Problem Set 2: The Language of Algebra

Using the Language of Algebra

Use Variables and Algebraic Symbols In the following exercises, translate from algebraic notation to words.

16 - 9

16 minus 9, the difference of sixteen and nine

25 - 7

5\cdot 6

5 times 6, the product of five and six

3\cdot 9

28\div 4

28 divided by 4, the quotient of twenty-eight and four

45\div 5

x+8

x plus 8, the sum of x and eight

x+11

\left(2\right)\left(7\right)

2 times 7, the product of two and seven

\left(4\right)\left(8\right)

14<21

fourteen is less than twenty-one

17<35

36\ge 19

thirty-six is greater than or equal to nineteen

42\ge 27

3n=24

3 times n equals 24, the product of three and n equals twenty-four

6n=36

y - 1>6

y minus 1 is greater than 6, the difference of y and one is greater than six

y - 4>8

2\le 18\div 6

2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six

3\le 20\div 4

a\ne 7\cdot 4

a is not equal to 7 times 4, a is not equal to the product of seven and four

a\ne 1\cdot 12

Identify Expressions and Equations In the following exercises, determine if each is an expression or an equation.

9\cdot 6=54

equation

7\cdot 9=63

5\cdot 4+3

expression

6\cdot 3+5

x+7

expression

x+9

y - 5=25

equation

y - 8=32

Simplify Expressions with Exponents In the following exercises, write in exponential form.

3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3

3⁷

4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4

x\cdot x\cdot x\cdot x\cdot x

x⁵

y\cdot y\cdot y\cdot y\cdot y\cdot y

In the following exercises, write in expanded form.

{5}^{3}

125

{8}^{3}

{2}^{8}

256

{10}^{5}

Simplify Expressions Using the Order of Operations In the following exercises, simplify.

$3+8\cdot 5$
$\text{(3+8)}\cdot \text{5}$

$2+6\cdot 3$
$\text{(2+6)}\cdot \text{3}$

{2}^{3}-12\div \left(9 - 5\right)

{3}^{2}-18\div \left(11 - 5\right)

3\cdot 8+5\cdot 2

4\cdot 7+3\cdot 5

2+8\left(6+1\right)

4+6\left(3+6\right)

4\cdot 12/8

2\cdot 36/6

6+10/2+2

9+12/3+4

\left(6+10\right)\div \left(2+2\right)

\left(9+12\right)\div \left(3+4\right)

20\div 4+6\cdot 5

33\div 3+8\cdot 2

20\div \left(4+6\right)\cdot 5

33\div \left(3+8\right)\cdot 2

{4}^{2}+{5}^{2}

{3}^{2}+{7}^{2}

{\left(4+5\right)}^{2}

{\left(3+7\right)}^{2}

3\left(1+9\cdot 6\right)-{4}^{2}

149

5\left(2+8\cdot 4\right)-{7}^{2}

2\left[1+3\left(10 - 2\right)\right]

5\left[2+4\left(3 - 2\right)\right]

Everyday Math

Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol

\text{(=},\text{<},\text{>)}

Spurs	Height	Heat	Height
Tim Duncan	$\text{83{\'\'} }$	Rashard Lewis	$\text{82{\'\'} }$
Boris Diaw	$\text{80{\'\'} }$	LeBron James	$\text{80{\'\'} }$
Kawhi Leonard	$\text{79{\'\'} }$	Chris Bosh	$\text{83{\'\'} }$
Tony Parker	$\text{74{\'\'} }$	Dwyane Wade	$\text{76{\'\'} }$
Danny Green	$\text{78{\'\'} }$	Ray Allen	$\text{77{\'\'} }$

Height of Tim Duncan____Height of Rashard Lewis
Height of Boris Diaw____Height of LeBron James
Height of Kawhi Leonard____Height of Chris Bosh
Height of Tony Parker____Height of Dwyane Wade
Height of Danny Green____Height of Ray Allen

Elevation In Colorado there are more than

50

mountains with an elevation of over

14,000\text{feet.}

The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

Mountain	Elevation
Mt. Elbert	$\text{14,433{\'} }$
Mt. Massive	$\text{14,421{\'} }$
Mt. Harvard	$\text{14,420{\'} }$
Blanca Peak	$\text{14,345{\'} }$
La Plata Peak	$\text{14,336{\'} }$
Uncompahgre Peak	$\text{14,309{\'} }$
Crestone Peak	$\text{14,294{\'} }$
Mt. Lincoln	$\text{14,286{\'} }$
Grays Peak	$\text{14,270{\'} }$
Mt. Antero	$\text{14,269{\'} }$

Elevation of La Plata Peak____Elevation of Mt. Antero Elevation of Blanca Peak____Elevation of Mt. Elbert Elevation of Gray’s Peak____Elevation of Mt. Lincoln Elevation of Mt. Massive____Elevation of Crestone Peak Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

Writing Exercises

Explain the difference between an expression and an equation. Why is it important to use the order of operations to simplify an expression?

Self Check

1. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

2. If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

7x+8\text{when}x=2

9x+7\text{when}x=3

5x - 4\text{when}x=6

8x - 6\text{when}x=7

{x}^{2}\text{when}x=12

144

{x}^{3}\text{when}x=5

{x}^{5}\text{when}x=2

{x}^{4}\text{when}x=3

{3}^{x}\text{when}x=3

{4}^{x}\text{when}x=2

{x}^{2}+3x - 7\text{when}x=4

{x}^{2}+5x - 8\text{when}x=6

2x+4y - 5\text{when}x=7,y=8

6x+3y - 9\text{when}x=6,y=9

{\left(x-y\right)}^{2}\text{when}x=10,y=7

{\left(x+y\right)}^{2}\text{when}x=6,y=9

225

{a}^{2}+{b}^{2}\text{when}a=3,b=8

{r}^{2}-{s}^{2}\text{when}r=12,s=5

2l+2w\text{when}l=15,w=12

2l+2w\text{when}l=18,w=14

Identify Terms, Coefficients, and Like Terms In the following exercises, list the terms in the given expression.

15{x}^{2}+6x+2

15x², 6x, 2

11{x}^{2}+8x+5

10{y}^{3}+y+2

10y³, y, 2

9{y}^{3}+y+5

In the following exercises, identify the coefficient of the given term.

8a

13m

5{r}^{2}

6{x}^{3}

In the following exercises, identify all sets of like terms.

{x}^{3},8x,14,8y,5,8{x}^{3}

x³, 8x³ and 14, 5

6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}

9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}

16ab and 4ab; 16b² and 9b²

3,25{r}^{2},10s,10r,4{r}^{2},3s

Simplify Expressions by Combining Like Terms In the following exercises, simplify the given expression by combining like terms.

10x+3x

13x

15x+4x

17a+9a

26a

18z+9z

4c+2c+c

6y+4y+y

9x+3x+8

12x + 8

8a+5a+9

7u+2+3u+1

10u + 3

8d+6+2d+5

7p+6+5p+4

12p + 10

8x+7+4x - 5

10a+7+5a - 2+7a - 4

22a + 1

7c+4+6c - 3+9c - 1

3{x}^{2}+12x+11+14{x}^{2}+8x+5

17x² + 20x + 16

5{b}^{2}+9b+10+2{b}^{2}+3b - 4

Translate English Phrases into Algebraic Expressions In the following exercises, translate the given word phrase into an algebraic expression. The sum of 8 and 12 8 + 12 The sum of 9 and 1 The difference of 14 and 9 14 − 9 8 less than 19 The product of 9 and 7 9 ⋅ 7 The product of 8 and 7 The quotient of 36 and 9 36 ÷ 9 The quotient of 42 and 7 The difference of

x

and

4

x − 4

3[/latex] less than [latex]x

The product of

6

and

y

6y The product of

9

and

y

The sum of

8x

and

3x

8x + 3x The sum of

13x

and

3x

The quotient of

y

and

3

\frac{y}{3}

The quotient of

y

and

8

Eight times the difference of

y

and nine 8 (y − 9) Seven times the difference of

y

and one Five times the sum of

x

and

y

5 (x + y) Nine times five less than twice

x

In the following exercises, write an algebraic expression. Adele bought a skirt and a blouse. The skirt cost

\text{$15}

more than the blouse. Let

b

represent the cost of the blouse. Write an expression for the cost of the skirt. b + 15 Eric has rock and classical CDs in his car. The number of rock CDs is

3

more than the number of classical CDs. Let

c

represent the number of classical CDs. Write an expression for the number of rock CDs. The number of girls in a second-grade class is

4

less than the number of boys. Let

b

represent the number of boys. Write an expression for the number of girls. b − 4 Marcella has

6

fewer male cousins than female cousins. Let

f

represent the number of female cousins. Write an expression for the number of boy cousins. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let

n

represent the number of nickels. Write an expression for the number of pennies. 2n − 7 Jeannette has

\text{$5}

and

\text{$10}

bills in her wallet. The number of fives is three more than six times the number of tens. Let

t

represent the number of tens. Write an expression for the number of fives.

Everyday Math

In the following exercises, use algebraic expressions to solve the problem. Car insurance Justin’s car insurance has a

\text{$750}

deductible per incident. This means that he pays

\text{$750}

and his insurance company will pay all costs beyond

\text{$750.}

If Justin files a claim for

\text{$2,100,}

how much will he pay, and how much will his insurance company pay? He will pay $750. His insurance company will pay $1350. Home insurance Pam and Armando’s home insurance has a

\text{$2,500}

deductible per incident. This means that they pay

\text{$2,500}

and their insurance company will pay all costs beyond

\text{$2,500.}

If Pam and Armando file a claim for

\text{$19,400,}

how much will they pay, and how much will their insurance company pay?

Writing Exercises

Explain why "the sum of x and y" is the same as "the sum of y and x," but "the difference of x and y" is not the same as "the difference of y and x." Try substituting two random numbers for

x

and

y

to help you explain. Explain the difference between

\text{"4}

times the sum of

x

and

y\text{"}

and "the sum of

4

times

x

and

y\text{."}

Self Check

1. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

2. After reviewing this checklist, what will you do to become confident for all objectives?

Subtraction Property of Equality

Determine Whether a Number is a Solution of an Equation In the following exercises, determine whether each given value is a solution to the equation.

x+13=21

ⓐ

x=8

ⓑ

x=34

ⓐ yes ⓑ no

y+18=25

ⓐ

y=7

ⓑ

y=43

m - 4=13

ⓐ

m=9

ⓑ

m=17

ⓐ no ⓑ yes

n - 9=6

ⓐ

n=3

ⓑ

n=15

3p+6=15

ⓐ

p=3

ⓑ

p=7

ⓐ yes ⓑ no

8q+4=20

ⓐ

q=2

ⓑ

q=3

18d - 9=27

ⓐ

d=1

ⓑ

d=2

ⓐ no ⓑ yes

24f - 12=60

ⓐ

f=2

ⓑ

f=3

8u - 4=4u+40

ⓐ

u=3

ⓑ

u=11

ⓐ no ⓑ yes

7v - 3=4v+36

ⓐ

v=3

ⓑ

v=11

20h - 5=15h+35

ⓐ

h=6

ⓑ

h=8

ⓐ no ⓑ yes

18k - 3=12k+33

ⓐ

k=1

ⓑ

k=6

Model the Subtraction Property of Equality In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality. The image is divided in half vertically. On the left side is an envelope with 2 counters below it. On the right side is 5 counters.

The image is divided in half vertically. On the left side is an envelope with 2 counters below it. On the right side is 5 counters.

x + 2 = 5; x = 3 The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 7 counters.

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 6 counters.

x + 3 = 6; x = 3 The image is divided in half vertically. On the left side is an envelope with 5 counters below it. On the right side is 9 counters.

Solve Equations using the Subtraction Property of Equality In the following exercises, solve each equation using the subtraction property of equality.

a+2=18

a = 16

b+5=13

p+18=23

p = 5

q+14=31

r+76=100

r = 24

s+62=95

16=x+9

x = 7

17=y+6

93=p+24

p = 69

116=q+79

465=d+398

d = 67

932=c+641

Solve Equations using the Addition Property of Equality In the following exercises, solve each equation using the addition property of equality.

y - 3=19

y = 22

x - 4=12

u - 6=24

u = 30

v - 7=35

f - 55=123

f = 178

g - 39=117

19=n - 13

n = 32

18=m - 15

10=p - 38

p = 48

18=q - 72

268=y - 199

y = 467

204=z - 149

Translate Word Phrase to Algebraic Equations In the following exercises, translate the given sentence into an algebraic equation. The sum of

8

and

9

is equal to

17

. 8 + 9 = 17 The sum of

7

and

9

is equal to

16

. The difference of

23

and

19

is equal to

4

. 23 − 19 = 4 The difference of

29

and

12

is equal to

17

. The product of

3

and

9

is equal to

27

. 3 ⋅ 9 = 27 The product of

6

and

8

is equal to

48

. The quotient of

54

and

6

is equal to

9

. 54 ÷ 6 = 9 The quotient of

42

and

7

is equal to

6

. Twice the difference of

n

and

10

gives

52

. 2(n − 10) = 52 Twice the difference of

m

and

14

gives

64

. The sum of three times

y

and

10

100

. 3y + 10 = 100 The sum of eight times

x

and

4

68

. Translate to an Equation and Solve In the following exercises, translate the given sentence into an algebraic equation and then solve it. Five more than

p

is equal to

21

. p + 5 = 21; p = 16 Nine more than

q

is equal to

40

. The sum of

r

and

18

73

. r + 18 = 73; r = 55 The sum of

s

and

13

68

. The difference of

d

and

30

is equal to

52

. d − 30 = 52; d = 82 The difference of

c

and

25

is equal to

75

12

less than

u

89

. u − 12 = 89; u = 101

19

less than

w

56

325

less than

c

gives

799

. c − 325 = 799; c = 1124

299

less than

d

gives

850

Everyday Math

Insurance Vince’s car insurance has a

\text{$500}

deductible. Find the amount the insurance company will pay,

p

, for an

\text{$1800}

claim by solving the equation

500+p=1800

. $1300 Insurance Marta’s homeowner’s insurance policy has a

\text{$750}

deductible. The insurance company paid

\text{$5800}

to repair damages caused by a storm. Find the total cost of the storm damage,

d

, by solving the equation

d - 750=5800

. Sale purchase Arthur bought a suit that was on sale for

\text{$120}

off. He paid

\text{$340}

for the suit. Find the original price,

p

, of the suit by solving the equation

p - 120=340

. $460 Sale purchase Rita bought a sofa that was on sale for

\text{$1299}

. She paid a total of

\text{$1409}

, including sales tax. Find the amount of the sales tax,

t

, by solving the equation

1299+t=1409

Writing Exercises

x=1

a solution to the equation

8x - 2=16 - 6x?

How do you know? Write the equation

y - 5=21

in words. Then make up a word problem for this equation.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Identify Multiples of Numbers

In the following exercises, list all the multiples less than $50$ for the given number.

$2$

2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48

$3$

$4$

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

$5$

$6$

6, 12, 18, 24, 30, 36, 42, 48

$7$

$8$

8, 16, 24, 32, 40, 48

$9$

$10$

10, 20, 30, 40

$12$

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by $2,3,4,5,6,\text{and}10$ .

$84$

Divisible by 2, 3, 4, 6

$96$

$75$

Divisible by 3, 5

$78$

$168$

Divisible by 2, 3, 4, 6

$264$

$900$

Divisible by 2, 3, 4, 5, 6, 10

$800$

$896$

Divisible by 2, 4

$942$

$375$

Divisible by 3, 5

$750$

$350$

Divisible by 2, 5, 10

$550$

$1430$

Divisible by 2, 5, 10

$1080$

$22,335$

Divisible by 3, 5

$39,075$

Find All the Factors of a Number

In the following exercises, find all the factors of the given number.

$36$

1, 2, 3, 4, 6, 9, 12, 18, 36

$42$

$60$

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

$48$

$144$

1, 2, 3, 4, 6, 8, 12, 18, 24, 36, 48, 72,144

$200$

$588$

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588

$576$

Identify Prime and Composite Numbers

In the following exercises, determine if the given number is prime or composite.

$43$

prime

$67$

$39$

composite

$53$

$71$

prime

$119$

$481$

composite

$221$

$209$

composite

$359$

$667$

composite

$1771$

Everyday Math

Banking Frank’s grandmother gave him $\text{$100}$ at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $\text{$15}$ to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after graduation	Total number of dollars Frank put in the account	Simplified Total
$0$	$100$	$100$
$1$	$100+15$	$115$
$2$	$100+15\cdot 2$	$130$
$3$	$100+15\cdot 3$
$4$	$100+15\cdot \left[\right]$
$5$	$100+\left[\right]$
$6$
$20$
$x$

This table has nine rows and three columns. The first row is a header row that labels each column. The first column is labeled

Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $\text{$75}$ to open the account. Every week, she added $\text{$20}$ to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after opening the account	Total number of dollars Gina put in the account	Simplified Total
$0$	$75$	$75$
$1$	$75+20$	$95$
$2$	$75+20\cdot 2$	$115$
$3$	$75+20\cdot 3$
$4$	$75+20\cdot \left[\right]$
$5$	$75+\left[\right]$
$6$
$20$
$x$

Writing Exercises

If a number is divisible by $2$ and by $3$ , why is it also divisible by $6?$

What is the difference between prime numbers and composite numbers?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

$86$

2 ⋅ 43

$78$

$132$

2 ⋅ 2 ⋅ 3 ⋅ 11

$455$

$693$

3 ⋅ 3 ⋅ 7 ⋅ 11

$420$

$115$

5 ⋅ 23

$225$

$2475$

3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11

1560

In the following exercises, find the prime factorization of each number using the ladder method.

$56$

2 ⋅ 2 ⋅ 2 ⋅ 7

$72$

$168$

2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7

$252$

$391$

17 ⋅ 23

$400$

$432$

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3

$627$

$2160$

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5

$2520$

In the following exercises, find the prime factorization of each number using any method.

$150$

2 ⋅ 3 ⋅ 5 ⋅ 5

$180$

$525$

3 ⋅ 5 ⋅ 5 ⋅ 7

$444$

$36$

2 ⋅ 2 ⋅ 3 ⋅ 3

$50$

$350$

2 ⋅ 5 ⋅ 5 ⋅ 7

$144$

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.

$8,12$

$4,3$

$6,15$

$12,16$

$30,40$

120

$20,30$

$60,75$

300

$44,55$

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

$8,12$

$12,16$

$24,30$

120

$28,40$

$70,84$

420

$84,90$

In the following exercises, find the least common multiple (LCM) using any method.

$6,21$

$9,15$

$24,30$

120

$32,40$

Everyday Math

Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

Grocery shopping Paper plates are sold in packages of $12$ and party cups come in packs of $8$ . What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

Writing Exercises

Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?

Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Guide allo studio > Prealgebra

Problem Set 2: The Language of Algebra

Using the Language of Algebra

Everyday Math

Writing Exercises

Self Check

Evaluate Algebraic Expressions

Everyday Math

Writing Exercises

Self Check

Subtraction Property of Equality

Everyday Math

Writing Exercises

Self Check

Identify Multiples of Numbers

Everyday Math

Writing Exercises

Self Check

Find the Prime Factorization of a Composite Number

Everyday Math

Writing Exercises

Self Check