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Study Guides > ALGEBRA / TRIG I

Problem Set: The Language of Algebra

Using the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.
  1. [latex]16 - 9[/latex]

    Answer: 16 minus 9, the difference of sixteen and nine

  2. [latex]25 - 7[/latex]
  3. [latex]5\cdot 6[/latex]

    Answer: 5 times 6, the product of five and six

  4. [latex]3\cdot 9[/latex]
  5. [latex]28\div 4[/latex]

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

  6. [latex]45\div 5[/latex]
  7. [latex]x+8[/latex]

    Answer: x plus 8, the sum of x and eight

  8. [latex]x+11[/latex]
  9. [latex]\left(2\right)\left(7\right)[/latex]

    Answer: 2 times 7, the product of two and seven

  10. [latex]\left(4\right)\left(8\right)[/latex]
  11. [latex]14<21[/latex]

    Answer: fourteen is less than twenty-one

  12. [latex]17<35[/latex]
  13. [latex]36\ge 19[/latex]

    Answer: thirty-six is greater than or equal to nineteen

  14. [latex]42\ge 27[/latex]
  15. [latex]3n=24[/latex]

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

  16. [latex]6n=36[/latex]
  17. [latex]y - 1>6[/latex]

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

  18. [latex]y - 4>8[/latex]
  19. [latex]2\le 18\div 6[/latex]

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

  20. [latex]3\le 20\div 4[/latex]
  21. [latex]a\ne 7\cdot 4[/latex]

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

  22. [latex]a\ne 1\cdot 12[/latex]

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.
  1. [latex]9\cdot 6=54[/latex]

    Answer: equation

  2. [latex]7\cdot 9=63[/latex]
  3. [latex]5\cdot 4+3[/latex]

    Answer: expression

  4. [latex]6\cdot 3+5[/latex]
  5. [latex]x+7[/latex]

    Answer: expression

  6. [latex]x+9[/latex]
  7. [latex]y - 5=25[/latex]

    Answer: equation

  8. [latex]y - 8=32[/latex]

Simplify Expressions with Exponents

In the following exercises, write in exponential form.
  1. [latex]3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3[/latex]

    Answer: 37

  2. [latex]4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4[/latex]
  3. [latex]x\cdot x\cdot x\cdot x\cdot x[/latex]

    Answer: x5

  4. [latex]y\cdot y\cdot y\cdot y\cdot y\cdot y[/latex]

Simplify Expressions with Exponents

In the following exercises, write in expanded form.
  1. [latex]{5}^{3}[/latex]

    Answer: 125

  2. [latex]{8}^{3}[/latex]
  3. [latex]{2}^{8}[/latex]

    Answer: 256

  4. [latex]{10}^{5}[/latex]

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.
  1. [latex]3+8\cdot 5[/latex]

    Answer: 43

  2. [latex]\text{(3+8)}\cdot \text{5}[/latex]

    Answer: 55

  3. [latex]2+6\cdot 3[/latex]
  4. [latex]\text{(2+6)}\cdot \text{3}[/latex]
  5. [latex]{2}^{3}-12\div \left(9 - 5\right)[/latex]

    Answer: 5

  6. [latex]{3}^{2}-18\div \left(11 - 5\right)[/latex]
  7. [latex]3\cdot 8+5\cdot 2[/latex]

    Answer: 34

  8. [latex]4\cdot 7+3\cdot 5[/latex]
  9. [latex]2+8\left(6+1\right)[/latex]

    Answer: 58

  10. [latex]4+6\left(3+6\right)[/latex]
  11. [latex]4\cdot 12/8[/latex]

    Answer: 6

  12. [latex]2\cdot 36/6[/latex]
  13. [latex]6+10/2+2[/latex]

    Answer: 13

  14. [latex]9+12/3+4[/latex]
  15. [latex]\left(6+10\right)\div \left(2+2\right)[/latex]

    Answer: 4

  16. [latex]\left(9+12\right)\div \left(3+4\right)[/latex]
  17. [latex]20\div 4+6\cdot 5[/latex]

    Answer: 35

  18. [latex]33\div 3+8\cdot 2[/latex]
  19. [latex]20\div \left(4+6\right)\cdot 5[/latex]

    Answer: 10

  20. [latex]33\div \left(3+8\right)\cdot 2[/latex]
  21. [latex]{4}^{2}+{5}^{2}[/latex]

    Answer: 41

  22. [latex]{3}^{2}+{7}^{2}[/latex]
  23. [latex]{\left(4+5\right)}^{2}[/latex]

    Answer: 81

  24. [latex]{\left(3+7\right)}^{2}[/latex]
  25. [latex]3\left(1+9\cdot 6\right)-{4}^{2}[/latex]

    Answer: 149

  26. [latex]5\left(2+8\cdot 4\right)-{7}^{2}[/latex]
  27. [latex]2\left[1+3\left(10 - 2\right)\right][/latex]

    Answer: 50

  28. [latex]5\left[2+4\left(3 - 2\right)\right][/latex]

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 [latex]\text{(=},\text{<},\text{>)}[/latex].
Spurs Height Heat Height
Tim Duncan 83" Rashard Lewis 82"
Boris Diaw 80" LeBron James 80"
Kawhi Leonard 79" Chris Bosh 83"
Tony Parker 74" Dwyane Wade 76"
Danny Green 78" Ray Allen 77"
  1. Height of Tim Duncan____Height of Rashard Lewis
  2. Height of Boris Diaw____Height of LeBron James
  3. Height of Kawhi Leonard____Height of Chris Bosh
  4. Height of Tony Parker____Height of Dwyane Wade
  5. Height of Danny Green____Height of Ray Allen

Elevation

In Colorado there are more than [latex]50[/latex] mountains with an elevation of over [latex]14,000\text{ feet.}[/latex] The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.
Mountain Elevation
Mt. Elbert 14,433'
Mt. Massive 14,421'
Mt. Harvard 14,420'
Blanca Peak 14,345'
La Plata Peak 14,336'
Uncompahgre Peak 14,309'
Crestone Peak 14,294'
Mt. Lincoln 14,286'
Grays Peak 14,270'
Mt. Antero 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. [practice-area rows="4"][/practice-area] Why is it important to use the order of operations to simplify an expression? [practice-area rows="4"][/practice-area]

Evaluating, Simplifying, and Translating Algebraic Expressions

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.
  1. [latex]7x+8\text{ when }x=2[/latex]

    Answer: 22

  2. [latex]9x+7\text{ when }x=3[/latex]
  3. [latex]5x - 4\text{ when }x=6[/latex]

    Answer: 26

  4.  [latex]8x - 6\text{ when }x=7[/latex]
  5. [latex]{x}^{2}\text{ when }x=12[/latex]

    Answer: 144

  6. [latex]{x}^{3}\text{ when }x=5[/latex]
  7. [latex]{x}^{5}\text{ when }x=2[/latex]

    Answer: 32

  8. [latex]{x}^{4}\text{ when }x=3[/latex]
  9. [latex]{3}^{x}\text{ when }x=3[/latex]

    Answer: 27

  10. [latex]{4}^{x}\text{ when }x=2[/latex]
  11. [latex]{x}^{2}+3x - 7\text{ when }x=4[/latex]

    Answer: 21

  12. [latex]{x}^{2}+5x - 8\text{ when }x=6[/latex]
  13. [latex]2x+4y - 5\text{ when }x=7,y=8[/latex]

    Answer: 41

  14. [latex]6x+3y - 9\text{ when }x=6,y=9[/latex]
  15. [latex]{\left(x-y\right)}^{2}\text{ when }x=10,y=7[/latex]

    Answer: 9

  16. [latex]{\left(x+y\right)}^{2}\text{ when }x=6,y=9[/latex]

    Answer: 225

  17. [latex]{a}^{2}+{b}^{2}\text{ when }a=3,b=8[/latex]

    Answer: 73

  18. [latex]{r}^{2}-{s}^{2}\text{ when }r=12,s=5[/latex]
  19. [latex]2l+2w\text{ when }l=15,w=12[/latex]

    Answer: 54

  20. [latex]2l+2w\text{ when }l=18,w=14[/latex]

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.
  1. [latex]15{x}^{2}+6x+2[/latex]

    Answer: 15x2, 6x, 2

  2. [latex]11{x}^{2}+8x+5[/latex]
  3. [latex]10{y}^{3}+y+2[/latex]

    Answer: 10y3, y, 2

  4. [latex]9{y}^{3}+y+5[/latex]
In the following exercises, identify the coefficient of the given term.
  1. [latex]8a[/latex]

    Answer: 8

  2. [latex]13m[/latex]
  3. [latex]5{r}^{2}[/latex]

    Answer: 5

  4. [latex]6{x}^{3}[/latex]
In the following exercises, identify all sets of like terms.
  1. [latex]{x}^{3},8x,14,8y,5,8{x}^{3}[/latex]

    Answer: x3, 8x3 and 14, 5

  2. [latex]6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}[/latex]
  3. [latex]9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}[/latex]

    Answer: 16ab and 4ab; 16b2 and 9b2

  4. [latex]3,25{r}^{2},10s,10r,4{r}^{2},3s[/latex]

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.
  1. [latex]10x+3x[/latex]

    Answer: 13x

  2. [latex]15x+4x[/latex]
  3. [latex]17a+9a[/latex]

    Answer: 26a

  4. [latex]18z+9z[/latex]
  5. [latex]4c+2c+c[/latex]

    Answer: 7c

  6. [latex]6y+4y+y[/latex]
  7. [latex]9x+3x+8[/latex]

    Answer: 12x + 8

  8. [latex]8a+5a+9[/latex]
  9. [latex]7u+2+3u+1[/latex]

    Answer: 10u + 3

  10. [latex]8d+6+2d+5[/latex]
  11. [latex]7p+6+5p+4[/latex]

    Answer: 12p + 10

  12. [latex]8x+7+4x - 5[/latex]
  13. [latex]10a+7+5a - 2+7a - 4[/latex]

    Answer: 22a + 1

  14. [latex]7c+4+6c - 3+9c - 1[/latex]
  15. [latex]3{x}^{2}+12x+11+14{x}^{2}+8x+5[/latex]

    Answer: 17x2 + 20x + 16

  16. [latex]5{b}^{2}+9b+10+2{b}^{2}+3b - 4[/latex]

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.
  1. The sum of 8 and 12

    Answer: 8 + 12

  2. The sum of 9 and 1
  3. The difference of 14 and 9

    Answer: 14 − 9

  4. 8 less than 19
  5. The product of 9 and 7

    Answer: 9 ⋅ 7

  6. The product of 8 and 7
  7. The quotient of 36 and 9

    Answer: 36 ÷ 9

  8. The quotient of 42 and 7
  9. The difference of [latex]x[/latex] and [latex]4[/latex]

    Answer: x − 4

  10. [latex]3[/latex] less than [latex]x[/latex]
  11. The product of [latex]6[/latex] and [latex]y[/latex]

    Answer: 6y

  12. The product of [latex]9[/latex] and [latex]y[/latex]
  13. The sum of [latex]8x[/latex] and [latex]3x[/latex]

    Answer: 8x + 3x

  14. The sum of [latex]13x[/latex] and [latex]3x[/latex]
  15. The quotient of [latex]y[/latex] and [latex]3[/latex]

    Answer: [latex]\Large\frac{y}{3}[/latex]

  16. The quotient of [latex]y[/latex] and [latex]8[/latex]
  17. Eight times the difference of [latex]y[/latex] and nine

    Answer: 8 (y − 9)

  18. Seven times the difference of [latex]y[/latex] and one
  19. Five times the sum of [latex]x[/latex] and [latex]y[/latex]

    Answer: 5 (x + y)

  20. Nine times five less than twice [latex]x[/latex]

Translate English Phrases into Algebraic Expressions

In the following exercises, write an algebraic expression.
  1. Adele bought a skirt and a blouse. The skirt cost [latex]\$15[/latex] more than the blouse. Let [latex]b[/latex] represent the cost of the blouse. Write an expression for the cost of the skirt.

    Answer: b + 15

  2. Eric has rock and classical CDs in his car. The number of rock CDs is [latex]3[/latex] more than the number of classical CDs. Let [latex]c[/latex] represent the number of classical CDs. Write an expression for the number of rock CDs.
  3. The number of girls in a second-grade class is [latex]4[/latex] less than the number of boys. Let [latex]b[/latex] represent the number of boys. Write an expression for the number of girls.

    Answer: b − 4

  4. Marcella has [latex]6[/latex] fewer male cousins than female cousins. Let [latex]f[/latex] represent the number of female cousins. Write an expression for the number of boy cousins.
  5. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let [latex]n[/latex] represent the number of nickels. Write an expression for the number of pennies.

    Answer: 2n − 7

  6. Jeannette has [latex]\$5[/latex] and [latex]\$10[/latex] bills in her wallet. The number of fives is three more than six times the number of tens. Let [latex]t[/latex] 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 [latex]\$750[/latex] deductible per incident. This means that he pays [latex]\$750[/latex] and his insurance company will pay all costs beyond [latex]\$750[/latex]. If Justin files a claim for [latex]\$2,100[/latex], how much will he pay, and how much will his insurance company pay?

Answer: He will pay $750. His insurance company will pay $1350.

Home insurance

Pam and Armando’s home insurance has a [latex]\$2,500[/latex] deductible per incident. This means that they pay [latex]\$2,500[/latex] and their insurance company will pay all costs beyond [latex]\$2,500[/latex]. If Pam and Armando file a claim for [latex]\$19,400[/latex], 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 [latex]x[/latex] and [latex]y[/latex] to help you explain. [practice-area rows="4"][/practice-area] Explain the difference between "[latex]4[/latex] times the sum of [latex]x[/latex] and [latex]y[/latex]" and "the sum of [latex]4[/latex] times [latex]x[/latex] and [latex]y[/latex]." [practice-area rows="4"][/practice-area]

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.

Exercise 1

[latex-display]x+13=21[/latex-display]
  1. [latex]x=8[/latex]

    Answer: yes

  2. [latex]x=34[/latex]

    Answer: no

Exercise 2

[latex-display]y+18=25[/latex-display]
  1. [latex]y=7[/latex]
  2. [latex]y=43[/latex]

Exercise 3

[latex-display]m - 4=13[/latex-display]
  1. [latex]m=9[/latex]

    Answer: no

  2. [latex]m=17[/latex]

    Answer: yes

Exercise 4

[latex-display]n - 9=6[/latex-display]
  1. [latex]n=3[/latex]
  2. [latex]n=15[/latex]

Exercise 5

[latex-display]3p+6=15[/latex-display]
  1. [latex]p=3[/latex]

    Answer: yes

  2. [latex]p=7[/latex]

    Answer: no

Exercise 6

[latex-display]8q+4=20[/latex-display]
  1. [latex]q=2[/latex]
  2. [latex]q=3[/latex]

Exercise 7

[latex-display]18d - 9=27[/latex-display]
  1. [latex]d=1[/latex]

    Answer: no

  2. [latex]d=2[/latex]

    Answer: yes

Exercise 8

[latex-display]24f - 12=60[/latex-display]
  1. [latex]f=2[/latex]
  2. [latex]f=3[/latex]

Exercise 9

[latex-display]8u - 4=4u+40[/latex-display]
  1. [latex]u=3[/latex]

    Answer: no

  2. [latex]u=11[/latex]

    Answer: yes

Exercise 10

[latex-display]7v - 3=4v+36[/latex-display]
  1. [latex]v=3[/latex]
  2. [latex]v=11[/latex]

Exercise 11

[latex-display]20h - 5=15h+35[/latex-display]
  1. [latex]h=6[/latex]

    Answer: no

  2. [latex]h=8[/latex]

    Answer: yes

Exercise 12

[latex-display]18k - 3=12k+33[/latex-display]
  1. [latex]k=1[/latex]
  2. [latex]k=6[/latex]

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.

Exercise 1

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.

Answer: x + 2 = 5; x = 3

Exercise 2

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.

Exercise 3

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.

Answer: x + 3 = 6; x = 3

Exercise 4

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.
  1. [latex]a+2=18[/latex]

    Answer: a = 16

  2. [latex]b+5=13[/latex]
  3. [latex]p+18=23[/latex]

    Answer: p = 5

  4. [latex]q+14=31[/latex]
  5. [latex]r+76=100[/latex]

    Answer: r = 24

  6. [latex]s+62=95[/latex]
  7. [latex]16=x+9[/latex]

    Answer: x = 7

  8. [latex]17=y+6[/latex]
  9. [latex]93=p+24[/latex]

    Answer: p = 69

  10. [latex]116=q+79[/latex]
  11. [latex]465=d+398[/latex]

    Answer: d = 67

  12. [latex]932=c+641[/latex]

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.
  1. [latex]y - 3=19[/latex]

    Answer: y = 22

  2. [latex]x - 4=12[/latex]
  3. [latex]u - 6=24[/latex]

    Answer: u = 30

  4. [latex]v - 7=35[/latex]
  5. [latex]f - 55=123[/latex]

    Answer: f = 178

  6. [latex]g - 39=117[/latex]
  7. [latex]19=n - 13[/latex]

    Answer: n = 32

  8. [latex]18=m - 15[/latex]
  9. [latex]10=p - 38[/latex]

    Answer: p = 48

  10. [latex]18=q - 72[/latex]
  11. [latex]268=y - 199[/latex]

    Answer: y = 467

  12. [latex]204=z - 149[/latex]

Translate Word Phrase to Algebraic Equations

In the following exercises, translate the given sentence into an algebraic equation.
  1. The sum of [latex]8[/latex] and [latex]9[/latex] is equal to [latex]17[/latex].

    Answer: 8 + 9 = 17

  2. The sum of [latex]7[/latex] and [latex]9[/latex] is equal to [latex]16[/latex].
  3. The difference of [latex]23[/latex] and [latex]19[/latex] is equal to [latex]4[/latex].

    Answer: 23 − 19 = 4

  4. The difference of [latex]29[/latex] and [latex]12[/latex] is equal to [latex]17[/latex].
  5. The product of [latex]3[/latex] and [latex]9[/latex] is equal to [latex]27[/latex].

    Answer: 3 ⋅ 9 = 27

  6. The product of [latex]6[/latex] and [latex]8[/latex] is equal to [latex]48[/latex].
  7. The quotient of [latex]54[/latex] and [latex]6[/latex] is equal to [latex]9[/latex].

    Answer: 54 ÷ 6 = 9

  8. The quotient of [latex]42[/latex] and [latex]7[/latex] is equal to [latex]6[/latex].
  9. Twice the difference of [latex]n[/latex] and [latex]10[/latex] gives [latex]52[/latex].

    Answer: 2(n − 10) = 52

  10. Twice the difference of [latex]m[/latex] and [latex]14[/latex] gives [latex]64[/latex].
  11. The sum of three times [latex]y[/latex] and [latex]10[/latex] is [latex]100[/latex].

    Answer: 3y + 10 = 100

  12. The sum of eight times [latex]x[/latex] and [latex]4[/latex] is [latex]68[/latex].

Translate to an Equation and Solve

In the following exercises, translate the given sentence into an algebraic equation and then solve it.
  1. Five more than [latex]p[/latex] is equal to [latex]21[/latex].

    Answer: p + 5 = 21; p = 16

  2. Nine more than [latex]q[/latex] is equal to [latex]40[/latex].
  3. The sum of [latex]r[/latex] and [latex]18[/latex] is [latex]73[/latex].

    Answer: r + 18 = 73; r = 55

  4. The sum of [latex]s[/latex] and [latex]13[/latex] is [latex]68[/latex].
  5. The difference of [latex]d[/latex] and [latex]30[/latex] is equal to [latex]52[/latex].

    Answer: d − 30 = 52; d = 82

  6. The difference of [latex]c[/latex] and [latex]25[/latex] is equal to [latex]75[/latex].
  7. [latex]12[/latex] less than [latex]u[/latex] is [latex]89[/latex].

    Answer: u − 12 = 89; u = 101

  8. [latex]19[/latex] less than [latex]w[/latex] is [latex]56[/latex].
  9. [latex]325[/latex] less than [latex]c[/latex] gives [latex]799[/latex].

    Answer: c − 325 = 799; c = 1124

  10. [latex]299[/latex] less than [latex]d[/latex] gives [latex]850[/latex].

Everyday Math

Insurance

Vince’s car insurance has a [latex]\$500[/latex] deductible. Find the amount the insurance company will pay, [latex]p[/latex], for an [latex]\$1800[/latex] claim by solving the equation [latex]500+p=1800[/latex].

Answer: $1300

Insurance

Marta’s homeowner’s insurance policy has a [latex]\$750[/latex] deductible. The insurance company paid [latex]\$5800[/latex] to repair damages caused by a storm. Find the total cost of the storm damage, [latex]d[/latex], by solving the equation [latex]d - 750=5800[/latex].

Sale purchase

Arthur bought a suit that was on sale for [latex]\$120[/latex] off. He paid [latex]\$340[/latex] for the suit. Find the original price, [latex]p[/latex], of the suit by solving the equation [latex]p - 120=340[/latex].

Answer: $460

Sale purchase

Rita bought a sofa that was on sale for [latex]\$1299[/latex]. She paid a total of [latex]\$1409[/latex], including sales tax. Find the amount of the sales tax, [latex]t[/latex], by solving the equation [latex]1299+t=1409[/latex].

Writing Exercises

Is [latex]x=1[/latex] a solution to the equation [latex]8x - 2=16 - 6x?[/latex] How do you know? [practice-area rows="4"][/practice-area] Write the equation [latex]y - 5=21[/latex] in words. Then make up a word problem for this equation. [practice-area rows="4"][/practice-area]

Finding Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than [latex]50[/latex] for the given number.

  1. [latex]2[/latex]

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

  2. [latex]3[/latex]
  3. [latex]4[/latex]

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

  4. [latex]5[/latex]
  5. [latex]6[/latex]

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

  6. [latex]7[/latex]
  7. [latex]8[/latex]

    Answer: 8, 16, 24, 32, 40, 48

  8. [latex]9[/latex]
  9. [latex]10[/latex]

    Answer: 10, 20, 30, 40

  10. [latex]12[/latex]

Use Common Divisibility Tests

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

  1. [latex]84[/latex]

    Answer: Divisible by 2, 3, 4, 6

  2. [latex]96[/latex]
  3. [latex]75[/latex]

    Answer: Divisible by 3, 5

  4. [latex]78[/latex]
  5. [latex]168[/latex]

    Answer: Divisible by 2, 3, 4, 6

  6. [latex]264[/latex]
  7. [latex]900[/latex]

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

  8. [latex]800[/latex]
  9. [latex]896[/latex]

    Answer: Divisible by 2, 4

  10. [latex]942[/latex]
  11. [latex]375[/latex]

    Answer: Divisible by 3, 5

  12. [latex]750[/latex]
  13. [latex]350[/latex]

    Answer: Divisible by 2, 5, 10

  14. [latex]550[/latex]
  15. [latex]1430[/latex]

    Answer: Divisible by 2, 5, 10

  16. [latex]1080[/latex]
  17. [latex]22,335[/latex]

    Answer: Divisible by 3, 5

  18. [latex]39,075[/latex]

Find All the Factors of a Number

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

  1. [latex]36[/latex]

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

  2. [latex]42[/latex]
  3. [latex]60[/latex]

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

  4. [latex]48[/latex]
  5. [latex]144[/latex]

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

  6. [latex]200[/latex]
  7. [latex]588[/latex]

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

  8. [latex]576[/latex]

Identify Prime and Composite Numbers

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

  1. [latex]43[/latex]

    Answer: prime

  2. [latex]67[/latex]
  3. [latex]39[/latex]

    Answer: composite

  4. [latex]53[/latex]
  5. [latex]71[/latex]

    Answer: prime

  6. [latex]119[/latex]
  7. [latex]481[/latex]

    Answer: composite

  8. [latex]221[/latex]
  9. [latex]209[/latex]

    Answer: composite

  10. [latex]359[/latex]
  11. [latex]667[/latex]

    Answer: composite

  12. [latex]1771[/latex]

Everyday Math

Banking

Frank’s grandmother gave him [latex]\$100[/latex] at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added [latex]\$15[/latex] 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
[latex]0[/latex] [latex]100[/latex] [latex]100[/latex]
[latex]1[/latex] [latex]100+15[/latex] [latex]115[/latex]
[latex]2[/latex] [latex]100+15\cdot 2[/latex] [latex]130[/latex]
[latex]3[/latex] [latex]100+15\cdot 3[/latex] [practice-area rows="1"][/practice-area]
[latex]4[/latex] [latex]100+15\cdot \left[\right][/latex] [practice-area rows="1"][/practice-area]
[latex]5[/latex] [latex]100+\left[\right][/latex] [practice-area rows="1"][/practice-area]
[latex]6[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
[latex]20[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
[latex]x[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]

Answer: 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 [latex]\$75[/latex] to open the account. Every week, she added [latex]\$20[/latex] 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
[latex]0[/latex] [latex]75[/latex] [latex]75[/latex]
[latex]1[/latex] [latex]75+20[/latex] [latex]95[/latex]
[latex]2[/latex] [latex]75+20\cdot 2[/latex] [latex]115[/latex]
[latex]3[/latex] [latex]75+20\cdot 3[/latex] [practice-area rows="1"][/practice-area]
[latex]4[/latex] [latex]75+20\cdot \left[\right][/latex] [practice-area rows="1"][/practice-area]
[latex]5[/latex] [latex]75+\left[\right][/latex] [practice-area rows="1"][/practice-area]
[latex]6[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
[latex]20[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
[latex]x[/latex] [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]

Writing Exercises

If a number is divisible by [latex]2[/latex] and by [latex]3[/latex], why is it also divisible by [latex]6?[/latex] [practice-area rows="4"][/practice-area] What is the difference between prime numbers and composite numbers? [practice-area rows="4"][/practice-area]

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

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

  1. [latex]86[/latex]

    Answer: 2 ⋅ 43

  2. [latex]78[/latex]
  3. [latex]132[/latex]

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 11

  4. [latex]455[/latex]
  5. [latex]693[/latex]

    Answer: 3 ⋅ 3 ⋅ 7 ⋅ 11

  6. [latex]420[/latex]
  7. [latex]115[/latex]

    Answer: 5 ⋅ 23

  8. [latex]225[/latex]
  9. [latex]2475[/latex]

    Answer: 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11

  10. 1560

Find the Prime Factorization of a Composite Number

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

  1. [latex]56[/latex]

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 7

  2. [latex]72[/latex]
  3. [latex]168[/latex]

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7

  4. [latex]252[/latex]
  5. [latex]391[/latex]

    Answer: 17 ⋅ 23

  6. [latex]400[/latex]
  7. [latex]432[/latex]

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3

  8. [latex]627[/latex]
  9. [latex]2160[/latex]

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5

  10. [latex]2520[/latex]

Find the Prime Factorization of a Composite Number

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

  1. [latex]150[/latex]

    Answer: 2 ⋅ 3 ⋅ 5 ⋅ 5

  2. [latex]180[/latex]
  3. [latex]525[/latex]

    Answer: 3 ⋅ 5 ⋅ 5 ⋅ 7

  4. [latex]444[/latex]
  5. [latex]36[/latex]

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 3

  6. [latex]50[/latex]
  7. [latex]350[/latex]

    Answer: 2 ⋅ 5 ⋅ 5 ⋅ 7

  8. [latex]144[/latex]

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.
  1. [latex]8,12[/latex]

    Answer: 24

  2. [latex]4,3[/latex]
  3. [latex]6,15[/latex]

    Answer: 30

  4. [latex]12,16[/latex]
  5. [latex]30,40[/latex]

    Answer: 120

  6. [latex]20,30[/latex]
  7. [latex]60,75[/latex]

    Answer: 300

  8. [latex]44,55[/latex]

Find the Least Common Multiple (LCM) of Two Numbers

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

  1. [latex]8,12[/latex]

    Answer: 24

  2. [latex]12,16[/latex]
  3. [latex]24,30[/latex]

    Answer: 120

  4. [latex]28,40[/latex]
  5. [latex]70,84[/latex]

    Answer: 420

  6. [latex]84,90[/latex]

Find the Least Common Multiple (LCM) of Two Numbers

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

  1. [latex]6,21[/latex]

    Answer: 42

  2. [latex]9,15[/latex]
  3. [latex]24,30[/latex]

    Answer: 120

  4. [latex]32,40[/latex]

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!)

Answer: 40

Grocery shopping

Paper plates are sold in packages of [latex]12[/latex] and party cups come in packs of [latex]8[/latex]. 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? [practice-area rows="4"][/practice-area] Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why? [practice-area rows="4"][/practice-area]

Chapter Review Exercises

Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.
  1. [latex]3\cdot 8[/latex]

    Answer: the product of 3 and 8

  2. [latex]12-x[/latex]
  3. [latex]24\div 6[/latex]

    Answer: the quotient of 24 and 6

  4. [latex]9+2a[/latex]
  5. [latex]50\ge 47[/latex]

    Answer: 50 is greater than or equal to 47

  6. [latex]3y<15[/latex]
  7. [latex]n+4=13[/latex]

    Answer: The sum of n and 4 is equal to 13

  8. [latex]32-k=7[/latex]

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.
  1. [latex]5+u=84[/latex]

    Answer: equation

  2. [latex]36 - 6s[/latex]
  3. [latex]4y - 11[/latex]

    Answer: expression

  4. [latex]10x=120[/latex]

Simplify Expressions with Exponents

In the following exercises, write in exponential form.
  1. [latex]2\cdot 2\cdot 2[/latex]

    Answer: 23

  2. [latex]a\cdot a\cdot a\cdot a\cdot a[/latex]
  3. [latex]x\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]

    Answer: x6

  4. [latex]10\cdot 10\cdot 10[/latex]

Simplify Expressions with Exponents

In the following exercises, write in expanded form.
  1. [latex]{8}^{4}[/latex]

    Answer: 8 ⋅ 8 ⋅ 8 ⋅ 8

  2. [latex]{3}^{6}[/latex]
  3. [latex]{y}^{5}[/latex]

    Answer: y ⋅ y ⋅ y ⋅ y ⋅ y

  4. [latex]{n}^{4}[/latex]

Simplify Expressions with Exponents

In the following exercises, simplify each expression.
  1. [latex]{3}^{4}[/latex]

    Answer: 81

  2. [latex]{10}^{6}[/latex]
  3. [latex]{2}^{7}[/latex]

    Answer: 128

  4. [latex]{4}^{3}[/latex]

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.
  1. [latex]10+2\cdot 5[/latex]

    Answer: 20

  2. [latex]\left(10+2\right)\cdot 5[/latex]
  3. [latex]\left(30+6\right)\div 2[/latex]

    Answer: 18

  4. [latex]30+6\div 2[/latex]
  5. [latex]{7}^{2}+{5}^{2}[/latex]

    Answer: 74

  6. [latex]{\left(7+5\right)}^{2}[/latex]
  7. [latex]4+3\left(10 - 1\right)[/latex]

    Answer: 31

  8. [latex]\left(4+3\right)\left(10 - 1\right)[/latex]

Evaluate, Simplify, and Translate Expressions

Evaluate an Expression

In the following exercises, evaluate the following expressions.
  1. [latex]9x - 5\text{ when }x=7[/latex]

    Answer: 58

  2. [latex]{y}^{3}\text{ when }y=5[/latex]
  3. [latex]3a - 4b[/latex] when [latex]a=10,b=1[/latex]

    Answer: 26

  4. [latex]bh\text{ when }b=7,h=8[/latex]

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.
  1. [latex]12{n}^{2}+3n+1[/latex]

    Answer: 12n2,3n, 1

  2. [latex]4{x}^{3}+11x+3[/latex]

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the coefficient of each term.
  1. [latex]6y[/latex]

    Answer: 6

  2. [latex]13{x}^{2}[/latex]
In the following exercises, identify the like terms.
  1. [latex]5{x}^{2},3,5{y}^{2},3x,x,4[/latex]

    Answer: 3, 4, and 3x, x

  2. [latex]8,8{r}^{2},\text{8}r,3r,{r}^{2},3s[/latex]

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.
  1. [latex]15a+9a[/latex]

    Answer: 24a

  2. [latex]12y+3y+y[/latex]
  3. [latex]4x+7x+3x[/latex]

    Answer: 14x

  4. [latex]6+5c+3[/latex]
  5. [latex]8n+2+4n+9[/latex]

    Answer: 12n + 11

  6. [latex]19p+5+4p - 1+3p[/latex]
  7. [latex]7{y}^{2}+2y+11+3{y}^{2}-8[/latex]

    Answer: 10y2 + 2y + 3

  8. [latex]13{x}^{2}-x+6+5{x}^{2}+9x[/latex]

Translate English Phrases to Algebraic Expressions

In the following exercises, translate the following phrases into algebraic expressions.
  1. the difference of [latex]x[/latex] and [latex]6[/latex]

    Answer: x − 6

  2. the sum of [latex]10[/latex] and twice [latex]a[/latex]
  3. the product of [latex]3n[/latex] and [latex]9[/latex]

    Answer: 3n ⋅ 9

  4. the quotient of [latex]s[/latex] and [latex]4[/latex]
  5. [latex]5[/latex] times the sum of [latex]y[/latex] and [latex]1[/latex]

    Answer: 5(y + 1)

  6. [latex]10[/latex] less than the product of [latex]5[/latex] and [latex]z[/latex]

Translate English Phrases to Algebraic Expressions

In the following exercises, write the algebraic expressions that can be found in each sentence.
  1. Jack bought a sandwich and a coffee. The cost of the sandwich was [latex]\$3[/latex] more than the cost of the coffee. Call the cost of the coffee [latex]c[/latex]. Write an expression for the cost of the sandwich.

    Answer: c + 3

  2. The number of poetry books on Brianna’s bookshelf is [latex]5[/latex] less than twice the number of novels. Call the number of novels [latex]n[/latex]. Write an expression for the number of poetry books.

Subtraction Property of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.
Exercise 1
[latex-display]y+16=40[/latex-display]
  1. [latex]24[/latex]

    Answer: yes

  2. [latex]56[/latex]

    Answer: no

Exercise 2
[latex-display]d - 6=21[/latex-display]
  1.  [latex]15[/latex]
  2. [latex]27[/latex]
Exercise 3
[latex-display]4n+12=36[/latex-display]
  1. [latex]6[/latex]

    Answer: yes

  2. [latex]12[/latex]

    Answer: No

Exercise 4
[latex-display]20q - 10=70[/latex-display]
  1. [latex]3[/latex]
  2. [latex]4[/latex]
Exercise 5
[latex-display]15x - 5=10x+45[/latex-display]
  1.  [latex]2[/latex]

    Answer: no

  2. [latex]10[/latex]

    Answer: yes

Exercise 6
[latex-display]22p - 6=18p+86[/latex-display]
  1. [latex]4[/latex]
  2. [latex]23[/latex]

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality. This image is divided into two parts: the first part shows an envelope and 3 blue counters and the next to it, the second part shows five counters.

Answer: x + 3 = 5; x = 2

This image is divided into two parts: the first part shows an envelope and 4 blue counters and next to it, the second part shows 9 counters.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.
  1. [latex]c+8=14[/latex]

    Answer: 6

  2. [latex]v+8=150[/latex]
  3. [latex]23=x+12[/latex]

    Answer: 11

  4. [latex]376=n+265[/latex]

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.
  1. [latex]y - 7=16[/latex]

    Answer: 23

  2. [latex]k - 42=113[/latex]
  3. [latex]19=p - 15[/latex]

    Answer: 34

  4. [latex]501=u - 399[/latex]

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.
  1. The sum of [latex]7[/latex] and [latex]33[/latex] is equal to [latex]40[/latex].

    Answer: 7 + 33 = 44

  2. The difference of [latex]15[/latex] and [latex]3[/latex] is equal to [latex]12[/latex].
  3. The product of [latex]4[/latex] and [latex]8[/latex] is equal to [latex]32[/latex].

    Answer: 4 ⋅ 8 = 32

  4. The quotient of [latex]63[/latex] and [latex]9[/latex] is equal to [latex]7[/latex].
  5. Twice the difference of [latex]n[/latex] and [latex]3[/latex] gives [latex]76[/latex].

    Answer: 2(n − 3) = 76

  6. The sum of five times [latex]y[/latex] and [latex]4[/latex] is [latex]89[/latex].

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.
  1. Eight more than [latex]x[/latex] is equal to [latex]35[/latex].

    Answer: x + 8 = 35; x = 27

  2. [latex]21[/latex] less than [latex]a[/latex] is [latex]11[/latex].
  3. The difference of [latex]q[/latex] and [latex]18[/latex] is [latex]57[/latex].

    Answer: q − 18 = 57; q = 75

  4. The sum of [latex]m[/latex] and [latex]125[/latex] is [latex]240[/latex].

Mixed Practice

In the following exercises, solve each equation.
  1. [latex]h - 15=27[/latex]

    Answer: h = 42

  2. [latex]k - 11=34[/latex]
  3. [latex]z+52=85[/latex]

    Answer: z = 33

  4. [latex]x+93=114[/latex]
  5. [latex]27=q+19[/latex]

    Answer: q = 8

  6. [latex]38=p+19[/latex]
  7. [latex]31=v - 25[/latex]

    Answer: v = 56

  8. [latex]38=u - 16[/latex]

Finding Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than [latex]50[/latex] for each of the following.
  1. [latex]3[/latex]

    Answer: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

  2. [latex]2[/latex]
  3. [latex]8[/latex]

    Answer: 8, 16, 24, 32, 40, 48

  4. [latex]10[/latex]

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
  1. [latex]96[/latex]

    Answer: 2, 3, 6

  2. [latex]250[/latex]
  3. [latex]420[/latex]

    Answer: 2, 3, 5, 6, 10

  4. [latex]625[/latex]

Find All the Factors of a Number

In the following exercises, find all the factors of each number.
  1. [latex]30[/latex]

    Answer: 1, 2, 3, 5, 6, 10, 15, 30

  2. [latex]70[/latex]
  3. [latex]180[/latex]

    Answer: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

  4. [latex]378[/latex]

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.
  1. [latex]19[/latex]

    Answer: prime

  2. [latex]51[/latex]
  3. [latex]121[/latex]

    Answer: composite

  4. [latex]219[/latex]

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.
  1. [latex]84[/latex]

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 7

  2. [latex]165[/latex]
  3. [latex]350[/latex]

    Answer: 2 ⋅ 5 ⋅ 5 ⋅ 7

  4. [latex]572[/latex]

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.
  1. [latex]9,15[/latex]

    Answer: 45

  2. [latex]12,20[/latex]
  3. [latex]25,35[/latex]

    Answer: 350

  4. [latex]18,40[/latex]

Everyday Math

Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month. [practice-area rows="4"][/practice-area]

Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.
  1. [latex]6\cdot 4[/latex]
  2. [latex]15-x[/latex]

    Answer: fifteen minus x

In the following exercises, identify each as an expression or equation.
  1. [latex]5\cdot 8+10[/latex]
  2. [latex]x+6=9[/latex]

    Answer: equation

  3. [latex]3\cdot 11=33[/latex]
  4. Write [latex]n\cdot n\cdot n\cdot n\cdot n\cdot n[/latex] in exponential form.

    Answer: n6

  5. Write [latex]{3}^{5}[/latex] in expanded form and then simplify.

    Answer: 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243

In the following exercises, simplify, using the order of operations.
  1. [latex]4+3\cdot 5[/latex]
  2. [latex]\left(8+1\right)\cdot 4[/latex]

    Answer: 36

  3. [latex]1+6\left(3 - 1\right)[/latex]
  4. [latex]\left(8+4\right)\div 3+1[/latex]

    Answer: 5

  5. [latex]{\left(1+4\right)}^{2}[/latex]
  6. [latex]5\left[2+7\left(9 - 8\right)\right][/latex]

    Answer: 45

In the following exercises, evaluate each expression.
  1. [latex]8x - 3\text{ when }x=4[/latex]
  2. [latex]{y}^{3}\text{ when }y=5[/latex]

    Answer: 125

  3. [latex]6a - 2b\text{ when }a=5,b=7[/latex]
  4. [latex]hw\text{ when }h=12,w=3[/latex]

    Answer: 36

Simplify by combining like terms.
  1.  [latex]6x+8x[/latex]
  2. [latex]9m+10+m+3[/latex]
In the following exercises, translate each phrase into an algebraic expression.
  1. [latex]5[/latex] more than [latex]x[/latex]

    Answer: x + 5

  2. the quotient of [latex]12[/latex] and [latex]y[/latex]
  3. three times the difference of [latex]a\text{ and }b[/latex]

    Answer: 3(a − b)

  4. Caroline has [latex]3[/latex] fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, [latex]r[/latex]. Write an expression for the number of earrings on her left ear.
In the following exercises, solve each equation.
  1. [latex]n - 6=25[/latex]

    Answer: n = 31

  2. [latex]x+58=71[/latex]
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
  1. [latex]15[/latex] less than [latex]y[/latex] is [latex]32[/latex].

    Answer: y − 15 = 32; y = 47

  2. the sum of [latex]a[/latex] and [latex]129[/latex] is [latex]164[/latex].
  3. List all the multiples of [latex]4[/latex], that are less than [latex]50[/latex].

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

  4. Find all the factors of [latex]90[/latex].
  5. Find the prime factorization of [latex]1080[/latex].

    Answer: 23 ⋅ 33 ⋅ 5

  6. Find the LCM (Least Common Multiple) of [latex]24[/latex] and [latex]40[/latex].

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