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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!#$!%&' 15 %( &)*+,'( %( 2015
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CONTENIDO
OBJETIVO GENERAL................................................................................................................3
OBJETIVOS ESECIFICOS.......................................................................................................3
INTRODUCCIÓN........................................................................................................................4
DESIGUALDADES.....................................................................................................................5
DESIGUALDADES LINEALES................................................................................................
DESIGUALDADES TRIVIALES.........................................................................................../
DESIGUALDADES SIMULTNEAS......................................................................................10
NOTACIÓN DE INTERVALOS...........................................................................................10
SOLUCIÓN DE DESIGUALDADES SIMULTNEAS.........................................................11
DESIGUALDADES CON VALOR ABSOLUTO....................................................................13
DEFINICION DE VALOR ABSOLUTO.............................................................................13
CONCLUSIONES RECOMENDACIONES...........................................................................17
BIBLIOGRAFA.........................................................................................................................1
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OBJETIVO GENERAL.
C&"&)(' )&6'("%(' #& )&")(6*& 8+"%!("*!#( %( I"*('$!#& 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
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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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DESIGUALDADES
U"! desiguadad ( +" ("+")9!%& & ()+!)9(6#&:
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&'+( -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
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)!,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)! ( *'!%+)((" %(>!' ! #! $!'9!,#( ?.
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A#*('"!*9$! 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
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+#*96#9)!"%& *&%& (# 9(,'& 9@+9('%& ("*&")( 6!! %9$9%9("%& 9" !#*('!' (# ("*9%& %( #!%(9+!#%!%.
3P?Y1Q /
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3
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4
5 ?
E*! (?6'(9(')9)9& %( %(!''#&: R($(' 1 Y3
2 P? [ 1Q 2. D!' #! +)9
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DESIGUALDADES SIMULTÁNEAS
! H ? H ,S9"989)! +( *!"*& ! H ? )&& ? H ,. 6&' (>(6#&; (# )&">+"*& %( "('& '(!#( +(!*98!)("
2 H ? H 5
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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
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−1
2 PY2Q −1
2 P2?Q −1
2 P2Q
1 ? Y1
!' (?6'(!' (*! +)9
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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
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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$%e"a
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Cas$ 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
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L! +)9&" (: S\?_|2 x 1|?3 ^\−4
3 ;2 ^.
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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
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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$
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U$%&'%** T',$./%, *' E. .&*
CONCLUSIONES Y RECOMENDACIONES
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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