Classes of Real Numbers

Rational Numbers

In the following exercises, write as the ratio of two integers.

1. ⓐ $5$
2. ⓑ $3.19$

1. ⓐ $\Large\frac{5}{1}$
2. ⓑ $\Large\frac{319}{100}$

1. ⓐ $8$
2. ⓑ $-1.61$

1. ⓐ $-12$
2. ⓑ $9.279$

1. ⓐ $\Large\frac{-12}{1}$
2. ⓑ $\Large\frac{9279}{1000}$

1. ⓐ $-16$
2. ⓑ $4.399$

In the following exercises, determine which of the given numbers are rational and which are irrational.

$0.75$ , $0.22\stackrel{\text{-}}{\text{3}}$ , $\text{1.39174... }$

Rational: $0.75,0.22\stackrel{\text{-}}{\text{3}}$ . Irrational: $\text{1.39174... }$

$0.36$ , $\text{0.94729\ldots }$ , $2.52\stackrel{\text{-}}{\text{8}}$

$0.\stackrel{\text{-}}{45}$ , $\text{1.919293... }$ , $3.59$

Rational: $0.\stackrel{\text{-}}{45}$ , $3.59$ . Irrational: $\text{1.919293... }$

$0.1\stackrel{\text{-}}{\text{3}},\text{0.42982... }$ , $1.875$

In the following exercises, identify whether each number is rational or irrational.

1. ⓐ $\sqrt{25}$
2. ⓑ $\sqrt{30}$

1. ⓐ rational
2. ⓑ irrational

1. ⓐ $\sqrt{44}$
2. ⓑ $\sqrt{49}$

1. ⓐ $\sqrt{164}$
2. ⓑ $\sqrt{169}$

1. ⓐ irrational
2. ⓑ rational

1. ⓐ $\sqrt{225}$
2. ⓑ $\sqrt{216}$

Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

$-8$ , $0,\text{1.95286....}$ , $\Large\frac{12}{5}$ , $\sqrt{36}$ , $9$

$-9$ , $-3\Large\frac{4}{9}$ , $-\sqrt{9}$ , $0.4\stackrel{\text{-}}{09}$ , $\Large\frac{11}{6}$ , $7$

$-\sqrt{100}$ , $-7$ , $-\Large\frac{8}{3}$ , $-1$ , $0.77$ , $3\Large\frac{1}{4}$

 

 

Using the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

$\text{8+9=___}$

$\text{7+6=___}$

7 + 6 = 6 + 7

$\text{8}\left(-12\right)\text{=___}$

$\text{7}\left(-13\right)\text{=___}$

7(−13) = (−13)7

$\text{(-19)(-14)=___}$

$\left(-12\right)\left(-18\right)\text{=___}$

(−12)(−18) = (−18)(−12)

$\text{-11+8=___}$

$\text{-15+7=___}$

−15 + 7 = 7 + (−15)

$\text{x+4=___}$

$\text{y+1=___}$

y + 1 = 1 + y

$\text{-2a=___}$

$\text{-3m=___}$

−3m = m(−3)

In the following exercises, use the associative properties to rewrite the given expression.

$\left(11+9\right)\text{+14=___}$

$\left(21+14\right)\text{+9=___}$

(21 + 14) + 9 = 21 + (14 + 9)

$\left(12\cdot 5\right)\cdot \text{7=___}$

$\left(14\cdot 6\right)\cdot \text{9=___}$

(14 · 6) · 9 = 14(6 · 9)

$\left(-7+9\right)\text{+8=___}$

$\left(-2+6\right)\text{+7=___}$

(−2 + 6) + 7 = −2 + (6 + 7)

$\left(16\cdot\Large\frac{4}{5}\normalsize\right)\cdot \text{15=___}$

$\left(13\cdot\Large\frac{2}{3}\normalsize\right)\cdot \text{18=___}$

$\left(13\cdot\Large\frac{2}{3}\normalsize\right)\cdot 18=13\left(\Large\frac{2}{3}\normalsize\cdot 18\right)$

$3\left(4x\right)\text{=___}$

$4\left(7x\right)\text{=___}$

4(7x) = (4 · 7)x

$\left(12+x\right)\text{+28=___}$

$\left(17+y\right)\text{+33=___}$

(17 + y) + 33 = 17 + (y + 33)

Evaluate Expressions using the Commutative and Associative Properties

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

If $y=\Large\frac{5}{8}$, evaluate:

ⓐ $y+0.49+\left(-y\right)$
ⓑ $y+\left(-y\right)+0.49$

If $z=\Large\frac{7}{8}$, evaluate:

ⓐ $z+0.97+\left(-z\right)$
ⓑ $z+\left(-z\right)+0.97$

1. ⓐ 0.97
2. ⓑ 0.97

If $c=-\Large\frac{11}{4}$, evaluate:

ⓐ $c+3.125+\left(-c\right)$
ⓑ $c+\left(-c\right)+3.125$

If $d=-\Large\frac{9}{4}$, evaluate:

ⓐ $d+2.375+\left(-d\right)$
ⓑ $d+\left(-d\right)+2.375$

ⓐ 2.375
ⓑ 2.375

If $j=11$, evaluate:

ⓐ $\Large\frac{5}{6}\left(\frac{6}{5}\normalsize j\Large\right)$
ⓑ $\Large\left(\frac{5}{6}\normalsize\cdot\Large\frac{6}{5}\right)\normalsize j$

If $k=21$, evaluate:

ⓐ $\Large\frac{4}{13}\left(\frac{13}{4}\normalsize k\Large\right)$
ⓑ $\Large\left(\frac{4}{13}\normalsize\cdot\Large\frac{13}{4}\right)\normalsize k$

ⓐ 21
ⓑ 21

If $m=-25$, evaluate:

ⓐ $-\Large\frac{3}{7}\left(\frac{7}{3}\normalsize m\Large\right)$
ⓑ $\Large\left(-\frac{3}{7}\normalsize\cdot\Large\frac{7}{3}\right)\normalsize m$

If $n=-8$, evaluate:

ⓐ $-\Large\frac{5}{21}\left(\frac{21}{5}\normalsize n\Large\right)$
ⓑ $\Large\left(-\frac{5}{21}\normalsize\cdot\Large\frac{21}{5}\right)\normalsize n$

ⓐ −8
ⓑ −8

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

$-45a+15+45a$

$9y+23+\left(-9y\right)$

23

$\Large\frac{1}{2}\normalsize +\Large\frac{7}{8}\normalsize +\Large\left(-\frac{1}{2}\right)$

$\Large\frac{2}{5}\normalsize +\Large\frac{5}{12}\normalsize+\Large\left(-\frac{2}{5}\right)$

$\Large\frac{5}{12}$

$\Large\frac{3}{20}\normalsize\cdot\Large\frac{49}{11}\normalsize\cdot\Large\frac{20}{3}$

$\Large\frac{13}{18}\normalsize\cdot\Large\frac{25}{7}\normalsize\cdot\Large\frac{18}{13}$

$\Large\frac{25}{7}$

$\Large\frac{7}{12}\normalsize\cdot\Large\frac{9}{17}\normalsize\cdot\Large\frac{24}{7}$

$\Large\frac{3}{10}\normalsize\cdot\Large\frac{13}{23}\normalsize\cdot\Large\frac{50}{3}$

$\Large\frac{65}{23}$

$-24\cdot 7\cdot\Large\frac{3}{8}$

$-36\cdot 11\cdot\Large\frac{4}{9}$

−176

$\Large\left(\frac{5}{6}\normalsize +\Large\frac{8}{15}\right)\normalsize +\Large\frac{7}{15}$

$\Large\left(\frac{1}{12}\normalsize +\Large\frac{4}{9}\right)\normalsize +\Large\frac{5}{9}$

$\Large\frac{13}{12}$

$\Large\frac{5}{13}\normalsize +\Large\frac{3}{4}\normalsize +\Large\frac{1}{4}$

$\Large\frac{8}{15}\normalsize +\Large\frac{5}{7}\normalsize +\Large\frac{2}{7}$

$\Large\frac{23}{15}$

$\left(4.33p+1.09p\right)+3.91p$

$\left(5.89d+2.75d\right)+1.25d$

9.89d

$17\left(0.25\right)\left(4\right)$

$36\left(0.2\right)\left(5\right)$

36

$\left[2.48\left(12\right)\right]\left(0.5\right)$

$\left[9.731\left(4\right)\right]\left(0.75\right)$

29.193

$7\left(4a\right)$

$9\left(8w\right)$

72w

$-15\left(5m\right)$

$-23\left(2n\right)$

−46n

$12\left(\Large\frac{5}{6}\normalsize p\right)$

$20\left(\Large\frac{3}{5}\normalsize q\right)$

12q

$14x+19y+25x+3y$

$15u+11v+27u+19v$

42u + 30v

$43m+\left(-12n\right)+\text{ }\left(-16m\right)+\left(-9n\right)$

$-22p+17q+\left(-35p\right)+\left(-27q\right)$

−57p + (−10q)

$\Large\frac{3}{8}\normalsize g+\Large\frac{1}{12}\normalsize h+\Large\frac{7}{8}\normalsize g+\Large\frac{5}{12}\normalsize h$

$\Large\frac{5}{6}\normalsize a+\Large\frac{3}{10}\normalsize b+\Large\frac{1}{6}\normalsize a+\Large\frac{9}{10}\normalsize b$

$a+\Large\frac{6}{5}\normalsize b$

$6.8p+9.14q+\left(-4.37p\right)+\left(-0.88q\right)$

$9.6m+7.22n+\left(-2.19m\right)+\left(-0.65n\right)$

7.41m + 6.57n

 

 

Using the Distributive Property

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

$4\left(x+8\right)$

$3\left(a+9\right)$

3a + 27

$8\left(4y+9\right)$

$9\left(3w+7\right)$

27w + 63

$6\left(c - 13\right)$

$7\left(y - 13\right)$

7y − 91

$7\left(3p - 8\right)$

$5\left(7u - 4\right)$

35u − 20

$\Large\frac{1}{2}\normalsize\left(n+8\right)$

$\Large\frac{1}{3}\normalsize\left(u+9\right)$

$\Large\frac{1}{3}\normalsize u+3$

$\Large\frac{1}{4}\normalsize\left(3q+12\right)$

$\Large\frac{1}{5}\normalsize\left(4m+20\right)$

$\Large\frac{4}{5}\normalsize m+4$

$9\left(\Large\frac{5}{9}\normalsize y-\Large\frac{1}{3}\normalsize\right)$

$10\left(\Large\frac{3}{10}\normalsize x-\Large\frac{2}{5}\normalsize\right)$

3x − 4

$12\left(\Large\frac{1}{4}\normalsize+\Large\frac{2}{3}\normalsize r\right)$

$12\left(\Large\frac{1}{6}\normalsize +\Large\frac{3}{4}\normalsize s\right)$

2 + 9s

$r\left(s - 18\right)$

$u\left(v - 10\right)$

uv − 10u

$\left(y+4\right)p$

$\left(a+7\right)x$

ax + 7x

$-2\left(y+13\right)$

$-3\left(a+11\right)$

−3a − 33

$-7\left(4p+1\right)$

$-9\left(9a+4\right)$

−81a − 36

$-3\left(x - 6\right)$

$-4\left(q - 7\right)$

−4q + 28

$-9\left(3a - 7\right)$

$-6\left(7x - 8\right)$

−42x + 48

$-\left(r+7\right)$

$-\left(q+11\right)$

−q − 11

$-\left(3x - 7\right)$

$-\left(5p - 4\right)$

−5p + 4

$5+9\left(n - 6\right)$

$12+8\left(u - 1\right)$

8u + 4

$16 - 3\left(y+8\right)$

$18 - 4\left(x+2\right)$

−4x + 10

$4 - 11\left(3c - 2\right)$

$9 - 6\left(7n - 5\right)$

−42n + 39

$22-\left(a+3\right)$

$8-\left(r - 7\right)$

−r + 15

$-12-\left(u+10\right)$

$-4-\left(c - 10\right)$

−c + 6

$\left(5m - 3\right)-\left(m+7\right)$

$\left(4y - 1\right)-\left(y - 2\right)$

3y + 1

$5\left(2n+9\right)+12\left(n - 3\right)$

$9\left(5u+8\right)+2\left(u - 6\right)$

47u + 60

$9\left(8x - 3\right)-\left(-2\right)$

$4\left(6x - 1\right)-\left(-8\right)$

24x + 4

$14\left(c - 1\right)-8\left(c - 6\right)$

$11\left(n - 7\right)-5\left(n - 1\right)$

6n − 72

$6\left(7y+8\right)-\left(30y - 15\right)$

$7\left(3n+9\right)-\left(4n - 13\right)$

17n + 76

Evaluate Expressions Using the Distributive Property

In the following exercises, evaluate both expressions for the given value.

If $v=-2$, evaluate

ⓐ $6\left(4v+7\right)$
ⓑ $6\cdot 4v+6\cdot 7$

If $u=-1$, evaluate

ⓐ $8\left(5u+12\right)$
ⓑ $8\cdot 5u+8\cdot 12$

ⓐ 56
ⓑ 56

If $n=\Large\frac{2}{3}$, evaluate

ⓐ $3\left(n+\Large\frac{5}{6}\normalsize\right)$
ⓑ $3\cdot n+3\cdot\Large\frac{5}{6}$

If $y=\Large\frac{3}{4}$, evaluate

ⓐ $4\left(y+\Large\frac{3}{8}\normalsize\right)$
ⓑ $4\cdot y+4\cdot\Large\frac{3}{8}$

ⓐ $\Large\frac{9}{2}$
ⓑ $\Large\frac{9}{2}$

If $y=\Large\frac{7}{12}$, evaluate

ⓐ $-3\left(4y+15\right)$
ⓑ $3\cdot 4y+\left(-3\right)\cdot 15$

If $p=\Large\frac{23}{30}$, evaluate

ⓐ $-6\left(5p+11\right)$
ⓑ $-6\cdot 5p+\left(-6\right)\cdot 11$

ⓐ −89
ⓑ −89

If $m=0.4$, evaluate

ⓐ $-10\left(3m - 0.9\right)$
ⓑ $-10\cdot 3m-\left(-10\right)\left(0.9\right)$

If $n=0.75$, evaluate

ⓐ $-100\left(5n+1.5\right)$
ⓑ $-100\cdot 5n+\left(-100\right)\left(1.5\right)$

ⓐ −525
ⓑ −525

If $y=-25$, evaluate

ⓐ $-\left(y - 25\right)$
ⓑ $-y+25$

If $w=-80$, evaluate

ⓐ $-\left(w - 80\right)$
ⓑ $-w+80$

ⓐ 160
ⓑ 160

If $p=0.19$, evaluate

ⓐ $-\left(p+0.72\right)$
ⓑ $-p - 0.72$

If $q=0.55$, evaluate

ⓐ $-\left(q+0.48\right)$
ⓑ $-q - 0.48$

ⓐ −1.03
ⓑ −1.03

 

 

Properties of Identity, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

$101+0=101$

$\Large\frac{3}{5}\normalsize\left(1\right)=\Large\frac{3}{5}$

identity property of multiplication

$-9\cdot 1=-9$

$0+64=64$

identity property of addition

Use the Inverse Properties of Addition and Multiplication
In the following exercises, find the multiplicative inverse.

$8$

$14$

$\Large\frac{1}{14}$

$-17$

$-19$

$-\Large\frac{1}{19}$

$\Large\frac{7}{12}$

$\Large\frac{8}{13}$

$\Large\frac{13}{8}$

$-\Large\frac{3}{10}$

$-\Large\frac{5}{12}$

$-\Large\frac{12}{5}$

$0.8$

$0.4$

$\Large\frac{5}{2}$

$-0.2$

$-0.5$

−2

Use the Properties of Zero

In the following exercises, simplify using the properties of zero.

$48\cdot 0$

$\Large\frac{0}{6}$

0

$\Large\frac{3}{0}$

$22\cdot 0$

0

$0\div\Large\frac{11}{12}$

$\Large\frac{6}{0}$

undefined

$\Large\frac{0}{3}$

$0\div\Large\frac{7}{15}$

0

$0\cdot\Large\frac{8}{15}$

$\left(-3.14\right)\left(0\right)$

0

$5.72\div 0$

$\Large\frac{\frac{1}{10}}{0}$

undefined

Simplify Expressions using the Properties of Identities, Inverses, and Zero

In the following exercises, simplify using the properties of identities, inverses, and zero.

$19a+44 - 19a$

$27c+16 - 27c$

16

$38+11r - 38$

$92+31s - 92$

31s

$10\left(0.1d\right)$

$100\left(0.01p\right)$

p

$5\left(0.6q\right)$

$40\left(0.05n\right)$

2n

$\Large\frac{0}{r+20}$ , where $r\ne -20$

$\Large\frac{0}{s+13}$ , where $s\ne -13$

0

$\Large\frac{0}{u - 4.99}$ , where $u\ne 4.99$

$\Large\frac{0}{v - 65.1}$ , where $v\ne 65.1$

0

$0\div \left(x-\Large\frac{1}{2}\normalsize\right)$ , where $x\ne\Large\frac{1}{2}$

$0\div \left(y-\Large\frac{1}{6}\normalsize\right)$ , where $y\ne\Large\frac{1}{6}$

0

$\Large\frac{32 - 5a}{0}$ , where $32 - 5a\ne 0$

$\Large\frac{28 - 9b}{0}$ , where $28 - 9b\ne 0$

undefined

$\Large\frac{2.1+0.4c}{0}$ , where $2.1+0.4c\ne 0$

$\Large\frac{1.75+9f}{0}$ , where $1.75+9f\ne 0$

undefined

$\left(\Large\frac{3}{4}\normalsize +\Large\frac{9}{10}\normalsize m\right)\div 0$ , where $\Large\frac{3}{4}\normalsize +\Large\frac{9}{10}\normalsize m\ne 0$

$\left(\Large\frac{5}{16}\normalsize n-\Large\frac{3}{7}\normalsize\right)\div 0$ , where $\Large\frac{5}{16}\normalsize n-\Large\frac{3}{7}\normalsize \ne 0$

undefined

$\Large\frac{9}{10}\normalsize\cdot\Large\frac{10}{9}\normalsize\left(18p - 21\right)$

$\Large\frac{5}{7}\normalsize\cdot\Large\frac{7}{5}\normalsize\left(20q - 35\right)$

20q − 35

$15\cdot\Large\frac{3}{5}\normalsize\left(4d+10\right)$

$18\cdot\Large\frac{5}{6}\normalsize\left(15h+24\right)$

225h + 360

 

 

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

$4x - 2=6$

ⓐ $x=-2$
ⓑ $x=-1$
ⓒ $x=2$

1. ⓐ no
2. ⓑ yes
3. ⓒ no

$4y - 10=-14$

ⓐ $y=-6$
ⓑ $y=-1$
ⓒ $y=1$

$9a+27=-63$

ⓐ $a=6$
ⓑ $a=-6$
ⓒ $a=-10$

1. ⓐ no
2. ⓑ no
3. ⓒ yes

$7c+42=-56$

ⓐ $c=2$
ⓑ $c=-2$
ⓒ $c=-14$

Solve Equations Using the Addition and Subtraction Properties of Equality

In the following exercises, solve for the unknown.

$n+12=5$

−7

$m+16=2$

$p+9=-8$

−17

$q+5=-6$

$u - 3=-7$

−4

$v - 7=-8$

$h - 10=-4$

6

$k - 9=-5$

$x+\left(-2\right)=-18$

−16

$y+\left(-3\right)=-10$

$r-\left(-5\right)=-9$

−14

$s-\left(-2\right)=-11$

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve it.

3x = 6; x = 2

2x = 8; x = 4

Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

$5x=45$

9

$4p=64$

$-7c=56$

−8

$-9x=54$

$-14p=-42$

3

$-8m=-40$

$-120=10q$

−12

$-75=15y$

$24x=480$

20

$18n=540$

$-3z=0$

0

$4u=0$

Translate to an Equation and Solve

In the following exercises, translate and solve.

Four more than $n$ is equal to 1.

n + 4 = 1; n = −3

Nine more than $m$ is equal to 5.

The sum of eight and $p$ is $-3$ .

8 + p = −3; p = −11

The sum of two and $q$ is $-7$ .

The difference of $a$ and three is $-14$ .

a − 3 = −14; a = −11

The difference of $b$ and $5$ is $-2$ .

The number −42 is the product of −7 and $x$ .

−42 = −7x; x = 6

The number −54 is the product of −9 and $y$ .

The product of $f$ and −15 is 75.

f(−15) = 75; f = 5

The product of $g$ and −18 is 36.

−6 plus $c$ is equal to 4.

−6 + c = 4; c = 10

−2 plus $d$ is equal to 1.

Nine less than $n$ is −4.

m − 9 = −4; m = 5

Thirteen less than $n$ is $-10$ .

Mixed Practice

In the following exercises, solve.

1. ⓐ $x+2=10$
2. ⓑ $2x=10$

1. ⓐ 8
2. ⓑ 5

1. ⓐ $y+6=12$
2. ⓑ $6y=12$

1. ⓐ $-3p=27$
2. ⓑ $p - 3=27$

1. ⓐ −9
2. ⓑ 30

1. ⓐ $-2q=34$
2. ⓑ $q - 2=34$

$a - 4=16$

20

$b - 1=11$

$-8m=-56$

7

$-6n=-48$

$-39=u+13$

−52

$-100=v+25$

$11r=-99$

−9

$15s=-300$

$100=20d$

5

$250=25n$

$-49=x - 7$

−42

$64=y - 4$

 

 

Chapter Review Exercises