10.8: Exercise Supplement
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Variables, Constants, and Real Numbers
For problems 1-5, next to each real number, note all subsets of the real numbers to which it belongs by writing N for natural numbers, W for whole numbers, or Z for integers. Some numbers may belong to more than one subset.
Exercise \(\PageIndex{1}\)
61
- Answer
-
N, W, Z
Exercise \(\PageIndex{2}\)
-14
Exercise \(\PageIndex{3}\)
0
- Answer
-
W, Z
Exercise \(\PageIndex{4}\)
1
Exercise \(\PageIndex{5}\)
Write all the integers that are strictly between −4 and 3
- Answer
-
{-3, -2, -1, 0, 1, 2}
Exercise \(\PageIndex{6}\)
Write all the integers that are between and including -6 and -1
For each pair of numbers in problems 7-10, write the appropriate symbol (<, >, =) in place of the □.
Exercise \(\PageIndex{7}\)
-5 □ -1
- Answer
-
<
Exercise \(\PageIndex{8}\)
0 □ 2
Exercise \(\PageIndex{9}\)
-7 □ 0
- Answer
-
<
Exercise \(\PageIndex{10}\)
-1 □ 0
For problems 11-15, what numbers can replace x so that each statement is true?
Exercise \(\PageIndex{11}\)
\(-5 \le x \le -1\), \(x\) is an integer
- Answer
-
{-5, -4, -3, -2, -1}
Exercise \(\PageIndex{12}\)
\(-10 < x \le 0\), \(x\) is a whole number.
Exercise \(\PageIndex{13}\)
\(0 \le x < 5\), \(x\) is a natural number.
- Answer
-
{1, 2, 3, 4}
Exercise \(\PageIndex{14}\)
\(-3 < x < 3\), \(x\) is a natural number
Exercise \(\PageIndex{15}\)
\(-8 < x \le -2\), \(x\) is a whole number
- Answer
-
none
For problems 16-20, how many units are there between the given pair of numbers?
Exercise \(\PageIndex{16}\)
0 and 4
Exercise \(\PageIndex{17}\)
-1 and 3
- Answer
-
4
Exercise \(\PageIndex{18}\)
-7 and -4
Exercise \(\PageIndex{19}\)
-6 and 0
- Answer
-
6
Exercise \(\PageIndex{20}\)
-1 and 1
Exercise \(\PageIndex{21}\)
A number is positive if it is directly preceded by a sign or no sign at all.
- Answer
-
+ (plus)
Exercise \(\PageIndex{22}\)
A number is negative if it is directly preceded by a sign.
Signed Numbers
For problems 23-26, how should each number be read?
Exercise \(\PageIndex{23}\)
-8
- Answer
-
negative eight
Exercise \(\PageIndex{24}\)
-(-4)
Exercise \(\PageIndex{25}\)
-(-1)
- Answer
-
negative negative one or opposite negative one
Exercise \(\PageIndex{26}\)
-2
For problems 27-31, write each expression in words.
Exercise \(\PageIndex{27}\)
1 + (-7)
- Answer
-
one plus negative seven
Exercise \(\PageIndex{28}\)
-2 - (-6)
Exercise \(\PageIndex{29}\)
-1 - (+4)
- Answer
-
negative one minus four
Exercise \(\PageIndex{30}\)
-(-(-3))
Exercise \(\PageIndex{31}\)
0 - (-11)
- Answer
-
zero minus negative eleven
For problems 32-36, rewrite each expression in simpler form.
Exercise \(\PageIndex{32}\)
-(-4)
Exercise \(\PageIndex{33}\)
-(-15)
- Answer
-
15
Exercise \(\PageIndex{34}\)
-[-(-7)]
Exercise \(\PageIndex{35}\)
1 - (-18)
- Answer
-
19 or 1 + 18
Exercise \(\PageIndex{36}\)
0 - (-1)
Absolute Value
For problems 37-52, determine each value.
Exercise \(\PageIndex{37}\)
|9|
- Answer
-
9
Exercise \(\PageIndex{38}\)
|16|
Exercise \(\PageIndex{39}\)
|-5|
- Answer
-
5
Exercise \(\PageIndex{40}\)
|-8|
Exercise \(\PageIndex{41}\)
-|-2|
- Answer
-
-2
Exercise \(\PageIndex{42}\)
-|-1|
Exercise \(\PageIndex{43}\)
-(-|12|)
- Answer
-
12
Exercise \(\PageIndex{44}\)
-(-|90|)
Exercise \(\PageIndex{45}\)
-(-|-16|)
- Answer
-
16
Exercise \(\PageIndex{46}\)
-(-|0|)
Exercise \(\PageIndex{47}\)
\(|-4|^2\)
- Answer
-
16
Exercise \(\PageIndex{48}\)
\(|-5|^2\)
Exercise \(\PageIndex{49}\)
\(|-2|^3\)
- Answer
-
8
Exercise \(\PageIndex{50}\)
\(|-(3 \cdot 4)|\)
Exercise \(\PageIndex{51}\)
|-5| + |-2|
- Answer
-
7
Exercise \(\PageIndex{52}\)
|-7| - |-10|
Addition, Subtraction, Multiplication and Division of Signed Numbers ([link],[link],[link])
For problems 53-71, perform each operation.
Exercise \(\PageIndex{53}\)
-6 + 4
- Answer
-
-2
Exercise \(\PageIndex{54}\)
-10 + 8
Exercise \(\PageIndex{55}\)
-1 - 6
- Answer
-
-7
Exercise \(\PageIndex{56}\)
8 - 12
Exercise \(\PageIndex{57}\)
0 - 14
- Answer
-
-14
Exercise \(\PageIndex{58}\)
\(5 \cdot (-2)\)
Exercise \(\PageIndex{59}\)
\(-8 \cdot (-6)\)
- Answer
-
48
Exercise \(\PageIndex{60}\)
\((-3) \cdot (-9)\)
Exercise \(\PageIndex{61}\)
\(14 \cdot (-3)\)
- Answer
-
-42
Exercise \(\PageIndex{62}\)
\(5 \cdot (-70)\)
Exercise \(\PageIndex{63}\)
\(-18 \div -6\)
- Answer
-
3
Exercise \(\PageIndex{64}\)
\(72 \div - 12\)
Exercise \(\PageIndex{65}\)
\(-16 \div -16\)
- Answer
-
1
Exercise \(\PageIndex{66}\)
\(0 \div -8\)
Exercise \(\PageIndex{67}\)
\(-5 \div 0\)
- Answer
-
not defined
Exercise \(\PageIndex{68}\)
\(\dfrac{-15}{-3}\)
Exercise \(\PageIndex{69}\)
\(\dfrac{-28}{7}\)
- Answer
-
-4
Exercise \(\PageIndex{70}\)
\(\dfrac{-120}{-|2|}\)
Exercise \(\PageIndex{71}\)
\(\dfrac{|-66|}{-|-3|}\)
- Answer
-
-22