.3 × 10⁷ miles. The distance from the Earth to Mars is approximately 142,000,000 miles. What is the approximate distance from the sun to Mars?

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SOLVING COMPLEX LINEAR EQUATIONS NOTES

Combining Like Terms Examples

Like terms are variables that are the same AND raised to the same power.

Distributive Property Examples

Simplifying Expressions Examples

Two Step Equations with Variables on Both sides

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SOLVING COMPLEX LINEAR EQUATIONS

Solve the following equations.

2(x + 1) – 7 = 5

Solutions________

4(y + 3) – 2y = 7

5(y + 2) – 4(y – 1) = 6

5(2 – x) – 3(4 – 2x) = 20

2m + 4 – 3m = 8(m – 1)

3m + 12 = 2(m – 3) + 4

Solutions________ Solutions________ Solutions________

-8n + 4(1 + 5n) = -6n – 14

-6n – 20 = -2n + 4(1 – 3n)

-3(x - 1) + 8(x - 3) = 6x + 7 – 5x

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LINEAR EQUATIONS WITH NO, ONE, OR INFINITE SOLUTIONS NOTES

https://www.youtube.com/watch?v=68GvUlnhe10

Mini-Review: Distributive Property

4(-4 – 8m) 3 – (6k + 3)

How to determine if there is NO solution.

__________________________________________

EXAMPLE: 154 = -4(8 + 6r) + 24

How to determine if there is ONE solution.

____________________________________________

EXAMPLE: -21 – 8a = -1 + 6(4 – 5a)

How to determine if there are INFINITE solutions.

__________________________________________

EXAMPLE: -28 = -7(3x + 4) + 21x

Let’s TRY it!