trabajo de investigacion de matematica i[1]

8/17/2019 Trabajo de Investigacion de Matematica i[1]

1/19

1

FACULTAD DE INFORMÁTICA Y CIENCIAS APLICADAS

MATERIA:

MATEMÁTICA I

SECCIÓN:

13

TEMA:

INTERVALOS Y DESIGUALDADES LINEALES Y VALOR ABSOLUTO

PROFESOR: ING. RENE ARMANDO CUBIAS AVILÉS

ESTUDIANTES: BORJA MORALES NOÉ EDUARDO 22-3247-2015

CONTRERAS RIVAS VICTOR MANUEL 25-4041-2015

RIVAS MONGE FRED ALDAIR

S!" S!#$!%&' 15 %( &)*+,'( %( 2015


2/19

2

CONTENIDO

OBJETIVO GENERAL................................................................................................................3

OBJETIVOS ESECIFICOS.......................................................................................................3

INTRODUCCIÓN........................................................................................................................4

DESIGUALDADES.....................................................................................................................5

DESIGUALDADES LINEALES................................................................................................

DESIGUALDADES TRIVIALES.........................................................................................../

DESIGUALDADES SIMULTNEAS......................................................................................10

NOTACIÓN DE INTERVALOS...........................................................................................10

SOLUCIÓN DE DESIGUALDADES SIMULTNEAS.........................................................11

DESIGUALDADES CON VALOR ABSOLUTO....................................................................13

DEFINICION DE VALOR ABSOLUTO.............................................................................13

CONCLUSIONES RECOMENDACIONES...........................................................................17

BIBLIOGRAFA.........................................................................................................................1


3/19

3

OBJETIVO GENERAL.

C&"&)(' )&6'("%(' #& )&")(6*& 8+"%!("*!#( %( I"*('$!#& D(9+!#%!%(;

"()(!'9& 6!'! ("8'("*!' !*98!)*&'9!("*( !9"!*+'! %( "9$(# +6('9&'.

OBJETIVOS ESPECIFICOS

D(89"9' (# 9"989)!%& %( %(9+!#%!%( #9"(!#(.

D(89"9' )!%! *96& %( %(9+!#%!%.

#!"*(!' #! +)9


4/19

4

INTRODUCCIÓN

L! %(9+!#%!%( >+(!" +" ' 8+"%!("*!# (" !*(=*9)!. E?9*(" #9,'& )&6#(*&%(%9)!%& ! + (*+%9&; (" #! )&6(*(")9! 9"*('"!)9&"!#( %( 6'&,#(! !6!'()(" )&"8'()+(")9!.

T&%& +)9&"9*! (?6('*& %(,( (*!' 8!9#9!'9@!%& )&" $!'9! %( (##! )&" #! *)"9)!("('!#( 6!'! + !"(>&.

E" #& +( 9+( ( +6&"( +( (# #()*&' %&9"! #! 6'&69(%!%( ,=9)! %( #!%(9+!#%!%( ("*'( "('& '(!#(.

L! %(9+!#%!% 8+"%!("*!# !*98()! 6&' )+!#+9(' "('& '(!#; %( #! )+!# (" )9('*&("*9%& ( %('9$!" *&%! #! %(=; ( (")9##!("*( ?2 0;


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5

DESIGUALDADES

U"! desiguadad ( +" ("+")9!%& & ()+!)9(6#&:


6/19

6

&'+( -10 - P-30Q -10 30 20

DESIGUALDADES LINEALES

E##! &" #! +( ( 6+(%(" ()'9,9' (" #! 8&'! ax + b > 0 ; ()Q %&"%( a y b &" )&"*!"*(;Pa ≠ 0 Q. R($(' +"! %(9+!#%!% ( )&"(+9' *&%& #& $!#&'( x +( !*98!)(" (*='(#!)9+"*& +)9 0. -3 -2 -1 0 1 2 3L! (?6'(9


7/19

7

4Q ? !. L! +)9


8/19

8

E1e"2$ 5.

!Q −5 < 4 6&' #! '(#! 2 *("(& +( (8()*9$!("*(−5

−5> 4

−5 ; (*& ( 1 Z 5

4 .

T!,9" -5P-2Q 4P-2Q; ( %()9' 10 Y #& )+!# !,(& +( ( )9('*& "& #! %(9+!#%!%(" (# &*'& ("*9%&.

*4 L! %(9+!#%!% Y3? H 1 ( (+9$!#("*(−3 x

−3> 1

−3 ; ( %()9' x Y1

3 .

,4 L! %(9+!#%!% x

2 4 ( (+9$!#("*( ! 2 x

2 2.4; ( %()9' x .

A2i,a,i-' de a %ega 5.3 S9 +" "('& 6&9*9$& (*= +#*96#9)!"%& P%9$9%9("%&Q+" #!%& %( #! %(9+!#%!% 6!! !# &*'& #!%& %9$9%9("%& P+#*96#9)!"%&Q 9"

)!,9!' (# ("*9%& %( #! %(9+!#%!%. S9 +" "('& NEGATIVO (*=MULTILICANDO P%9$9%9("%&Q +" #!%& %( #! %(9+!#%!% 6!! !# &*'& #!%&%9$9%9("%& P+#*96#9)!"%&Q (# ("*9%& %( #! %(9+!#%!% SE INVIERTE.

O,('$( )&& +*9#9@!"%& #! '(#! 2 #&'!& *'!"8&'!' (" (# (>(6#& 2, 2)%(9+!#%!%( #9"(!#( (" &*'! (+9$!#("*( )+! +)9&"( ('!" ($9%("*(. V(!&(>(6#& = )&6#9)!%& 6!'! '($(' %(9+!#%!%( #9"(!#(. N+(*'! *)"9)! ( *'!%+)((" %(>!' ! #! $!'9!,#( ?.

E1e"2$ 7.- R($(' 3P? Y1Q /.S$u,i-'

A#*('"!*9$! 1: U"! (*'!*(9! ! (6#(!' ( '($(' 6'9('& #& 6!'"*(9 %9*'9,+("%& (# 3.3? [ 3 /

L+(& %(>!& #& *'9"& (" ? (" +" #!%& #! )&"*!"*( (" (# &*'& #!%&.E# 3 (*= '(*!"%& 6!! +!"%& 9" !#*('!' (# ("*9%& %( #! %(9+!#%!%.

3? / 33? 12

A&'! 3 (*= +#*96#9)!"%&; 6!! %9$9%9("%& 9" !#*('!' (# ("*9%& %( #! %(9+!#%!%

x 12

3

x 4E?6'(!'(& #! +)9( "& &" #! "9)! !#*('"!*9$! 6!'! %(6(>!' #! $!'9!,#(. C&& 3 (*=


9/19

9

+#*96#9)!"%& *&%& (# 9(,'& 9@+9('%& ("*&")( 6!! %9$9%9("%& 9" !#*('!' (# ("*9%& %( #!%(9+!#%!%.

3P?Y1Q /

? Y 1 9

3

? Y 1 31 (*= '(*!"%& ("*&")( 6!! +!"%& 9" !#*('!' (# ("*9%& %( #! %(9+!#%!%

? 3 1

? 4

E*= )#!'& +( (# )&">+"*& +)9


10/19

10

Y3? 10 2? 10 [ 2? 3? 4 5?

4

5 ?

E*! (?6'(9(')9)9& %( %(!''#&: R($(' 1 Y3

2 P? [ 1Q 2. D!' #! +)9


11/19

11

DESIGUALDADES SIMULTÁNEAS

! H ? H ,S9"989)! +( *!"*& ! H ? )&& ? H ,. 6&' (>(6#&; (# )&">+"*& %( "('& '(!#( +(!*98!)("

2 H ? H 5

E #! 9"*('())9+"*& \?]2 H ?^ \ ?]? H 5^. !'! '!89)!' +" )&">+"*& %( "('&'(!#(; &)+'()(& #& 6+"*& (" #! #"(! %( "('& '(!#( +( )&''(6&"%!" !# )&">+"*&%!%&.U*9#9@!"%& %(9+!#%!%( ,=9)! %(9+!#%!%( 9+#*!"(!; 6&%(& %()'9,9' )9('*&)&">+"*& %( "('& '(!#( ##!!%& 9"*('$!#&. A (*& 9"*('$!#& )&''(6&"%(" +"!"&*!)9


12/19

12

Y 2? H 1

−8

2 2 x

2 H18

2

Y4 ? H / A; #! +)9&"( %( #! %(9+!#%!% %!%! &" *&%& #& %(# 9"*('$!#& XY4; /X.

= =

Y5 Y4 Y3 Y2 Y1 0 1 2 3 4 5 7 / 10

A6e%'a6i+a 5: PR($(' 6&' (6!'!%& #+(& %(*('9"!' #! 6!'*( )&" %( #& )&">+"*&+)9


13/19

13

−1

2 PY2Q −1

2 P2?Q −1

2 P2Q

1 ? Y1

!' (?6'(!' (*! +)9


14/19

14

DESIGUALDADES CON VALOR ABSOLUTO

L! '(+)9&" %( ()+!)9&"( %( %(9+!#%!%( %( 6'9(' '!%& )&" $!#&' !,+*&;'(+9('( %( %& 6'&)(%99("*& PC!& 1 C!& 2Q; (" +( ( +*9#9@!" #! 9! #(( %(+"! ()+!)9&" %( +"! 9"()+!)9&" #9"(!# "&'!#.

DEFINICIO>N DE VALOR ABSOLUTO

E# $!#&' !,+*& %( +" "+ ('& '(!# x ( %("&*! 6&' |x| ( %(89"( )&& 9+(:.

x, 79 x ≥ 0

| x| N−x, 79 x < 0

V('(& !&'! +"& *(&'(! +( "& ('$9'!" 6!'! '($(' #! ()+!)9&"(.

Te$%e"a 0 !'! )+!#+9(' "+('& '(!# ?:1. |x| ≥ 0

2. |x| N 0 ←→ x N 0

3. |x|2 N x 2

4. √ x2 N |x|

5. −|x| ≤ x ≤ |x|

Te$%e"a 5 !'! )+!#+9('! "+('& '(!#( ? !:

|x| a ←→ a ≥ 0

y

x N a y/o x N −a


15/19

Te$%e"a 7 !'! )+!#+9('! "+('& '(!#( x a:

|x| N |a| ←→ P x N a y/o x N −a

Te$%e"a : !'! )+!#+9('! "+('& '(!#( x y a:

1. |x| ≤ a ←→ −a ≤ x ≤ a

2. |x| ≥ a ←→ x ≤ −a ∨ a ≤ x

Te$%e"a


16/19

Cas$ 0

2x+1=x+3

2x-x=3-1x=2

Cas$ 5

2x+1=-(x+3)2x+1=-x-32x+x=-3-1

3x = -4

X=-4

3

T!"*& x N 3 )&& ?N2; )+6#(" #! 6'9('! %(79+!#%!%.

L! +)9&" (: S\?_|2 x 1|?3 ^\−4

3 ;2 ^.

E1e"2$ 7 R($(' #! ()+!)9&" |x − 2| 3 x − /.

S$u,i$>' 6&' (# T(&'(! 2; *("(& #! 9*+!)9&"(

| x − 2| N 3 x − / ←→

D( #! 6'9('! %(9+!#%!% *("(& +( 3 x − / ≥ 0 %(6(>!"%& &,*("(& x ≥ 3. E*!

%(9+!#%!% ( 8+"%!("*!# 6!'! (*!,#()(' +( $!#&'( 9'$(" )&& +)9&".

D( (*! %& +)9&"( &7

2 )+6#( +( (! !&' +( 3. L! +)9&"11

4 (

("&' +( 3 ( %()!'*!.

Cas$ 0

x-2=3x-!x-3x=-!+2

-2x=-"

x= -7

2

Cas$ 5

x-2=-(3x-!)

x-2=-3x+! x+3x=!+2 4x=11

x =11

4

3 x − / ≥

x − 2 N 3 x – / y/o x − 2 N − P3 x − /Q


17/19

S={x# |x − 2|= 3x-! } = {7

2 }

E1e"2$ : R($(' #! %(9+!#%!% |x 12| ≤ 4.

S$u,i$>' 6&' (# T(&'(! 4; 6!'*( 1; *("(& #! 9+9("*( &6)9&"(.

| x R 12| ≤ 4 ←→ − 4 ≤ x R 12 ≤ 4

R(*!"%& 12 ! #& *'( *('9"&; *("(&

−4 − 12 ≤ x R 12 − 12 ≤ 4 − 12

−1 ≤ x ≤ −

L& +( ( (+9$!#("*( ! x ∈ X−1, − #! +)9&" 89"!#("*( S X-1;-.

E1e"2$


18/19

U$%&'%** T',$./%, *' E. .&*

CONCLUSIONES Y RECOMENDACIONES


19/19

U$%&'%** T',$./%, *' E. .&*

BIBLIOGRAF?A

`&$!); M9)!(# L. M!*(!*9)!: !6#9)!)9&"( ! #! )9(")9! ()&"&9)!-

!%9"9*'!*9$!.

T!#&'; &a!'% T&! b!%(. M!*(!*9)! B!9)!: )&"

$()*&'( !*'9)(.

M!*(=*9)! I D(""9 c9## _J!)+(#9"( D(a!'; (+"%! (%9)9

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