, Numbers ,
Sets and Inequalities
Rational Numbers
1. NUMBERS -
r : Fn where m.net , n±o
Note : the decimal representation of a rational number is
Real Numbers IT I
:
always repeating eg
: =
0,50
R V3 : 2-3 =
0,6666 . . .
=
0,5
④
Integers
V5 ±
4
Zo
-3
"
v5
/
Irrational numbers IN Too
z
{
-
-
Cannot be expressed 3
- ,
Natural numbers 1,213.415.6 . . .
}
as a ratio of
integers 2 I
¥ rn
↳ %
no
2. SETS a collection of objects ,
called elements .
Ex S={ a. b. c. d } .
Here a ES ,
but e¢S .
0 denotes the set i. e. it has elements
-
empty ,
no
set operations
a) Union b) Intersection
S T S T S T
Metta took }
SUT :{ xlx c- Sor x ET } SAT -
-
{ xlxes and ✗ ET }
define
t
3. INTERVALS
Open intervals ( a. b) { xlacxcb } Look at table of intervals
}
=
a b
• •
page A- 4
Closed intervals [ a. b ] =
{ x1aExEb }
a b
Rules : ① If uab and bcc ,
than aac
② If ahb ,
than ate Cbtc
③ If acb and c >o than accbc
{④
,
*
If asb and cco ,
than ac > be
⑤ If aab and cod ,
than atccbtd
* ⑥ If osa< b. than ta > ¥
EX 1 tx ( 7×1-5 2 . XZ -
Tx + 12<-0 231-4×2 -11×-30<0
⇐> -4 [ 63C < Ix -
4) ( x 3) -
£0 ⇐> Gc 3) (x2 +7×+10 )
-
co
⇐7
-2g ( x <=> ( x -
3) ( ✗ + 5) (2+2)<0
3Mt 4
Replace
,↳
values before t
/
g p after into factors -
g z
/ 5
-
z
+ +
31×14 (x 3) o t
pct d
- -
o o
x -3
- -
-
- -
,
o + o +
( -213 )
-
+
xE[3,4] ( xtz )
-
o
-
or ✗ C-
2-
-
-
"
-
↳ + o + o +
( ✗ + g) o
-
3
✗ C- ( -2g ; g) + -
t
- t - t
1- cs ,
-
5) u 1-2135
,Absolute Values
Absolute Value : The absolute value of a number a. denoted / at ,
is the distance from a too on the real
number line .
Distance are
always positive or 0 ,
so we have
191>-0 for number of Ex : 131--3 1-31--3 13-171--17-3
every a , ,
Tn 191=9 if
general a ≥o
lot = -
a if a Lo
"
Recall ✓
"
Ote :
means the positive sguar root of
Ta true if a -30
is
only
'
= a
[ take a =-3 than F-3) 2=3 = a
In the case where aco ,
Tai = -
a
↳ than -
a >0
THUS : Ma = lol for all values of a
Properties of absolute values
,§
① label = lallbl
② 1%1 ¥
}
=
③ tant = 1am , NEZ
* ④ 1×1 = a iff x ±a
* ⑤ 1×1 < a iff -
a < ✗ La / IN ≤ a iff -
a ≤✗≤ a
⑥ ,,, µ, ,,
* , ,
, , , , , , ,
, µ, ≥, a≥,
LD If and if
only
C- × 1×1=3 x = 1--3
1×1<3 -3 LxL3
1×1>3 ⇐ >
xc -3 or x >3
C- ✗ 5
Express 13×-21 without the absolute value
system
13×-21
{ 3×-2 if
=
3×-220
2- 32C if 3 >c- 240
{ 3×-2 if x ≥ 73
= 2- 3.x if x ( 43
C- ✗ 6 Solve 12×-51 =3 1×1=9 if x = ± a
27C -51 = 3
⇐> 2x 5 3 or 22C -5 =-3
=
-
⇐ > 7C = Li or 7C = I
x C- { 1,4 }
C-✗ 7 Solve 1×-51<2
1×1 <a
⇐>
-
2 ( x -
5 C- 2 0 -
a < TCL a
⇐ > Loc < 7
3
x E 13,7 ) 3 7
,46 22C -
I
✗ + I =3 121=9 iffsc = -1-9
23C -
I 27C
-
I
2+1 =3 or act 1
=
-3
l =
-
32 -3
⇐> 23C -1 32+3 or 22C -
=
⇐ > -4 = x or 2C : -45 and ✗ =/ -
I
56 051×-51<42 -
Both must be true
AND
-
asel 1×-51 > o case z be -51<112
'
iff x -5 >o or x -
5<0 ⇐ >
-
I 4×-54 12 1×1 >a iff
"
iff 2C 75 of scc 5 912 Csc C 12 x G- a or I >a
o o
±
Thus x c- ( 92,5 ) u 15 ,
"
12 )
14
'
12 5 5112
Ex Solve 13×+41 -
12×-11=1
start with
{ [ ;D
"
13×1-41 =
3×+4 ,
✗ c- 13
-3×-4 ,xE C- as ; -
%)
①
[ 12,00 )
{
12×-11
'
=
2×-1 , x c- -2×+1 ③
0
' 3×+4
-
zxtl ,
ICE 1- as 12 ) i
-3 >c -
c, •• 2 >c- I
0 ② •
413
'
12
-
"
①
Suppose c- 1- as 13 )
-
✗ , .
1×+41-12×-11=1 ⇐> (-3×-4)-(-2×+1)=1
>
⇐ -
2C -
5 =L
< x =
-
6
Note that -
G. c- too -413) ,
② Suppose ✗ c- [ -413 ,
'
12] ③
Suppose ✗ c- [ "z.es )
13×+41-12×-11=1 ⇐> (3)etc, ) -
(-2×+1) =\ 13×+41
-
12×-11--1 ⇐>
(321-4)-(2×-1)=1
⇐> 5×+3=1 ⇐ >
✗ +5 =
I
⇐ > >c = -215 ⇐ >
x =
-
a
[ ¥ 'z ) [
'
Note that -215 c- Note that -4¢ lz ,
as )
,
THUS 13×+41-12×-11=1 <= >
✗ c-
{ -6 ,
-215 }
, C- ✗ Solve the
inequality 2<-13×-5146
First solve the
inequality separately :
13×-51<6 13.x -51 22
⇐> -
6 53.x -516 <= > 3×-5>-2 or 32C -5£ -
2
⇐ > -1 432C 911 ⇐ > 32-77 or 32 £3
⇐ > -
'
13 Soc <
"
13 ⇐ > x
I 73 or 2C
It
( " "
x c- 13) C- I -0,1 ] [ 73 )
-
>
⇐ 3 ,
⇐ > x u ,
as
2<-13×-51 < 6
1- 13 [1-0,1]
173,2b¥ Can you
'
do this ?
"
⇐ > x c-
,
B) n u
⇐ > x c- ( -
'
13, I ] u [ >
13 ,
' '
13)
• • 0
'
I ' '
13 713 13
-
Proof
if xzo
{
"
loci =
-
x if oc co txt : Fi
for all x (R)
→
C- ✗ :
prove that
the lock = xz
Proof real number either
:
Suppose >c is a , xzo or xco
case 1 : ✗ 20
we have lock = x2 ( since bcl=x )
cases :
x co
1×12 =
1 xp-
= x2 ( since bcl = -
x )
therefore 1×12=70 for a " "
☐
[ ☐ end of proof
The
Triangle Inequality
If a ,b c- 112 ,
then latbl E Iat t Ibl
Always
*
smaller or
equal to
PROOF lal Etat & Ibl Elbl
:
Suppose a. b c- 1112 :
Case If b have the same then the either
, a and
sign sum on side is equal .
Case 2 If a andb have opposite signs ,
then
-
late a stat ①
and - Ibl Eb E Ibl ②
Adding ① ② that
and we
get
-
( lait lbl ) I atb E Iat t lbl -
c < x Cc
Put atb and
/
✗ "
y
=
=
then c = Iat t Ibl
lxtscifff.ro
that ← ✗ ,
teased
so - c ⇐
m
⑤ bet <C
lxl=ciHy#
and ④ loci =L iff x=±c
we have lxl Ec x = ± c
l
THUS latbl E Iat t Ibl
Sets and Inequalities
Rational Numbers
1. NUMBERS -
r : Fn where m.net , n±o
Note : the decimal representation of a rational number is
Real Numbers IT I
:
always repeating eg
: =
0,50
R V3 : 2-3 =
0,6666 . . .
=
0,5
④
Integers
V5 ±
4
Zo
-3
"
v5
/
Irrational numbers IN Too
z
{
-
-
Cannot be expressed 3
- ,
Natural numbers 1,213.415.6 . . .
}
as a ratio of
integers 2 I
¥ rn
↳ %
no
2. SETS a collection of objects ,
called elements .
Ex S={ a. b. c. d } .
Here a ES ,
but e¢S .
0 denotes the set i. e. it has elements
-
empty ,
no
set operations
a) Union b) Intersection
S T S T S T
Metta took }
SUT :{ xlx c- Sor x ET } SAT -
-
{ xlxes and ✗ ET }
define
t
3. INTERVALS
Open intervals ( a. b) { xlacxcb } Look at table of intervals
}
=
a b
• •
page A- 4
Closed intervals [ a. b ] =
{ x1aExEb }
a b
Rules : ① If uab and bcc ,
than aac
② If ahb ,
than ate Cbtc
③ If acb and c >o than accbc
{④
,
*
If asb and cco ,
than ac > be
⑤ If aab and cod ,
than atccbtd
* ⑥ If osa< b. than ta > ¥
EX 1 tx ( 7×1-5 2 . XZ -
Tx + 12<-0 231-4×2 -11×-30<0
⇐> -4 [ 63C < Ix -
4) ( x 3) -
£0 ⇐> Gc 3) (x2 +7×+10 )
-
co
⇐7
-2g ( x <=> ( x -
3) ( ✗ + 5) (2+2)<0
3Mt 4
Replace
,↳
values before t
/
g p after into factors -
g z
/ 5
-
z
+ +
31×14 (x 3) o t
pct d
- -
o o
x -3
- -
-
- -
,
o + o +
( -213 )
-
+
xE[3,4] ( xtz )
-
o
-
or ✗ C-
2-
-
-
"
-
↳ + o + o +
( ✗ + g) o
-
3
✗ C- ( -2g ; g) + -
t
- t - t
1- cs ,
-
5) u 1-2135
,Absolute Values
Absolute Value : The absolute value of a number a. denoted / at ,
is the distance from a too on the real
number line .
Distance are
always positive or 0 ,
so we have
191>-0 for number of Ex : 131--3 1-31--3 13-171--17-3
every a , ,
Tn 191=9 if
general a ≥o
lot = -
a if a Lo
"
Recall ✓
"
Ote :
means the positive sguar root of
Ta true if a -30
is
only
'
= a
[ take a =-3 than F-3) 2=3 = a
In the case where aco ,
Tai = -
a
↳ than -
a >0
THUS : Ma = lol for all values of a
Properties of absolute values
,§
① label = lallbl
② 1%1 ¥
}
=
③ tant = 1am , NEZ
* ④ 1×1 = a iff x ±a
* ⑤ 1×1 < a iff -
a < ✗ La / IN ≤ a iff -
a ≤✗≤ a
⑥ ,,, µ, ,,
* , ,
, , , , , , ,
, µ, ≥, a≥,
LD If and if
only
C- × 1×1=3 x = 1--3
1×1<3 -3 LxL3
1×1>3 ⇐ >
xc -3 or x >3
C- ✗ 5
Express 13×-21 without the absolute value
system
13×-21
{ 3×-2 if
=
3×-220
2- 32C if 3 >c- 240
{ 3×-2 if x ≥ 73
= 2- 3.x if x ( 43
C- ✗ 6 Solve 12×-51 =3 1×1=9 if x = ± a
27C -51 = 3
⇐> 2x 5 3 or 22C -5 =-3
=
-
⇐ > 7C = Li or 7C = I
x C- { 1,4 }
C-✗ 7 Solve 1×-51<2
1×1 <a
⇐>
-
2 ( x -
5 C- 2 0 -
a < TCL a
⇐ > Loc < 7
3
x E 13,7 ) 3 7
,46 22C -
I
✗ + I =3 121=9 iffsc = -1-9
23C -
I 27C
-
I
2+1 =3 or act 1
=
-3
l =
-
32 -3
⇐> 23C -1 32+3 or 22C -
=
⇐ > -4 = x or 2C : -45 and ✗ =/ -
I
56 051×-51<42 -
Both must be true
AND
-
asel 1×-51 > o case z be -51<112
'
iff x -5 >o or x -
5<0 ⇐ >
-
I 4×-54 12 1×1 >a iff
"
iff 2C 75 of scc 5 912 Csc C 12 x G- a or I >a
o o
±
Thus x c- ( 92,5 ) u 15 ,
"
12 )
14
'
12 5 5112
Ex Solve 13×+41 -
12×-11=1
start with
{ [ ;D
"
13×1-41 =
3×+4 ,
✗ c- 13
-3×-4 ,xE C- as ; -
%)
①
[ 12,00 )
{
12×-11
'
=
2×-1 , x c- -2×+1 ③
0
' 3×+4
-
zxtl ,
ICE 1- as 12 ) i
-3 >c -
c, •• 2 >c- I
0 ② •
413
'
12
-
"
①
Suppose c- 1- as 13 )
-
✗ , .
1×+41-12×-11=1 ⇐> (-3×-4)-(-2×+1)=1
>
⇐ -
2C -
5 =L
< x =
-
6
Note that -
G. c- too -413) ,
② Suppose ✗ c- [ -413 ,
'
12] ③
Suppose ✗ c- [ "z.es )
13×+41-12×-11=1 ⇐> (3)etc, ) -
(-2×+1) =\ 13×+41
-
12×-11--1 ⇐>
(321-4)-(2×-1)=1
⇐> 5×+3=1 ⇐ >
✗ +5 =
I
⇐ > >c = -215 ⇐ >
x =
-
a
[ ¥ 'z ) [
'
Note that -215 c- Note that -4¢ lz ,
as )
,
THUS 13×+41-12×-11=1 <= >
✗ c-
{ -6 ,
-215 }
, C- ✗ Solve the
inequality 2<-13×-5146
First solve the
inequality separately :
13×-51<6 13.x -51 22
⇐> -
6 53.x -516 <= > 3×-5>-2 or 32C -5£ -
2
⇐ > -1 432C 911 ⇐ > 32-77 or 32 £3
⇐ > -
'
13 Soc <
"
13 ⇐ > x
I 73 or 2C
It
( " "
x c- 13) C- I -0,1 ] [ 73 )
-
>
⇐ 3 ,
⇐ > x u ,
as
2<-13×-51 < 6
1- 13 [1-0,1]
173,2b¥ Can you
'
do this ?
"
⇐ > x c-
,
B) n u
⇐ > x c- ( -
'
13, I ] u [ >
13 ,
' '
13)
• • 0
'
I ' '
13 713 13
-
Proof
if xzo
{
"
loci =
-
x if oc co txt : Fi
for all x (R)
→
C- ✗ :
prove that
the lock = xz
Proof real number either
:
Suppose >c is a , xzo or xco
case 1 : ✗ 20
we have lock = x2 ( since bcl=x )
cases :
x co
1×12 =
1 xp-
= x2 ( since bcl = -
x )
therefore 1×12=70 for a " "
☐
[ ☐ end of proof
The
Triangle Inequality
If a ,b c- 112 ,
then latbl E Iat t Ibl
Always
*
smaller or
equal to
PROOF lal Etat & Ibl Elbl
:
Suppose a. b c- 1112 :
Case If b have the same then the either
, a and
sign sum on side is equal .
Case 2 If a andb have opposite signs ,
then
-
late a stat ①
and - Ibl Eb E Ibl ②
Adding ① ② that
and we
get
-
( lait lbl ) I atb E Iat t lbl -
c < x Cc
Put atb and
/
✗ "
y
=
=
then c = Iat t Ibl
lxtscifff.ro
that ← ✗ ,
teased
so - c ⇐
m
⑤ bet <C
lxl=ciHy#
and ④ loci =L iff x=±c
we have lxl Ec x = ± c
l
THUS latbl E Iat t Ibl
