Rules for
for Numbers
Numbers
Rules
Thereal
realnumbers
numbersare
aregoverned
governedfor
collection
rulesthat
thathave
havetotododowith
with
The
bybya acollection
ofofrules
Rules
Numbers
Rules
addition,multiplication,
multiplication,and
andinequalities.
inequalities. InInthe
therules
rulesbelow,
below,x,x,y,y,z zCCJR.JR. (In
(In
addition,
The
real
are
by
other
words,
and
arereal
realnumbers.)
numbers.)
The
real numbers
numbers
are
governed
by aa collection
collection of
of rules
rules that
that have
have to
to do
do with
with
other
words,
.x,.x,y,y,and
z zgoverned
are
addition,
multiplication,
and
inequalities.
In
the
rules
below,
x,
y,
z
⇥
R.
addition, multiplication, and inequalities. In the rules below, x, y, z 2 R. (In
(In
Rules
of
addition.
Rules
of
addition.
other
other words,
words, x,
x, y,
y, and
and zz are
are real
real numbers.)
numbers.)
(Lawofofassociativity)
associativity)
• •(x(x++y)y)++z z==xx++(y(y++z)z) (Law
Rules
of
addition.
Rules
of
addition.
(Lawofofcommutativity)
commutativity)
• •xx++yy==y y++xx(Law
•• (x
+
y)
+
zz =
xx +
(y
+
z)
(x
+
y)
+
=
+
(y
+
z) (Law
(Law of
of associativity)
associativity)
(Lawofofidentity)
identity)
• •.x.x++0 0==xx(Law
• x—x
y ===yy0+
x
of
commutativity)
x+
+
x (Law
(Law
of
commutativity)
0+(Law
(Law
inverses)
• ••—x
++yii=
ofofinverses)
•• xx +
+ 00 =
= xx (Law
(Law of
of identity)
identity)
Rules•of
Rules
xmultiplication.
•ofmultiplication.
x+
+ xx =
= 00 (Law
(Law of
of inverses)
inverses)
(xy)z==x(yz)
x(yz) (Law
(Lawofofassociativity)
associativity)
• •(xy)z
Rules
of
multiplication.
Rules
of
multiplication.
xy==yx
yx(Law
(Lawofofcommutativity)
commutativity)
• •xy
•• (xy)z
=
x(yz)
(Law
of
(xy)z
=
x(yz)
(Law
of associativity)
associativity)
(Lawofofidentity)
identity)
• •xlxl==xx(Law
•• xy
(Law
of
xy =
= yx
yx then
(Law~x
of commutativity)
commutativity)
(Lawofofinverses)
inverses)
• •IfIfxx~~0 0then
~x ==1 1 (Law
•• x1
x1 =
= xx (Law
(Law of
of identity)
identity)
11
Distributive
Law.
There
arule
ruleof
that
combinesaddition
additionand
andmultiplica
multiplica
Distributive
There
that
combines
•• IfIf xx ⇤=
00 then
inverses)
6=Law.
then
x=
=is11isa(Law
(Law
of
inverses)
xxx
tion: the
thedistributive
distributivelaw.
law. Of
Ofallallthe
therules
ruleslisted
listedsosofar,
far,it’s
it’sarguably
arguablythe
themost
most
tion:
Distributive
Distributive Law.
Law. There
There isis aa rule
rule that
that combines
combines addition
addition and
and multiplicamultiplicaimportant.
important.
tion:
tion: the
the distributive
distributive law.
law. Of
Of all
all the
the rules
rules listed
listed so
so far,
far, it’s
it’s arguably
arguably the
the most
most
important.
important.
x(y++z)z)==xy
xy++xz
xz(Distributive
(DistributiveLaw)
Law)
• •x(y
•• x(y
x(y +
+ z)
z) =
= xy
xy +
+ xz
xz (Distributive
(Distributive Law)
Law)
44
46
Here are some other forms of the distributive law that you will have to be
comfortable with:
•
•
•
•
•
(y + z)x = yx + zx
x(y z) = xy xz
(x + y)(z + w) = xz + xw + yz + yw
x(y + z + w) = xy + xz + xw
x(y1 + y2 + y3 + · · · + yn ) = xy1 + xy2 + xy3 + · · · + xyn
Examples. Sometimes you’ll have to use the distributive law in the “forwards” direction, as in the following three examples:
• 3(y + z) = 3y + 3z
• ( 2)(4y 5z) = ( 2)4y ( 2)5z = 8y + 10z
• 2(3x 2y + 4z) = 6x 4y + 8z
Sometimes you’ll have to use the distributive law in “reverse”. This process
is sometimes called factoring out a term. The three equations below are
examples of factoring out a 4, factoring out a 3, and factoring out a 2.
• 4y 4z = 4(y + z)
• 3x + 6y = 3(x + 2y)
• 10x 8y + 4z = 2(5x 4y + 2z)
*
*
*
*
*
*
*
*
*
*
*
*
*
In addition to the algebra rules above, the real numbers are governed by
laws of inequalities.
Rules of inequalities.
• If x > 0 and y > 0 then x + y > 0
• If x > 0 and y > 0 then xy > 0
• If x 2 R, then either x > 0, or x < 0, or x = 0
*
*
*
*
*
*
*
7
*
*
*
*
*
*
Intervals
The following chart lists 8 important types of subsets of R. Any set of one
of these types is called an interval.
Name of
interval
Those x 2 R
contained
in the interval
drawing of interval
[a,[a,
b] {a,b]
a bxx 
x bb }}b } (
[a,
b] =
= {=
{ xx{x
2aR
R|RI
| xaa 


b]
2

Q
b
a xx 
x bb }}b } (
[a,{a,b]
b] =
= {=
{ xx{x
2R
R |RI
| aa 


[a,
b]
2
Q
b
a xx 
x bb }}b } (
[a,{a,b]
b] =
= {=
{ xx{x
2R
R |RI
| aa 


[a,
b]
2
Q
b
a xx 
x bb }}b } (
[a,{a,b]
b] =
= {=
{ xx{x
2R
R |RI
| aa 


[a,
b]
2
Oj.rn1u,imuwsvemO
a xxx<x
{x
(a,(a,b)
b)=
={=
2R
R|RI
|aaaa
x
(a,
b)
=
{{xxxx{x
2
R
<
<
Q
b
{a,b]
a<
x<bbb<b }}}}bb}} .((
[a,
b]
=
{=
2
R
||RI


b]
2
(a,[a,
b)
a
<
x
<
b
Oj.rn1u,imuwsvemO
(a,b)
a
<x
=
{x
(a,
b)
=
{
x
2
R
|
a
<
x
<
b
}
0..
<
b}
(a,
b)
=
{
x
2
R
|
a
<
x
<
b
}
.(
Q
RI
b
{a,b]
{x
x bbb}}}b}
[a,b)
b] =
= {{=
2R
R |RI
| aa 


}b } (
[a,
b]
xx{x
2
xxx <

Oj.rn1u,imuwsvemO
aaa x
<x
(a,
b)
=
{{{x
2
R
<
<
(a,
2
0..Q
{a,b]
bb
x b<
[a,(a,b)
b] =
==
x{x
2R
R |RI

x

[a,
b]
=
{=
xx
2
R
||| aaaa <

x
bb }}b } .((
Oj.rn1u,imuwsvemO
(a,b)
a
<x
=
{x
(a,
b)
=
{
x
2
R
|
a
<
x
<
b
}
<
b}
Q
(a,
b)
=
{
x
2
R
|
a
<
x
<
b
}
.(
0..
RI
bb
{a,b]
ax
x bbb}}}}b } ((
[a,[a,b)={xERIa<x<b}
b] =
= {=
{x
x{x
2R
R ||RI
| aa 

x<

[a,
b]
2

411.IJIL.*1L_1*_O
[a,
b)
=
xx
2
R

xx
<
[a,
b)
2
Oj.rn1u,imuwsvemO
(a,b)
a xx<x
{x
(a,
b) =
= {=
2R
R ||RI
| aaaa 
<
< bb<
b }}b} .(
(a,
b)
=
{{{xx
2
R
<
<
0..Q
bb
0..411.IJIL.*1L_1*_O
[a,
b)
=
{
x
2
R
|
a

x
<
b
}
(
[a,
b)
=
{
x
2
R
|
a

x
<
b
}
[a,b)={xERIa<x<b}
b
Oj.rn1u,imuwsvemO
0..
(a,b)
ab x<x
(a,b)
b)=
==
R |x
| aa<
<
x<
< b<
b }}b} .(
(a,
b)
=
{{ xx{x
2
R
<
<
bb
RI
[a, [a,
b)
a2R

411.IJIL.*1L_1*_O
0..
[a,
b)
22
2
Oj.rn1u,imuwsvemO
(a,b)
axxx<x
(a,[a,b)={xERIa<x<b}
b)=
={{=
2R
R|||RI
|aaaa
<
x<
<bbb<
b}}}}b} (.(
(a,
b)
=
{{xxxx{x
R
<
<
0..
0..
411.IJIL.*1L_1*_O
[a,
b)
=
{=
xxx{x
22
R
||RI

xx
<
[a,
b)
2
x
<
[a,b)={xERIa<x<b}
bbb
Oj.rn1u,imuwsvemO
(a,b)
ax
<x
(a,(a,b]
b)=
={=
2R
R||RI
|aaaaaa
<
x
<bbbbb<
b}}}}}}b}
b} (.(
(a,
b)
=
{{x
R
<
<
0..
{x
a
<x
(a,
b]
=
{
x
2
R
|
<
x

(a,
b]
=
{
x
2
R
<
0..411.IJIL.*1L_1*_O
[a,[a,b)={xERIa<x<b}
b) =
= {{ xx 2
2R
R || aa 
 xx <
< bb }}
(
[a,
b)
bbb
a.
0..
(a,b]
=
{x
a
<x
(a,
b]
=
{
x
2
R
|
a
<
x

b
}
b}
(a,
b]
=
{
x
2
R
|
a
<
x

b
}
RI
0.. 411.IJIL.*1L_1*_O
[a,
b)
=
{
x
2
R
|
a

x
<
b
}
(
[a,
b)
=
{
x
2
R
|
a

x
<
b
}
[a,b)={xERIa<x<b}
b
a.0..411.IJIL.*1L_1*_O
{x
ax<x
(a,
b]
=
2
R
|||aaaa<
<
xxx

(a,
b]
{{{{xxxx
2
R
bb
[a,(a,b]
b)=
==
2
R|RI
<bbbb}}}}b} (
[a,
b)
=
2
R

<
[a,b)={xERIa<x<b}
(a,(a,
b]
a
<
x

b
a.
(a,b]
=
{x
a
<x
(a,
b]
=
{
x
2
R
|
a
<
x

b
}
b}
b]
=
{
x
2
R
|
a
<
x

b
}
RI
b
0..411.IJIL.*1L_1*_O
[a,[a,oo)={x
b) =
=={{{{xxxx2
22R
RERI

< bb }}
(
[a,
b)
|| aa 
xx <
[a,b)={xERIa<x<b}
ax<x
[a,
1)
R|RI
[a,
1)
a.0..
(a,b]
=
a
(a,
b] =
==
22
RR
| a|a| aa<
<
xxx
}}x}
(a,
b]
{{ xx{x
2
R
bb }}b} M$W4*W
bbb
M$W4*W
[a,oo)={x
a
[a,
1)
=
{
x
2
R
|
a

x
}
x}
[a,
1)
=
{
x
2
R
|
a

x
}
ERI
a.
(a,b]
b
=
a
<x
(a,1)
b] =
=
22R
RR
| aa <
<
x

(a,
b]
{{ xx{x
2
|RI
bb }}b}
axx<x
[a,
1)
==
x2
[a,
=
}}x}
ERI
(a,b]
a.
{x
a
(a,[a,oo)={x
b] =
=
22
RR
|||aaaa<
<
xxx

(a,
b]
{{{{xxx
R
|RI
bb }}b} M$W4*W
b
[a,oo)={x
ax<x
[a,
1)
==
x2
2R
R|RI

[a,
1)
=
{{xxx
2
R
||aaaa|<
xx
}}bx}
a.
ERI
(a,b]
{x
ax
(a,(-oo,b]={xERIxb}
b] =
=
{=
2
R
|R
<
x

(a,
b]
{=
bb }}b} M$W4*W
b
(1)
1,
b]
{
x
2
x

}
[a,[a,
a

x
([a,
1,
b]
{
x
2
R
|

b
}
M$W4*W
[a,oo)={x
a xx }}x}
1) =
= {{ xx 2
2R
R || aa 

1)
a.
ERI
1,
b]
=
{
x
2
R
|
x

b
}
(([a,
1,
b]
=
{
x
2
R
|
x

b
}
(-oo,b]={xERIxb}
bb
M$W4*W
[a,oo)={x
a
[a,
1)
=
{
x
2
R
|
a

x
}
x}
1)
=
{
x
2
R
|
a

x
}
ERI
1,
b]=
={{{{xxxx2
2R
R|| |a|axx
xxbb}}x}
(([a,
1,
}}
M$W4*W
[a,oo)={x
a
[a,(-oo,b]={xERIxb}
1)b]
==
22
RR

b
1)
ERI
1,
b]
=
{
x
2
R
|
x

b
}
(((a,
1,
b]
=
{
x
2
R
|
x

b
}
(-oo,b]={xERIxb}
M$W4*W
[a,oo)={x
b
a
[a,
1)
=
{
x
2
R
|
a

x
}
x}
[a,
1)
=
{
x
2
R
|
a

x
}
ERI
1)b]
==
22
RR
<
1)
(_
(a,oo)={XERIa<x}
( (-oo,b]={xERIxb}
1,
b]=
={{{x
{xxx2
2R
R|| |a|axx<
xxb}b}}}
((a,
1,
b
0
( (a,
1,
b]
x

b
(a,
1)
=
{
x
2
R
|
a
<
x
}
1)
=
{
x
2
R
|
a
<
x
}
(_
(a,oo)={XERIa<x}
(
1,
b]
=
{
x
2
R
|
x

b
}
(
1,
b]
=
{
x
2
R
|
x

b
}
(-oo,b]={xERIxb}
b
(a,
1)
==
22
RR
(a,
1)
=
{{ xx
2
R
|| aa|| <
xx }}bb }}
0
(_
1, b]
b]
=
x
2
R
x

(( (a,oo)={XERIa<x}
1,
{{ x
x<
(-oo,b]={xERIxb}
b
(a,
1)
=
{
x
2
R
|
a
<
x
}
0
(a,
1)
=
{
x
2
R
|
a
<
x
}
(_
(a,oo)={XERIa<x}
1,b)
b] =
= {{{{xx
x2
2R
R |||| xx
x<
bbbb}}}}
(( (—oo,b)={XERIX<b}
1,
b]
=
2
R

(-oo,b]={xERIxb}
((a,
1,
b)
=
x
2
R
x
<
b
((a,
1,
1) =
= {{ xx 2
2R
R || aa <
< xx }}
0
1)
(_
(a,oo)={XERIa<x}
b
1,
b)=
={{{{xxxx2
2R
R|| |a|axx<
<xxbb}}}}
(((a,
1,
b)
=
2
R
<
(—oo,b)={XERIX<b}
0
(a,
1)
=
2
R
<
1)
(_
(a,oo)={XERIa<x}
1,
b)
=
{
x
2
R
|
x
<
b
}
(a,(((a,
1)
a
<
x
1,
b)
=
{
x
2
R
|
x
<
b
}
(—oo,b)={XERIX<b}
(a,(a,oo)={XERIa<x}
1) =
= {{ xx 2
2R
R || aa <
< xx }}
1)
(_
0
1,
b)
=
{
x
2
R
|
x
<
b
}
(((a,
1,
b)
=
{
x
2
R
|
x
<
b
}
*
*
*
*
*
*
*
* 0
(a,(—oo,b)={XERIX<b}
1)
=
{
x
2
R
|
a
<
x
}
1)
=
{
x
2
R
|
a
<
x
}
(_
(a,oo)={XERIa<x}
*
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(( (—oo,b)={XERIX<b}
1,
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|
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*b)
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* R
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** = {**x 2 **R | x**< b**} ** ** ** * ** * ** * ** * ** *
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(( *(—oo,b)={XERIX<b}
1,
=* {**x *2 **R *| x**<* b**} * ** * ** * ** * ** * ** * ** * ** *
*1,
** =
1,
b)
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R*|| xx <
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(( (—oo,b)={XERIX<b}
* b)
* {
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(
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x
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*
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Set
Empty
The
Empty
Set
*that
* the
**{4,
**9}
** is
**set
*that
**contains
** ** the
*the
* numbers
* 4,* 4,
* and
* 9.
The
set
{4,
7,
9}
is
the
set
contains
numbers
7,
and
9.
The
setset
The
set
{4,
the
set
that
contains
7,
9}
The
set
7,
is
the
numbers
7,
and
The
set
The
*
*
*
*
*
*
*
* * * 4,
* 7,
* 9.
*
The
The
Empty
Set
Set
Empty
The
Empty
Set
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
is
The
set
{4,
7,
9}
is
the
set
that
contains
the
numbers
4,
7,
and
9.
The
The
set
8 * {4}
the
that
7,
9}
The
set
{4,*{4,
7,**only
9}
is
set
that
contains
the
numbers
4,**4,
7,
and
9.
the
numbers
4,
7,
and
The
9.The
*
* set
*the
*contains
*
* *
*only
* 4,
* the
*
{4,
{4,
7}7}
contains
only
4* and
7,
set
{4}
contains
only
4,
and
the
set
{set
only
4and
and
contains
{4,
7}
contains
4the
7,
set
{4}
contains
only
and
set
}}set
the
set
contains
7,the
and
the
set{set
The
The
Empty
Set
Set
Empty
The
Empty
Set
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
{}
is
The
set
{4,
7,
9}
is
the
set
that
contains
the
numbers
4,
7,
and
9.
The
set
The
set
{4,
the
set
that
contains
7,
9}
The
set
{4,
7,
9}
is
the
set
that
contains
the
numbers
4,
7,
and
9.
The
set
the
numbers
4,
7,
and
The
9.
set
{4,
{4,
7}
contains
only
4
and
7,
the
set
{4}
contains
only
4,
and
the
set
{
}
only
7}
4
and
contains
The
The
Empty
Set
Set
{4,
7}
contains
only
4
and
7,
the
set
{4}
contains
only
4,
and
the
set
{
}
the
set
Empty
{4}
contains
7,
only
The
Empty
Set
4,
and
the
set
contains,
well,
nothing.
contains,
nothing.
well,
contains,
well,
{}
is
The
set
{4,
7,nothing.
9}
isSet
set
that
contains
the
numbers
4,4,
7,4,and
and
9.the
The
The
set
{4,
the
set
that
contains
7,
9}
The
{4,
7,
9}
is
set
that
contains
the
numbers
4,
7,
and
9.
set
the
numbers
4,
7,
and
The
9.The
{4,
{4,
7}
contains
only
4the
7,
the
set
{4}
contains
only
4,
and
the
set
{ }}set
only
7}set
4and
and
contains
{4,
7}
contains
only
4the
and
7,
set
contains
only
the
set
{set
the
set{4}
{4}
contains
7,the
only
and
set
The
The
Empty
Set
Empty
The
Empty
Set
contains,
well,
nothing.
{
}
contains,
nothing.
well,
contains,
well,
nothing.
It
probably
seems
silly
to
you
now,
but
the
set
that
has
nothing
inthe
it,
that
It
seems
silly
to
now,
It
probably
seems
silly
to
you
now,
but
the
set
that
has
nothing
in
it,
that
is
you
The
set
{4,{4,
7,
9}
isSet
set
that
contains
the
numbers
4,
7,4,
and
9.
The
The
set
but
the
set
that
set
has
that
contains
nothing
7,
9}
in
it,
The
set
{4,
7,
9}
is
set
that
contains
the
numbers
4,
7,
and
9.
set
the
numbers
4,
7,
and
The
9.The
The
The
Empty
Set
{4,
Empty
{4,
7}
contains
only
4the
and
7,
the
set
{4}
contains
only
4,
and
the
set
{that
only
7}probably
4the
and
contains
The
Empty
Set
{4,
7}
contains
only
4the
and
7,
set
{4}
contains
only
4,
and
the
set
{set
}}set
the
set
{4}
contains
7,the
only
and
set
contains,
well,
nothing.
contains,
nothing.
well,
contains,
well,
nothing.
{
}
It
probably
seems
silly
to
you
now,
but
set
that
has
nothing
in
it,
that
It
probably
seems
silly
to
now,
It
probably
seems
to
you
now,
but
the
set
that
has
nothing
in
it,
that
you
but
the
set
that
in
that
is
The
set
{4,
7,
9}
issilly
the
set
that
contains
the
numbers
4,4,
7,4,nothing
and
9.
The
set
The
set
{4,
the
set
that
contains
7,
9}
The
set
7,
9}
is
the
set
that
contains
the
numbers
7,
and
9.
The
set
the
numbers
7,
and
The
9.
set
is
the
set
{{4,
},
is
an
important
example
of
acontains
set,
and
it’s
a4,
set
that
comes
is
the
set
is
an
important
is
the
set
{well,
},
is
an
important
example
of
athe
set,
and
it’s
ahas
set
that
comes
example
{4,
of
The
{4,
7}
contains
only
7,
the
set
{4}
contains
only
4,
and
the
setit,
{up
only
set,
The
Empty
Set
7}
4and
and
it’s
Set
contains
a contains
Empty
and
a4,
set
that
{4,
7}
contains
only
44is
and
7,
the
set
{4}
only
and
the
set
{up
}}{up
the
set
comes
The
Empty
Set
{4}
7,
only
and
the
set
{
},
contains,
nothing.
contains,
nothing.
well,
contains,
well,
nothing.
}
The
set
{4,
7,
9}
is
the
set
that
contains
the
numbers
4,
7,
and
9.
The
set
The
set
{4,
the
set
that
contains
7,
9}
The
set
{4,
7,
9}
is
the
set
that
contains
the
numbers
4,
7,
and
9.
The
set
the
numbers
It
probably
seems
silly
to
you
now,
but
the
set
that
has
nothing
in
it,
that
It
probably
seems
4,
7,
and
silly
to
The
now,
It
probably
seems
silly
to
you
now,
but
the
set
that
has
nothing
in
it,
that
9.
you
set
but
the
set
that
has
nothing
in
it,
that
is
the
set
{
},
is
an
important
example
of
a
set,
and
it’s
a
set
that
comes
up
is
the
set
is
an
is
the
set
{in
},
isnothing.
an
important
of
aenough
set,
and
it’s
a set
that
comes
up
example
of
it’s
a set,
and
agiven
set
that
comes
{4,
{4,
7}7}
contains
only
4 Itcomes
7,example
the
set
{4}
contains
only
4,
and
the
setset
only
4and
and7,
contains
{4,
7}
contains
only
4important
and
set
{4}
contains
only
4,
the
set
{{ }}up
the
set
{4}
contains
7,the
only
4,and
and
frequently
in
It
comes
up
often
enough
that
we’ve
given
it
name.
math.
frequently
comes
frequently
It
up
often
enough
that
we’ve
it
aathe
name.
{ inmath.
},math.
contains,
well,
up
often
contains,
nothing.
that
we’ve
well,
given
it
tt
ttt
ttt
Notice that for every interval listed in the chart on the previous page, the
least of the two numbers written in the interval is always written on the left,
just as they appear in the real number line. For example, 3 < 7, so 3 is
drawn on the left of 7 in the real number line, and the following intervals
are legitimate intervals to write: [3, 7], (3, 7), [3, 7), and (3, 7]. You must not
write an interval such as (7, 3), because the least of the two numbers used in
an interval must be written on the left.
Similarly, (1, 2) or [5, 1) are not proper ways of writing intervals.
Notice that in all of the examples of intervals, round parentheses next
to numbers correspond to less-than symbols < and are drawn with little
circles. Squared parentheses next to numbers correspond to less-than-orequal-to symbols  and are drawn with giant dots. A little circle drawn on
a number means that number is not part of the interval. A giant dot drawn
on a number means that number is part of the interval.
9
Exercises
Decide whether the following statements are true or false.
1.) 5x + 5y = 5(x + y)
2.) 3x + y = 3(x + y)
3.) 4x
6y = 2(2x
3y)
4.) x + 7y = 7(x + y)
5.) 36x
9y + 81z = 3(12x
3y + 9z)
6.) 2 2 (2, 5]
7.) 0 2 ( 4, 0]
8.)
3 2 [ 3, 1)
9.) 156, 345, 678 2 ( 1, 1)
10.) 2 2 ( 1, 3]
11.) [7, 10) ✓ [7, 10]
12.) [ 17, 1) ✓ ( 17, 1)
13.) ( 4, 0] ✓ [ 4, 0)
14.) ( 1, 20] ✓ ( 1, 7]
15.) [0, 1) ✓ R
{⇡}
16.) { 3, 10, 7 } ✓ (2, 8)
17.) { 0, 2, 45 ,
p
2 } ✓ [0, 1)
10
For #18-21, match the numbered interval with its lettered picture.
19.)
0
(3, 5]
20.)
[3, 5)
21.) (3, 5)
:
[3, 5]
a
18.)
0
0
-
(nJ
B.)
a
A.)
D.)
1
Ill
4:
(A)
I
41
1
:
0
a
C.)
0
UI
25.) ( 1, 4)
41
(nJ
B.)
( 1, 4]
4:
U’
24.)
w
(4, 1)
U’
4:
4:
-
w
w
23.)
7%).
A.)
[4, 1)
7%).
22.)
0
a
For #22-25, match the numbered interval with its lettered picture.
D.)
11
UI
4:
w
Ill
4:
(A)
I
C.)