Practica Dirigida

PRACTICA DIRIGIDA 1. HALLAR EL CONJUNTO SOLUCION DE LAS SIGUIENTES INECUACIONES a) X-3 3

+ 5 ≤ 4

X + 2X+9 12 15

4X+3 - X ≤ 2X+9 12 12 15 3X+3 - 2X+9 12 15

≤ 0

45X+45 - 24X-108 ≤ 0 21X - 63 ≤ 0 X ≤ 3

;

X є ]-δ ; 3]

b) 11-3X < 1(5X+14) ^ 1(5X+14) ≥ 9(2+X) 2 3 3 2 22-3x < 5x+14 ^ 5x+14 ≥ 18 +9x 2 3 3 2 66-19x-28 < 0 ^ 2 <X

-26-17x ≥ 0

^

-26/17 ≥ X

. - 26/17

. 2

C .S : ] 2, +∞ c) x + 6 < 3x +11 2 5 4 5

^

5x + 12 - 15x - 44 10 20

3x + 11 > 2x - 14 4 5 5 <0 ^

15x + 44 - 10x + 14 > 0 20 5

100x + 240 - 150x - 440 < 0 ^ 75x + 220 - 200x + 280 > 0 -50x - 200 <0 ^ -125x + 500 > 0 -200 < 50x 125x > 500 -4 < x ^ x > 4 .

.

-4

4

C . S : ] 4 , +∞

3. Hallar el conjunto solución de las siguientes inecuaciones : a)

( X3 - X2 - X +1 ) ( X - 6 )4 ( X - 3) (X3 + 8X2 + 14X + 12) (X -5 )3 (X – 7)1177 (X2 – 3)

≥ 0

(X (X2 - X – 1) + 1) (X – 3) (X (X2 + 8X + 14) + 12) (X – 7) (X + 3½) ( X - 3½)

≥ 0

(X + 1) (X – 3) (X + 12 (X – 7) (X + 3½) ( X - 3½) + . -12 C .S

b)

]-∞, 12

- . + - 3½ U

. -1

-

] - 3½, -1]

≥ 0

. + 3½

U

] 3½, 3 ]

( X3 - X2 - X +1 ) (X + 2)4 (X – 3) (X – 5) (X2 – 7) (X2 + 3)

>

0

(X (X2 - X – 1) + 1) (X – 3) (X – 5) (X - 7½) (X + 7½) (X2 + 3)

>

0

(X + 1) (X – 3) (X – 5) (X - 7½) (X + 7½) -

. + - 7½

C. S : ]- 7½ , -1> 3.1.

. 3

a)

. -1 U

-

X2 – 2X – 3 X2 – 11X – 28

> 0 ≥ 0 . -1

C. S : [ 4 , +∞ >

U

. + 7 ] 7, +∞

> 0

. + 7½

] 7½ , 3>

-

. 3 U

-

. + 5

] 5 , +∞ >

x > 3 ; x > -1 ; ] -1, +∞ > x ≥ 7 ; x ≥ 4 ; [ 4 , +∞ > . 4

b)

4X2 – 1 -2X2 + 5X – 3 . -3/2

> 0 ; X > 1/2 > 0 ; X > -3/2 , X > -1 . -1

. 1/2

C. S: ] ½ , +∞ > 5.Resolver las siguientes inecuaciones: a) ‌│ X2 – X │ > -2 2 > │ X2 – X │

٧

│ X2 – X │ > -2

0 > X2 – X – 2

X2 – X + 2 > 0

X>2 X > -1

٧

C. S : ] -1, +∞ > b.- │X2-4│

≥ -2X+4

X2-4 ≥ -2X-4 X2+2X-8 ≥ 0 X +4 X -2

X ≥ -4 X ≥ 2

[-4,+∞) V

X2-4 ≤ 2X-4 X2 2X ≤ 0 X2

≤

2X

X ≤ 2 (-∞,2] -∞

-4

0

2

+∞

C.S. (-∞,+∞) C.-

| X+8| ≤ 6 X

X2+8-6 ≤ 0 X X2+8-6X ≤ 0

-6 ≤ X2+8 ≤6 X

C. S . ǿ

-6 ≤ X2+8 X 0 ≤ X2+6X+8 X +4 X -2 X ≥ -4 X ≥ -2

C.S. [-4+∞)

d).- (X+3) -5 |x+3| +6 < 0 x+3-5x-15+ < 0 -4x-6 < 0 -6 < 4x X > -3 2 C.S. (-3,+∞) e).- |x+6| > |x+5| + |x-2| -(|x+9|+|x-2|) > |x-6 | > (|x+9|+|x<2|) -|2x-7| > x-6|

x-6 > 2x+7

-2x-7 > x-6

V

-13 > x

-1 > 3x X < 1 3 -∞

-13

-1

+∞

3 C.S. (-∞,-1) 3 f) │4x + 2 │ ≥ │x – 1 │+3 │x + 1│ 4x + 2 ≥ 2x + 3

v

-2x – 3 ≥ 4x + 2

C. S : [1/2 , +∞ > U < - ∞ , -5/6 ] X ≥ 1/2 g)

v

-5/6

≥

x

2 < │X2 – X – 1│< │X2 – X│ 2 <│X2 – X – 1│

V

│X2 – X – 1│< -2

0 < X2 – X -3

X2 – X +1 < 0

X > -1+ 13½ 2

V

X > -1 - 13½

ø

V

ø

2 C. S : < -1 - 13½ , +∞ > 2 2 < │X2 – X│ 0 < X2 – X – 2 ; X

-2 V X

1

C. S : < -2 , +∞ > C. S final :

h)

|

< -1 - 13½ , +∞ > 2

- 4| < -2x + 4 2x – 4 <

2x

<

2

<

-4

v v

x

v

- 4 < -2x + 4 + 2x - 8 (x + 4) (x – 2) < 0

2 < x X Є ( 2 ; +∞ )

v

X < -4 ; x < 2 X Є (-∞ ; 2) . 2

c. s : R - 2

i)

2x - 5 4-x

≥

1

2x - 5 4- x

≥

1

2x – 5

v

≥ 4–x

X

≥

2x - 5 4-x

v

2x - 5

9

≤ -1

≤ x

-4 + x ≤ 1

c. s : (-∞ ; 1] u [ 9 ; +∞ )

j)

3 – 3x X–1

> 2

3 – 3x X–1

> 2

v

3 – 3x X–1

3 – 3x > 2x - 2

v

3 – 3x < - 2x + 2

1 > x

v

1

< -2

< x

C. S : R–1

k)

3 4x + 1

≤

3 4x + 1

≤

3x + 3 2

≤ ≤

1 x+1 1 x+1

4x + 1 x

C , S : [ - 4/7 , +∞ )

3 4x + 1

v v v

≥ -( 1 ) x+1

3x + 3

≥

x

≥

-4x - 1 - 4/7

l)

x–3 3 + 2 x–3 2 - 5 x–3 - 6

≤

0

( x – 2 )2 – 2 x – 2 - 24 Sugerencia usar a ≤ 0 b •

A.

x–3

(a≥0 ^ b< 0 ) u (a≤0 ^ b>0)

ANALIZANDO : a ≥ o

- 5 x–3

-4 x – 3 x–3

≥

-6 ≥ 0 6

≥ -3/2

x

≥

3/2

v

x–3

v

≤ 3/2

x

≤

9/2

C.S: R

•

B. ANALIZANDO b < 0

-2 X – 2 < 24 X–2

< -12

X

C.S: •

v

< - 10

(- ∞ , -10 )

v

V

x

> 12

>

14

( 14 , +∞ )

C. ANALIZANDO a ≤ 0

-4 x–3 ≤ 6 X–3 ≤ 3/2 C. S :

•

x–2

v

- 4 x – 3 ≥ -6

v

x

≥

9/2

R

D. ANALIZANDO

-2 x – 2 > 24 X > -10

b> 0 v v

-2

x–2

x < 14

< 12

C . S : R - ( -10 , 14 ) ANALIZANDO TODOS LOS ENUNCIADOS: A^ B U

C ^ D

EL CONJUNTO SOLUCION GENERAL ES : R - ( -10 ; 14 )