#1

The number of marbles in 13 bags The number of marbles you had before purchasing more bags. The number of marbles in each of the 13 bags.

#2

The number of children Jamie has. The number of months. The monthly tuition for the science club The one-time materials fee. The total cost for one child to join the science club.

#3

Use the area model.

3𝑥

−2

2𝑥

6𝑥 2

−4𝑥

2𝑦

+5

+15𝑥

−10

6𝑥 2 + 11𝑥 − 10

5𝑦

+5

10𝑦 2

+10𝑦

−3 −15𝑦

−15

10𝑦 2 − 5𝑦 − 15

#4

Multiply Exponents

𝑦 6 ∗ 𝑦 4 = 𝑦10 Add Exponents

Multiply Exponents

𝑧6 ∗ 𝑧3 = 𝑧9 Add Exponents

To raise to a power; multiply exponents. To multiply; add exponents.

#5

3𝑎 ∙ 2 9 27 ∙ 4

=1

1 3𝑎 ∙ 2 ∙ 3 =1 27 ∙ 2 9 3𝑎 =1 9

Simplify the radicals.

Simplify the numerator and denominator. Re-write the equation with all terms in the same base.

3𝑎 = 30 2 3 Use the properties of exponents.

𝑎−2=0 𝑎=2

#6

3𝑑 ∙ 5 32

∙ 45 3𝑑

32

∙ 9

=3

Simplify the radicals.

=3

3𝑑 =3 32 ∙ 3

Use the properties of exponents.

3𝑑 =3 33

Use the properties of exponents again.

𝑑−3= 1 𝑑=4

#7

=

3 𝑔5

𝑎. 7 𝑥 𝑏.

5

𝑔3

𝑐.

𝑏

𝑥𝑎

#8

𝑎.

1 (7𝑦 3 )2

𝑏.

5 𝑘3

𝑐.

𝑎 𝑝𝑐

#9

𝑎. real; the 3 is irrational 𝑏. real; 4

3 − 5

=

2 3 5

rational

𝑐. real; irrational 𝑑. real;

7 9

2 9

9 9

+ = = 1 rational, integer, whole

# 10

Distribute

Distribute

20𝑤 − 10 = 8 + 2𝑤 − 12 + 12

2𝑡 + 8 = 7 + 5𝑡 − 20

Combine 2𝑡 + 8 = 5𝑡 − 13 Like 20𝑤 − 10 = 8 + 2𝑤 Terms −2𝑤 −2𝑤 −2𝑡 −2𝑡 2t from 8 = 3𝑡 − 13 Subtract 18𝑤 − 10 = 8 Subtract 2w each side of the from each side +10+10 +13 +13 equation. of the equation. 18𝑤 = 18 Divide each 21 = 3𝑡 each Add Divide 13 to each side of the 3 3 18 18 Add 10 to each side of the side of the equation by side of the equation by equation. 7 = 𝑡 𝑤 = 1 3. equation. 18.

# 11

# 12

23 + 3𝑥 = 5𝑥 + 7 −3𝑥 − 3𝑥 23 = 2𝑥 + 7 −7 −7 16 = 2𝑥 2 2 8=𝑥

8 hours

# 13

0.08𝑝 = 7.50 + 0.05𝑝 −0.05𝑝 − 0.05𝑝 0.03𝑝 = 7.50 0.03 0.03 250 = 𝑝

# 14

𝑥 + 2𝑥 − 6 = 36 +6 +6 𝑥 + 2𝑥 = 42 Combine like terms. 3𝑥 = 42 3 3 Divide each side of the 𝑥 = 14 equation by 3.

36 total card − 14 = 22 baseball cards

Add 6 to each side of the equation.

x represents the number of football cards!

# 15

21 + 7 287 𝑥+

𝑥21

2𝑥35 −7 2(21) − 7

Draw and label the triangle.

84 = 𝑥 + 2𝑥 − 7 + (𝑥 + 7) 84 = 4𝑥 4 4 21 = 𝑥

Combine Like Terms Divide each side of the equation by 4.

# 16

−10

1 6 26

3

1 1 𝑏. 17 ≥ −9 𝑐. 𝑎. −3𝑥 𝑞 + ≤𝑥>−9 Change the 3 2 3 2 1inequality +𝟗 +𝟗 − = −𝟑 −𝟑 𝟏 6 6 direction 6 when 𝑥𝟏≥ 26−≥ 𝑥−3 dividing by a 𝟑 𝟑 1 𝑞> 6

negative.

−1.5

56

(5) 𝑓. 𝑥1 + 12 − 2 𝑥 − 22 > 0 (5) 𝑑.𝑒. 4𝑥 𝑥 ≥<−2 −6− 𝑥 + 12 2𝑥 + 44 > 0 5 4 4 −1𝑥 + 56 > 0 𝑥𝑥 ≥ −10 <− −1.5 −56 −56 −1𝑥 > −56 −1 −1 𝑥 < 56

# 17

(4) 70 + 71 + 72 + 𝑥 4

≥ 73 (4)

70 + 71 + 72 + 𝑥 ≥ 292 213 + 𝑥 ≥ 292 −213 −213 𝑥 ≥ 79

Clear the denominator by multiplying each side of the equation by 4. Combine Like Terms. Subtract 213 from each side of the equation.

The plant height must be 79 inches or taller.

# 18

(6) 11 + 10 + 13 + 14 + 14 + 𝑥 ≤ 15(6) 6

11 + 10 + 13 + 14 + 14 + 𝑥 ≤ 90 62 + 𝑥 ≤ 90 −62 −62 𝑥 ≤ 28

Clear the denominator by multiplying each side of the equation by 6.

Combine Like Terms. Subtract 62 from each side of the equation.

Mrs. Hawk can have no more than 28 questions.

# 19

𝑎. 𝑎 − 𝑞 = 𝑎 + 𝑠𝑥 −𝑎 −𝑎 −𝑞 = 𝑠𝑥 𝑠 𝑠 𝑞 − =𝑥 𝑠 𝑐. 𝑎 − 𝑞 = 𝑎 + 𝑠𝑥 −𝑎 −𝑎 −𝑞 = 𝑠𝑥 −1 −1 𝑞 = −𝑠𝑥

(𝑚) 𝑓𝑦 (𝑚) 𝑏. 𝑘𝑥 − 𝑏𝑓 = 𝑚 𝑚(𝑘𝑥 − 𝑏𝑓) = 𝑓𝑦 𝑓 𝑓 𝑚(𝑘𝑥 − 𝑏𝑓) =𝑦 𝑓 (𝑚) 𝑓𝑦 (𝑚) 𝑑. 𝑘𝑥 − 𝑏𝑓 = 𝑚 𝑚(𝑘𝑥 − 𝑏𝑓) = 𝑓𝑦 𝑘𝑥 − 𝑏𝑓 𝑘𝑥 − 𝑏𝑓 𝑓𝑦 𝑚= 𝑘𝑥 − 𝑏𝑓

# 20

𝐴𝑥 + 3𝑦 = 48 𝐴(−3) + 3(14) = 48 −3𝐴 + 42 = 48 −42 − 42 −3𝐴 = 6 −3 − 3 𝐴 = −2

Substitute the values for x and y into the equation. Now solve for A.

# 21

Substitute the values for x and y into the equation.

2𝑥 + 𝐷𝑦 = 13 2(−18) + 𝐷(7) = 13 −36 + 7𝐷 = 13 +36 + 36 7𝐷 = 49 7 7 𝐷=7

Now solve for D.

# 22 This is not a function because –4 (domain) is assigned to more than one value in the range. This is a function because every element of the domain is assigned to exactly one element in the range. This is a function because every element of the domain is assigned to exactly one element in the range.

This is not a function because –3 is assigned to multiple values in the range.

# 23

Divide each side of the equation by 0.3.

13 = 0.3𝑚 + 1 −1 −1 12 = 0.3𝑚 0.3 0.3 40 = 𝑚

Subtract 1 from each side of the equation.

# 24

Divide each side of the equation by 25.

185 = 25𝑥 + 10 −10 −10 175 = 25𝑥 25 25 7=𝑥

Subtract 10 from each side of the equation.

# 25

𝑎. 3𝑥 3 + 2𝑥 2 − 3𝑥 − 2 + (𝑥 2 + 5𝑥 − 6) 3𝑥 3 + 3𝑥 2 + 2𝑥 −8

Combine Like Terms…

# 25

Distribute

𝑏. 12𝑦 2 + 25𝑦 − 6 − 9𝑥 2 + 3𝑥 − 4 + 11

12𝑦 2 + 25𝑦 − 6 + −9𝑥 2 − 3𝑥 + 4 + 11 12𝑦 2 − 9𝑥 2 + 25𝑦 − 3𝑥 + 9

Combine Like Terms…

# 25

𝑐. (2𝑘 − 5)(3𝑘 + 6) 3𝑘

+6

2𝑘

6𝑘 2

+12𝑘

−5

−15𝑘

−30

6𝑘 2 − 3𝑘 − 30

# 25

𝑑. (6𝑚 − 2)(2𝑚2 − 3𝑚 + 2) 2𝑚2

−3𝑚

+2

6𝑚

12𝑚3 −18𝑚2 +12𝑚

−2

Type equation here. −4 −4𝑚2 +6𝑚

12𝑚3 − 22𝑚2 + 18𝑚 − 4

# 26

𝑎. (3𝑥 + 5) + (−8𝑥 − 10) −𝟓𝒙 − 𝟓 𝑏. 3𝑥 + 5 − (−8𝑥 − 10) 3𝑥 + 5 + (+8𝑥 + 10)

(3𝑥 + 5) 𝑑. (−8𝑥 − 10) 𝟑𝒙 + 𝟓 −𝟐(𝟒𝒙 − 𝟓)

𝟏𝟏𝒙 + 𝟏𝟓 𝑐. 3𝑥 + 5 (−8𝑥 − 10) −24𝑥 2

−40𝑥

−30𝑥

−50

−𝟐𝟒𝒙𝟐 − 𝟕𝟎𝒙 − 𝟓𝟎

𝑒. a, b and c are polynomials

# 27

𝒈 𝒇 𝒙

= 𝟏. 𝟎𝟖(𝟎. 𝟕𝒙 + 𝟐𝟑

# 28

𝑔(𝑓 3 )

𝑓(𝑔 𝑥 )

= 𝑔(3 3 + 2)

= 𝑓(2𝑥 2 )

= 𝑔(11)

= 3 2𝑥 2 + 2

= 2 11

2

= 2(121) = 242

= 6𝑥 2 + 2

# 29

a. There are no games left in the warehouse after 27 shipments.

b. He has 9,000 games in the warehouse before making any shipments. c. The slope represents the decrease in the number of games in the warehouse per shipment. d. Domain 0 ≤ 𝑥 ≤ 27 and Range 0 ≤ 𝑦 ≤ 9,000

# 30

192

Domain 𝟎 ≤ 𝒙 ≤ 𝟖 and Range 𝟎 ≤ 𝒚 ≤ 𝟏𝟗𝟐

# 31

Graph the line with a y-intercept of 2 and a slope of

𝟓 . 𝟐

Translate the line 4 units left.

# 32

Graph the line with a y-intercept 𝟐 of −3 and a slope of . 𝟏

Translate the line 3 units up.

# 33

Initial value

𝑎. 𝑎𝑛 = 7 + (𝑛 − 1)(−3) Common difference

𝑏. 𝑎𝑛 = −11 + (𝑛 − 1)(8)

# 34

Find the slope…

𝑦2 − 𝑦1 𝑚= 𝑥2 − 𝑥1 151 − 79 72 6 = = =6 24 − 12 12 1

The special rate would be $6. The shipping would be $7.

Find the yintercept… 𝑦 = 𝑚𝑥 + 𝑏 151 = 6 24 + 𝑏 151 = 144 + 𝑏 −144 −144 7=𝑏

# 35

Find the slope…

𝑦2 − 𝑦1 𝑚= 𝑥2 − 𝑥1 40 − 30 10 5 = = 6−2 4 2

Find the yintercept… 𝑦 = 𝑚𝑥 + 𝑏

5 30 = 2 + 𝑏 2 30 = 5 + 𝑏 −5 −5 25 = 𝑏

He would make 2.5 cards per hour.

He started with 25 cards in the box.

# 36

The slope of the line of best 𝟏 fit is and it’s y-intercept is 0. 𝟑

𝟏 𝒇 𝒙 = 𝒙 𝟑

Every 3 days, 1 additional golf cart is repaired

# 37

The slope of the line of best fit is − 𝟓 and it’s y-intercept is 90.

𝒇 𝒙 = −𝟓𝒙 + 𝟗𝟎 Every day there are 5 less golf carts sold

# 38

(−𝟑) (−𝟑) (−𝟑)

Multiply each term in the first equation by −𝟑.

𝑎. 2𝑥 + 5𝑦 = 20 6𝑥 + 15𝑦 = 15 −6𝑥 − 15𝑦 = −60 6𝑥 + 15𝑦 = 15 0𝑥 + 0𝑦 = −45

No Solution

# 38

Multiply each term in the first equation by 𝟒.

𝑏. 9𝑥 + 18𝑦 = 6 12𝑥 + 24𝑦 = 8 Multiply each term in the second equation by −𝟑.

36𝑥 + 72𝑦 = 24 −36𝑥 − 72𝑦 = −24 0𝑥 + 0𝑦 = 0

Infinitely many solutions

# 38

Multiply each term in the first equation by 𝟔.

𝑐. 5𝑥 + 4𝑦 = 6 6𝑥 + 𝑦 = 13 (19) (19) 29 30𝑥 + 24 − = 36 (19) 19 570𝑥 − 696 = 684 +696 +696 570𝑥 = 1380 570 570 46 𝑥= 19

Multiply each term in the second equation by −𝟓.

30𝑥 + 24𝑦 = 36 −30𝑥 − 5𝑦 = −65 0𝑥 + 19𝑦 = −29 19 19 29 𝑦=− 19

# 39

28 − 16 = 12 blocks Let x be the number of bricks and y the number of blocks.

𝑥 + 𝑦 = 28 0.38𝑥 + 1.56𝑦 = 24.80

𝑦 = 28 − 𝑥

Substitute

0.38𝑥 + 1.56 28 − 𝑥 = 24.80

0.38𝑥 + 43.68 − 1.56𝑥 = 24.80 Combine −1.18𝑥 + 43.68 = 24.80 Like Terms −43.68 −43.68 Subtract 43.68 from each −1.18𝑥 = −18.88 side of the equation. −1.18 −1.18 Divide each side of the 𝑥 = 16 Distribute

equation by -1.18.

# 40

Let a = # adult tickets and c = # of child tickets Use substitution to replace c with 3a.

20𝑎 + 10𝑐 = 15,000 𝑐 = 3𝑎

20𝑎 + 10 3𝑎 = 15,000 20𝑎 + 30𝑎 = 15,000 50𝑎 = 15,000 50 50 𝑎 = 300

300 adult tickets were sold and now that can be substituted into either equation for a. 𝑐 = 3𝑎 𝑐 = 3(300) 𝑐 = 900

900 child tickets were sold.

# 41

Multiply the first equation by −𝟒𝟑. Let S = # of small dogs and L = # of large dogs Since we are looking for L, let’s eliminate S.

𝑆 + 𝐿 = 22 43𝑆 + 75𝐿 = 1234 Add the second equation.

−43𝑆 − 43𝐿 = −946 43𝑆 + 75𝐿 = 1234

+

32𝐿 = 288 32 32 𝐿=9 9 large dogs were groomed.

# 42

where C is the cost and d is the number of disks

13.75𝑑 = 3255 + 2.9𝑑 −2.9𝑑 −2.9𝑑 10.85𝑑 = 3255 10.85 10.85 𝑑 = 300

# 43

Answers will vary. Below are sample answers.

a. 4x + 6y = 8 b. b. 3x – 21y = 24 c. c.

5 x 2

+ 2y = 10

d. d. 4x + y =

1 3

# 44

𝑎. 7𝑥 + 6𝑦 > 30 −7𝑥 −7𝑥 6𝑦 > −7𝑥 + 30

6

6

6

7 𝑦 >− 𝑥+5 6

# 44

𝑏. 9𝑥 + 4𝑦 ≥ 28 −9𝑥 −9𝑥 4𝑦 ≥ −9𝑥 + 28 4 4 4 9 𝑦 ≥− 𝑥+7 4

# 44

𝑐. 3𝑥 + 7𝑦 < 14 −3𝑥 −3𝑥 7𝑦 < −3𝑥 + 14

7

7

7

3 𝑦 <− 𝑥+2 7

# 44

𝑑. 7𝑥 + 14𝑦 ≤ 3 −7𝑥 −7𝑥 14𝑦 ≤ −7𝑥 + 3 14 14 14 7 3 𝑦≤− 𝑥+ 14 14

1 3 𝑦≤− 𝑥+ 2 14

# 45

−5𝑥 − 4𝑦 ≤ 6 +5𝑥 +5𝑥 −4𝑦 ≤ 5𝑥 + 6 −4 −4 −4 5 1 𝑦 ≥− 𝑥−1 4 2 8𝑥 − 16𝑦 > 24 −8𝑥 −8𝑥 −16𝑦 > −8𝑥 + 24 −16 −16 −16 1 1 𝑦 < 𝑥−1 2 2

# 45

𝑥+𝑦 >1 −𝑥 −𝑥 𝑦 > −𝑥 + 1

5𝑥 + 2𝑦 < 10 −5𝑥 −5𝑥 2𝑦 < −5𝑥 + 10 2 2 2 5 𝑦 <− 𝑥+5 2

# 46

175 feet 25 seconds

# 47

about 4.5 seconds about 26 feet

# 48

𝑥2 + 𝑥 − 6

FACTOR

𝑥

−2

𝑥

𝑥2

−2𝑥

+3

3𝑥

−6

𝒙+𝟑

What are the factors of −𝟔 that also combine to + 𝟏?

−6𝑥 2

FACTOR

# 49

2𝑥 2 + 13𝑥 + 15

FACTOR

2𝑥

+3

𝑥

2𝑥 2

3𝑥

+5

10𝑥

+15

𝒙+𝟓

What are the factors of +𝟑𝟎 that also combine to + 𝟏𝟑?

+30𝑥 2

FACTOR

# 50

3𝑥

+7

𝑥

3𝑥 2

7𝑥

+5

15𝑥

+35

𝑎. 3𝑥 2 + 22𝑥 + 35 = 0

𝑥+5=0 −5 −5 𝑥 = −5

3𝑥 + 7 = 0 −7 −7 3𝑥 = −7 3 3 7 𝑥=− 3

What are the factors of 𝟏𝟎𝟓 that also combine to + 𝟐𝟐?

FACTOR

105𝑥 2

# 50

𝑏. 𝑥 2 − 𝑥 = 35 −35 −35 𝑥 2 − 𝑥 − 35 = 0

Not factorable.

−𝑏 ± 𝑏 2 − 4𝑎𝑐 𝑥= 2𝑎 1 ± −12 − 4(1)(−35) 𝑥= 2(1) 𝑥=

1 ± 1 + 140 2

𝑥=

1 ± 141 2

# 50

𝑐. 2𝑥 3 + 4𝑥 2 + 2𝑥 = 0

2𝑥(𝑥 2 + 2𝑥 + 1) = 0 2𝑥 = 0

𝑥 2 + 2𝑥 + 1 = 0

𝑥=0

(𝑥 + 1)(𝑥 + 1) = 0

𝑥+1=0 −1 −1 𝑥 = −1

# 50

𝑑. 𝑥 2 − 6𝑥 − 27 = 0 𝑥−9 𝑥+3 =0 𝑥−9=0 +9 +9 𝑥=9

𝑥+3=0 −3 −3 𝑥 = −3

# 50

𝑒. 45𝑥 2 − 20 = 0 +20 +20 45𝑥 2 = 20 45 45 20 5 ÷ = 45 5 4 2 𝑥 = 9 2 𝑥=± 3

𝑥2

# 50

𝑓. 5𝑥 3 − 20𝑥 = 0 𝑥(5𝑥 2 − 20) = 0

𝑥=0

5𝑥 2 − 20 = 0 +20 +20

5𝑥 2 = 20 5 5 𝑥2 = 4 𝑥 = ±2

# 51

𝑎. 3𝑥 2 − 6𝑥 − 24 2

3(𝑥 − 2𝑥 − 8)

𝟑(𝒙 + 𝟐)(𝒙 − 𝟒)

𝑥

+2

𝑥2

+2𝑥

−4 −4𝑥

−8

𝑥

What are the factors of −𝟖 that also combine to − 𝟐?

FACTOR

−8𝑥 2

# 51

Notice that EVERY term is a perfect square and the operation is subtraction.

𝑏. 4𝑥 2 − 9

This is difference of squares.

(𝟐𝒙 − 𝟑)(𝟐𝒙 + 𝟑)

# 51

𝑐. 7𝑥 2 + 9𝑥 + 2

𝑥

+1 (𝟕𝒙 + 𝟐)(𝒙 + 𝟏)

7𝑥

+2

7𝑥 2

+2𝑥

7𝑥

+2

What are the factors of 𝟏𝟒 that also combine to 𝟗?

FACTOR

+14𝑥 2

# 52

𝑎. 7𝑥 2 − 14 = 0 +14 +14 7𝑥 2 = 14 7 7 𝑥2 = 2 𝑥=± 2

𝑏. 𝑥 2 − 13 = 0 +13 +13 𝑥 2 = 13

𝑥 = ± 13

# 53

𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 + 𝑘 Factor 7 from the first two terms.

𝑎. 7𝑥 2 + 14𝑥 + 30 7(𝑥 2 + 2𝑥 +1 ) + 30 −7 𝑓 𝑥 = 7 𝑥 + 1 2 + 23

Complete the Square.

𝑥 +1 𝑥 𝑥 2 1𝑥

+1 1𝑥 +1

# 53

𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 + 𝑘

𝑏. 𝑥 2 − 6𝑥 − 28 (𝑥 2 −6𝑥 +9 ) − 28 −9 𝑓 𝑥 = 𝑥 − 3 2 − 37

Complete the Square.

𝑥 −3 𝑥 𝑥 2 −3𝑥

−3 −3𝑥 +9

# 54

𝑎. 𝑥 2 + 4𝑥 − 5 = 0 +5 +5 𝑥 2 + 4𝑥 +4 = 5 +4 (𝑥 + 2)2 = 9 𝑥 + 2 = ±3 𝑥+2=3 −2 −2 𝑥=1

𝑥 + 2 = −3 −2 −2 𝑥 = −5

𝑥 𝑥

+2

𝑥 2 +2𝑥

+2 +2𝑥 +4

# 54

𝑏. 4𝑥 2 + 8𝑥 − 48 = 0 4(𝑥 2 + 2𝑥 − 12) = 0 +12 +12 𝑥 2 + 2𝑥 +1 = 12 +1 (𝑥 + 1)2 = 13

𝑥 + 1 = ± 13 −1 −1 𝑥 = −1 ± 13

𝑥 𝑥

+1

𝑥 2 +1𝑥

+1 +1𝑥 +1

# 54

𝑥

𝑐. 𝑥 2 + 12𝑥 = 9 𝑥 2 + 12𝑥 +36 = 9 +36 (𝑥 + 6)2 = 45 𝑥 + 6 = ± 45 𝑥+6=± 9× 5 𝑥 + 6 = ±3 5 −6 −6 𝑥 = −6 ± 3 5

𝑥

+6

𝑥 2 +6𝑥

+6 +6𝑥 +36

# 55

𝑎.

−3 ± 32 − 4(1)(2) 2(1)

𝑏.

−8 ± 82 − 4(4)(−24) 2(4)

−3 ± 9 − 8 2

−8 ± 64 + 384 8

−3 ± 1 2

−8 ± 448 8

−3 + 1 = −1 2

−3 − 1 = −2 2

−8 + 8 7 = −1 ± 7 8

# 55

𝑐.

−27 ± 272 − 4(6)(30) 2(6)

𝑑.

−14 ± 142 − 4(1)(33) 2(1)

−27 ± 729 − 720 12

−14 ± 196 − 132 2

−27 ± 9 12

−14 ± 64 2

−27 + 3 = −2 12

−27 − 3 = −2.5 12

−14 + 8 = −3 2

−14 − 8 = −11 2

# 56

Vertex (2, 31); Min = 31 Vertex (–6, 4); Min = 4

Vertex is (−6,4)

𝑏 𝑥=− 2𝑎 −8 𝑥=− =2 2 2 𝑦 = 2 22 − 8 2 + 39 𝑦 = 8 − 16 + 39 𝑦 = 31