Inecuaciones

 ! " # $ % & ' (
I.
8 9 : ; 9 < = > =
?>
@ ; ?A : BC 9
D E
1.
NIVEL BÁSICO
3.
− 3 ( x + 1 ) < 2x + 2
6.
1 − 2x
x
−9 >
3
2
9.
12.
15.
4.
x > 3x − 2
G
x 2 + 13 x − 14 > 0
6 x 2 − 73 x + 12 < 0
18. x < x 2 + 2 x + 1
19.
21.
(x − 1)2 (x − 2)(x − 3)2
24.
−
27.
(x − 1)(1 − x ) 1
≤
7
30.
4
2
5x − 1
+
<
x + 3 x − 3 x2 − 9
33.
1<
36.
39.
42.
45.
3 x 2 − 7x + 8
2
x +1
2
3
>
2x − 1 3 x − 4
2x − 3
> x2 − 5
x +1
1
> −(x + 1)
x +3
7x − 3
x 3 −4
x +4 =2
51.
x−
54.
x
− (x + 3 ) ≤ 5
2
B9 E : A > : B; 9 E @
2−x
x +3
≤5−
3
8
3
5 1
x − x + > − 2x
4
6 3
−3 x > x − 8 ó (2 x − 3 ) + 1 > 6 x
x +6
1
<
16
2− x
 2 x + 1

 >0
 x −5 
1
≤4
3x − 2
32. x + 2 < 2
4
1
≤
x +3 4− x
2x − 1
1
>−
x −3
x
1
1
+
>0
3(x − 2) ( x + 1)(2 − x )
38. x − 2 < 2
46.
7 − 3x > 2
47.
49.
2x − 3 > 4
52.
3< x +5
55.
x −1 ≤ 2
y
x+4
35.
67.
28 x − 11 + 4 < 7 x
3
74.
x −1
x +1
- . / 0 1
≤0
58.
69.
72.
75.
41. 3 + 1 ≥ 4
x −2
44.
x − 2 (x − 1) ≥ 0
(x
)
− x − 20 30 − 7 x − x
2 3 . 4 . 5 . 6 4 3 5 / / 3 5 1 , 7 /
−
2
≥0
x +1
(x − 1)(1 − x ) ≤ 1
x2 + x − 6
3x − 4 − 2 ≤ 0
50.
2
x −1 ≥ 2
3
53.
x −3 =5
56.
x +1 ≥ 4
59.
3 x + 1 > 1,7
62.
3x + 3 − 5x > 4
65.
68.
8 − x ≥ 2x + 1
73.
76.
7
x >2
G
y
x −1 > 1
2x + 1 ≤ 3 + x − 3
3x + 2 ≤ 7
70.
11 − 2 x + x > 4
2
x +5
≤0
2x − 1
x +4
2−x ≤ 2
1
4x + 7 + − x − 2 >
2
71.
1
>4
x −2
29.
x − 7 > 4x + 7
K N I O
 2x 
 2x 

 + 2
−3 ≤ 0
 x −5
 x −5
x +2
<2
x +4
1
4
<
x − 1 (1 − x )(x − 5)
64.
M
2
20.
28.
3x − 5 < 1 − 4x
K L
1 2
x − 2x + 3 ≤ 0
4
x+
63.
66.
I J
x 2 − 3 x + 54 < 0
26.
61.
H
8.
23.
1
>0
x −3
2x − 3 ≤ 7 − x + 1
4 x − 3 x + 2  ≥ 5(x − 3 )
 6 
17.

 x
x
 − 1 + 1 > 0

 3
2
60.
<0
5.
1
1
≤
4−x
x+4
43.
x
−2(x − 2 ) > 3 x − 3
25.
40.
4
3
2.
11. 9 + 18y − 16y 2 < 0
14. 22 x 2 − 7 x − 2 ≥ 0
x
34.
2
≤2
3
1≤ x + 2 ≤ 2
:
22. 1 < 5
37.
x −1
@ BF A BE 9 < E @
x
<0
x +2
−7 x − 3 > 5,1
57.
2
<2
2x − 3 >
y
>0
31.
>0
48.
7.
& ( * % $ + % & " ( * ! ,
10. m 2 + 16 m − 192 ≤ 0
13. − x 2 − x + 2 > 0
16. 200c 2 + 80c + 8 ≥ 0
x (x − 7 ) > 8
4 x 2 + 8 x − 12 ≥ 0
x2 + x − 6
1
x +1≥
1− x
?> @
) *
3
x +1 x +1 ≤ 0
4 x + 3 > 5 − 2x
! " # $ % & ' (
x −2
≥0
78.
80. (1 − x ) 2 x − 9 > −5
81.
77.
x −1
& ( * % $ + % & " ( * ! ,
3 x 2 − 16 x − 12
x +4
79. x 2 > x
≥0
4 − x (x − 1) ≤ 4
(x − ) x + 1 > −2
2x − 3x − 2 ≤ 3
85.
x 2 − 6 x + 10 < 2
23 − 5 x − 2 x 2 > 19 − 3 x
87.
14 + 6 x − 4 x 2 ≥ 4 x 2 − 6
88.
4 x 2 + 4 x − 11 ≥ 9 − 2 x − 4 x 2
89.
x 2 + 3x + 2 ≤ 4
90.
91.
3x − 1
<2
x +1
92.
7−x
2
>
5x + 1 3
93.
94.
2x + 1
≤3
1− x
95.
5
x +7
>
10 x − 1 17
96.
x
86.
H
+ 3x + 2 ≤ 4
I J
K L
8
9 : K ; I O
;
2
82.
84.
83.
2
) *
3x +4
≤2
3x − 1
3 − 2x
≥4
x +2
(x − 2) x + 1 > −2
97. 3 x 2 − 11 x − 4 > 0
98. 2 x 2 − 5 x − 25 < 0
101.
99.
− (x + 2)(x + 5 ) < 2 x − 1
100. (x − 2)2 − 2
102.
8 x 2 + 14 x − 13 > 9
103. 3 x 2 − 11 x − 4 ≥ 0
105.
(
)
− x 2 − 2x + 1 < x 2 − 1 + 3
106.
2x − 3
x2 − 1
≥ 2 x −1
7x − 2x 2 − 4 ≤ 1
104. 5 x 2 − 2 x − 3 ≤ 0
≥2
II.
Escriba una inecuación cuadrática cuyo conjunto solución sea (−∞ ,2 ) < (5, ∞ )
III.
Escriba una inecuación cuadrática cuyo conjunto solución sea [−2,3]
IV.
ax
2
V.
VI.
El polinomio ax 2 + bx + c evaluado en 1 es 6. El conjunto solución de la inecuación
+ bx + c ≥ 0 es (−∞ ,−1] < [0, ∞ ) .
Determine las constantes a, b, c que satisfacen las condiciones dadas.
Es posible que una inecuación de la forma
x ∈ (−∞,−6] < (3, ∞ ) ? Justifique su respuesta.
VII.
=
VIII.
J K G K
1.
3.
> ?
@ A B ? C
L > ?
A D
B ? E FG
M K NA G ? D
B ?
B ?
O
H
NK D
ID F
I
D FP > F? Q R ? D
? E > K E FA Q ? D
(4m + 3)x + 5 x + 3 = 0
1.
X.
FQ K G
NA D
M K NA G ? D
B ?
x
L > ?
2
9 x + 9 x − 54
U ? R ? G C
FQ ?
@ K G K
L > ?
V K E ? Q
B ?
O
B A D
G K FE ? D
B ?
E K B K
4x − 1
NK D
? W @ G ? D FX Q
> Q
3.
D FP > F ? Q R ? D
FQ ? E > K E FA Q ? D
x ∈ℜ
1.
3.
5.
x 2 + 3x − m + 2 > 0
x − 3(m + 7 )x − 3m + 17 < 0
2
(m − 1)x 2 − 5 x + 6 < 0
- . / 0 1
2 3 . 4 . 5 . 6 4 3 5 / / 3 5 1 , 7 /
2.
4.
6.
G ? K N? D
B FS ? G ? Q R ? D T
mx 2 + 4 x − 2m + 1 = 0
2
2.
M K NA G ? D
R F? Q ? Q
2x 2 − 3x + 1 − m = 0
2.
4.
2
U ? R ? G C
tenga como conjunto solución
x (x − 2) < 0
x 2 − 8 x − 4m = 0
IX.
ax 2 + bx + c < 0
− 2 x 2 − x + 2m − 3 < 0
3 x 2 − (m − 1)x + 3 > 0
mx 2 − (1 + m )x + 1 > 0
Q Y C ? G A
3x
2
G ? K NT
−4
7
D A Q
M ? G B K B ? G K D
@ K G K
R A B A
! " # $ % & ' (
XI.
= > ? > @ A
> S Q @ > E BI C
1.
BC D @
es
∅
D H B@ A D F BI C
> E
E K MR F BI C
=
a
J > @ G D G > @ D
G >
MD
2.
BC > F R D F BI C
, a ≠ 0
a
1
x
5.
0 < x < 10 −3
7.
Y Z [ \ ] ^ _ ] ` \
a \ Z_ [ Xb ]
c d
x −a < b
8.
Y Z [ \ ] ^ _ ] ` \
a \ Z_ [ Xb ]
c d
x −a < b−a
XII.
e h c h
c d
Zh
10 3 <
, entonces,
c d a Xi _ h Zc h c
c d f d [ o h p
a1
c d
q d ` d f l
Zh
Xj k _ Xd f c h
X] h f
a2
Z\ a
n h f d a
a3
[ \ ]
` Xd ] d
L D ME D N
B > E
L D ME D
F K @ @ > P B @ MD
−5 ≤ x < 5
a2
2(1 − x ) > −2 x
b2
x < −7
a3
− 5 ≥ −3 x − 20 > −35
b3
x >2
b4
1< x < 2
b5
−5≥ x >5
b6
ℜ
b7
x >7
67
3
< 2x −
5
5
x +1 > x − 5
XIII.
t h f h
XIV.
Y ] [ _ d ] ` f d
k _ u
g h Z\ f d a
3x −m ≤ n
W
X
x <0
2<3
y
c d
d Z g h Z\ f
` d ] i h
Zh
v
c d
Zh
X] d [ _ h [ Xb ]
x
y
z
n h f h
a Xi _ Xd ] ` d
0 < x < 10 −4
6.
x < 10 −3
d a
d Z
a<b
R C D
2(-x ) < 3(− x )
, entonces,
[ \ l
\
XV.
e _ h Z d a
XVI.
Y ] [ \ ] ` f h f
d Z [ d ] ` f \
_ ] h
c d Z
x 2 < 10 −3
, entonces,
Z
(a − b ; a + b )
(− b; b )
X ] ` d f g h Z\
[ \ ] ^ _ ] ` \
k _ d
3x − 7 ≤ p − 3
2
d
] \
Z [ \ ] ^ _ ] ` \
f d n f d a d ] ` h [ Xb ]
a \ Z_ [ Xb ]
_ ] h
Zh
a Xi _ Xd ] ` d
` Xd ] d
a \ Z_ [ Xb ] w
c d
Zh a
k _ d
1.
c d
] { l
d f \ a
f d h Zd a
k _ d
a h ` Xa | h [ d
i f } | X[ h V
(1 − 2δ;1 + 2δ )
w
f d n f d a d ] ` d
2.
8
9
:
;
~
<
3.

€

‚
ƒ
„
4.
~
…
…

5.
…
~
…

€

‚
6.
…
…
~

€

d m n f d a X\ ] d a
` h s Zh p
2/3
X ] ` d f g h Z\
X] d [ _ h [ Xb ]
d
1
x
10 -4 <
, entonces,
X ] ` d f g h Z\
Y a
-10/3
…
Q K @
a5
b1
- . / 0 1
O
[ \ f f d a n \ ] c Xd ] ` d a r d ]
a4
3 − 2x < 1
a5
K
4.
b>0
e \ ]
a1
a4
V
a . a
3.
F D G D
& ( * % $ + % & " ( * ! ,
F K @ @ > F ? D N
T M F K C U R C ? K
x ≥x
E B
) *
‚
2 3 . 4 . 5 . 6 4 3 5 / / 3 5 1 , 7 /
…
…
…
~

€

‚
! " # $ % & ' (
XVII.
8
9: ;
< 9
< = >
? @ A B ; B @ A ; <
J K
L
M
x − 3 < 0,5
⇒
N
K
L
M
x <1 ⇒
1
<1
2x − 3
O
K
L
M
P
K
L
M
- . / 0 1
⇒
C ; D< ; <
D; <
x +2 ≥1
x2 + x +7
x2 +1
≤ 15
2 3 . 4 . 5 . 6 4 3 5 / / 3 5 1 , 7 /
& ( * % $ + % & " ( * ! ,
< 9: E 9@ > F @ <
5 x − 15 < 2,5
(x − 2) ∈ [− 3,−1] ⇒
x ≤2
=
) *
; C 9A G ; H 9= > @ < I
! " # $ % & ' (
8 9 : ; < 8 < ; =
C D E F
G HI
O N R N S
\ X L N S
X T O P VU TX
Z [ X
d
e X Q
d
K T N O P N
k Q
? @
A
& ( * % $ + % & " ( * ! ,
?
: 8 @
; >
? < B
8 < ; =
>
? @
J
K L M N L O P Q P
f
[ L
TN S
X L O X P N
U N S V W TX S
R X
L g \ X P N
L g \ X P N
X l U P X S Vm L
R VM _ N
>
) *
Z [ X
U Q P X S
R VM _ N
X L O X P N
X L O X P N S Y O Q TX S
X S O `
X L O P X
a b
]
Z [ X
TQ
S [ \ Q
R X
[ L N
R X
X
T TN S
]
^
[ L VR Q R X S
^ b c
X L O X P N c
S X P `
h a f
i
^ j
M N P P X S U N L R X
X S O `
R X
X L O P X
a b
]
Q
n TQ
S [ \ Q
^ b
n
S X
R X W X
R X
[ L
X L O X P N
]
^
[ L VR Q R X S
\ X L N S
Z [ X
X T O P VU TX
R X
X S
34 < x + (3 x − 5 ) < 54
o Q P Q
R Q P
S N T[ M Vm L
Q T U P N W TX \ Q
P X S N Tp X P
L X M [ Q M Vm L
34 < x + (3 x − 5 ) < 54
TQ
VL X M [ Q M Vm L
U TQ L O X Q R Q c
U TQ L O X Q R Q c
q
P [ U Q M Vm L
34 < 4 x − 5 < 54
R X
O t P \
VL N S
S X \ X u Q L O X S c
r s
o P N U VX R Q R
34 + 5 < 4 x − 5 + 5 < 54 + 5
R X
P [ U Q M Vm L
39 < 4 x < 59
N P R X L
R X
R X
O t P \
TQ
Q R VM Vm L c
VL N S
S X \ X u Q L O X S Y
VL p X P S N
Q R VO Vp N
]
\
m R [ TN
R X
r s
TQ
Q R VM Vm L c
o P N U VX R Q R
 1 39 <  1  4 x <  1 59
4
4
4
R X
o P N U VX R Q R
39 < x < 59
4
4
T P X U P X S X L O Q P
X L
TQ
P X M O Q
L [ \ t P VM Q
X T
N P R X L
VL p X P S N
R X
\
V L O X P p Q TN
TQ
\
[ TO VU T V M Q M V m L c
[ TO VU T V M Q O V p N
S N T[ M Vm L
R X
]
TQ
\
m R [ TN
R X
VL X M [ Q M Vm L
TQ
S X
\
[ TO VU T V M Q M V m L c
N W O VX L X v
r
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
0
w N \
N
TN S
L g \
x y Y U [ X S O N
o Q P Q
e
V
e
V
e
V
e
V
e
V
k N
]
R Q P
X P N S
Z [ X
TQ
1
2
3
4
W [ S M Q R N S
39 = 9,75
4
P X S U [ X S O Q
Q v
]
X T
5
6
R X W X L
S X P
\
X L O X P N
Q ] N P
X L M N L O P Q P
⇒
3 x − 5 = 3(10 ) − 5 = 30 − 5 = 25
x = 11
⇒
3 x − 5 = 3(11) − 5 = 33 − 5 = 28
x = 12
⇒
3 x − 5 = 3 (12 ) − 5 = 36 − 5 = 31
x = 13
⇒
3 x − 5 = 3(13 ) − 5 = 39 − 5 = 34
x = 14
⇒
3 x − 5 = 3(14 ) − 5 = 42 − 5 = 37
a x {
K S O Q S
x a
]
L N S
a b
{
U X P \
x b
S N T[ M VN L X S
L g \ X P N S
- . / 0 1
]
L N
VO X
M N L M T[ VP
Z [ X
8
9
X L O X P N S
O N R Q S
x = 10
Q L O X P VN P
7
TN S
O X L X \
U N S V W TX
TQ S
10
R X
N S
S X P `
U N S V W TX S
U Q P X S
11
12
Z [ X
x b
13
X T
] Q
X L O X P N S
\ X L N P
Z [ X
U Q P X u Q S
Z [ X
14
R X
15
X L O X P N
U Q P Q f
R X W X
S X P
X T
z | {
x z
59 = 14,75
4
X L O X P N S c
M [ \ U TX L
S N L v
x y
]
z ^
{
x x ]
a }
O VX L X L
P X U P X S X L O Q M Vm L
X L
X L O X P N S c
2 3 . 4 . 5 . 6 4 3 5 / / 3 5 1 , 7 /
TQ
P X M O Q
L [ \ t P VM Q
U [ X S O N
Z [ X
S N L
U Q P X u Q S
R X
! " # $ % & ' (
8 9 : ;
@
< =>
A B
) *
?
C D E FC G D H I
JD
C FK L F I M E I
FM I H L D H FN M
C I Q P M
JR C
S D JR Q I C
T R C F U JI C
V I
4x − 8
O
W
V D V D Y
X
Q R T FI V D V
V I
R Q V I M
V I
JD
[
L JE FT J F H D H F N M Y
Z
F[
7 < 4x < 9
T J FG FH D H FN M
V I
I ] T Q I C FR M I C Y
\
Q R T FI V D V
7 − 8 < 4x − 8 < 9 − 8
Z
−1 < 4 x − 8 < 1
\
F[
^ J Q I C L JE D V R
F M E I Q S D JR
H L P JI C
7<x<9
4
4
M I H L D H FN M
7<x<9
4
4
7
(4 ) < (4)x < (4 ) 9
4
4
D M E I Q FR Q
M R C
J JI S D
D
V I
H R M H JL FQ
I V Q R
V I
_ L I
V I
JD
D V FH FN M Y
I ] T Q I C FR M I C Y
JD
I ] T Q I C FN M
4x − 8
E I M V Q P
S D JR Q I C
Q I D JI C
I M
I J
T FC R Y
^ J
Y
(−1 , 1 )
D E D _ L FS D
Z
R Q V I M
T J FG FH D H FN M
` a
1.
& ( * % $ + % & " ( * ! ,
V I U I
[
R S I Q
L M
K Q D M
` b c d c d e
f
H D Q K D [ I M E R
V I
JFU Q R C
V I J T Q F[ I Q R
D J _ L FM E R
Z
JI E Q I Q R
V I
J D C H I M C R Q
V FH I g
h i D Q K D
[ P ]
F[ D
j k k
JFU Q D C l Y
F
H D V D
H D m D
V I
JFU Q R C
T I C D
n k
\
^ M H L I M E Q I
JFU Q D C Y
I J
M o [ I Q R
V I
H D m D C
_ L I
I V Q R
D E D _ L FS D
Z
2.
^ M
I J I m I Q H FH FR
C I
T Q I C I M E D
D q R Q D
JD
C FK L FI M E I
T L I V I
C FE L D H FN M Y
H R JR H D Q
I V Q R
C I
V I U I
C L U FQ
I M
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D J
D C H I M C R Q
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JD C
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3.
p
I J D C H I M C R Q Y
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I J
M o [
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p r k
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I Q R
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F[
R
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I J D C H I M C R Q Y
C L T I Q G FH FD J V I
Q D V FR
C F
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T I _ L I z R
C I
C D U I
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C F JR
D JE L Q D
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S D JR Q I C
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s
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L w
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4.
L M
M I K R H FR
L M D
G D [
F J FD
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