h) ecuaciones con mÁximo entero

7/28/2019 h) Ecuaciones con MXIMO ENTERO

1/6

Ecuaciones con Mximo entero:

1) [|3x|] = x +2

x + 2 Z ^ x +2 [|3x|] < x +3 --> x +2 3x < x +3 --> 2 2x < 3 --> 1

x < 3/2

x = 1

2) [|3x|] =2 x +2

2x+2 Z ^ 2x +2 [|3x|] 2x +2 3x < 2x +3 --> 2 x < 3

--> 4 2x < 5

2x = 4,5 --> x = 2, 5/2

3) [| |] = 5

5 10 |x 2| + 3 < 12 --> 7 |x 2| < 9

Por definicin de valor absoluto:

Para

7 x 2 < 9 --> 9 x < 11

1era Solucin: x [9, 11>

Para x < 2

7 -x + 2 < 9 --> 5 - x < 7 --> -7 < x 5

2da Solucin: x -1 - |x|< 0 --> 0 < |x| 1



2/6

Para

0 < x 1

1era Solucin: x

Solucin General: x ( [-1, 0> U 6 x < 7 --> 12

2x < 14

2x = 12, 13 --> x = 6, 13/2

6) [| |] = 2

Calculando Universo:

0 --> 3 x 0 --> -x -3 --> 3 x

x 2 < 3 --> 4 3 x < 9 --> 1 - x < 6 --> -1 x

> -6

x


3/6

2 < 3 --> 6 |x 1|-1 < 9 --> 7 |x 1| < 10


Para

7 x 1 < 10 --> 8 x < 11

1era Solucin: x [8, 11>

Para

7 -x + 1 < 10 --> 6 -x < 9 --> -6 x > -9

2da Solucin: x -1 ^ < 0 --> 0 ^ < 0 --> 0

^ < 0

0 (7 +5y)(2y) ^ ( (2y) < 0 --> x (< , -7/5] U ) --> x 0 z < 1 --> 0 2z z = 0, .

10) [|2x|] - |x 1| = 2x 3



4/6

Con x 1

[|2x|] x + 1 = 2x 3 --> [|2x|] = 3x 4

3x 4 Z ^ 3x - 4 2x < 3x -3 --> -4 -x < -3 --> 12 3x > 9

3x = 12, 11, 10 --> x = 4, 11/3, 10/3. (Solucin Parcial)

Con x [|2x|] = x 2

x 2 Z ^ x 2 2x < x 1 --> -2 x < -1

x = -2 (Solucin Parcial)

Solucin General: x = -2, 10/3, 11/3, 4.

11) [| x2 -2x|] = 3

3 x2 -2x < 4 --> 4 x2 -2x + 1 4 (x -1)2 2 x - 1< 5

a) 3 x < 1 + 5 ---> x [3, 1 + 5>

b) -1 x > 1 - 5 --> x U 0 - |x| < 1 -->0 |x| > -1

a) 0 x > -1 --> x x [0, 1>

Solucin General: x ()

13) | [|x2|] 1| = 3


5/6

[|x2|] 1 = 3 --> [|x2|] = -2 ^ [|x2|] = 4 --> -2 x2 -2

x2 2 x < 5

a) 2 x < 5 --> x [2, 5>

b) -2 x > - 5 x 9 3x + 10 < 10 ^ -9 3x + 10 -1 3x< 0 ^-19 3x < -18 --> -1/3 x< 0 ^-19/3 x< -6 --> x ([-

19/3, -6> U [-1/3, 0>)

15) [|3x|] = 5x+3

5x+3 Z ^ 5x +3 3x < 5x+4 --> 3 -2x < 4 --> -3 2x > -4 --> -3/2 x

>-2 --> -15/2 5x > -10

5x = -8, -9 --> x= -8/5, -9/5

16) [| n2 +4n|] = 4

4 n2 + 4n < 5 --> 8 n2 + 4n + 4 < 9 -->8 (n+2)2 < 9:

a) 2 n+2 2 - 2 n x [2 2, 1>

b) -2 n+2 >-3 --> -2 -2 n+2 >-5 --> x -3 -2x < -2 --> 3/2 x > 1 --> 9/2

3x > 3

3x = 4 --> x= 4/3

18) [|2x|] = 5 4x


6/6

x + 5 Z ^ 5 4x 2x < 6 4x --> 5 6x < 6 --> 5/6 x < 1 --> 20/6

4x < 4 -->-20/6 4x > -4

Analizando: -20/6 4x > -4 no existe en R--> x =

19) |[|x/2|]| =

Calculando el universo:

x 1 0 --> x 1 --> x [1, >

En la ecuacin:

[|x/2|]2 = x 11 --> [|x2|]/4 = x 1 --> 2x 2 x2 < 2x -1 --> -2 x2 -2x < -1

--> -1 (x -1)2 < 0

Analizando: -1 (x -1)2 < 0 no pertenece a los R. --> x =

Intersectando al Universo: x ([1, > ) =

20) [|3n -3|] = 8n +5

8n +5 R ^ 8n + 5 3n 3 < 8n + 6 --> 5 -5n -3 < 6 --> 8 -5n < 9

--> -8 5n > -9 --> -8/5 n > -9/5 --> -64/5 8n >-72/5

8n = -13, -14 --> n = -13/8, -7/4

h) ecuaciones con mÁximo entero

Documents