# Απόλυτη Τιμή Πραγματικού Αριθμού

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• pi: pi 12

• pi: pi 13

III. pi pi

/: :

:

pi pi pi pi 0.

pi .

pi .

pi pipi pi pi pi pi pi .

pi , 0 pi pi 0 1 , :

pi pi pi 0 pi _____.

pi pi pi _____ .

pi pi pi pi _____ .

pi

:

pi pi x pi 0, pi x d(x,0) | x | .

pi pi x y, | x y | d(x,y) .

3| 2 | , | 2 | , | 0 | , | 3 2 | ,2

+ = = = = =

pi pipi pi pi :

• pi: pi 14

pi , pi ________

pi , pi ________

pi pi , pi .

pi , ________

pipi, pi pi pi .

pi

x pi ( x pi , 0).

x:

:

1. pi pipi , pi pi:

x 0 pi |x| :

x 0< pi |x| :

2. pi pi pi pi |x|:

x 0

x 0 +

pi

1 : pi pi

. x : 2 2| x | x= .

pi:

x 0 | x | x= pi 2| x | x= (pi) x 0< | x | x= pi 2 2 2| x | ( x) x= = (pi)

1. x : | x | x | x |

pi: x 0 | x | x= pi x x x pi

.

x 0< | x | x= pi pi .

. | x y | | x | | y | = x,y .

pi:

pi :

2 2 2 2 2 2| xy | (xy) x y | x | | y |= = = | xy | 0 : | x y | | x | | y | =

: pi pi pi- x y pipi pi (: ( , ),( , ),( , ),( , )+ + + + ) - pi.

• pi: pi 16

. x | x |

,x,y ,y 0y | y |=

pi:

.

. x, y : | x y | | x | | y |+ + ( ) (1)

pi:

x y 0= = (1) . pi x y , ( pi ), pi pi . pi:

pi:

( )22 2 2 22

| x y | | x | | y | (x y) | x | 2 | x || y | | y |x

+ + + + +

22xy y+ + + 2x 22 | xy | y+ + 2 xy 2 | xy |

: xy | xy | pi 1.

:

. (1), , : | xy | xy= pi xy 0 x y .

. (1) pi pi : | x y | | x | | y | + 1.

. :

| x | | y | | x y | | x | | y |,x,y +

pi .

2 : pi -

pi pi pi pi pi- pi pi pi.

. 0 x : | x | x x = = =

pi:

:

1 : pi (+) .

• pi: pi 17

2 2 2 2| x | x 0 (x )(x ) 0= = + = pi pipi: x 0 x + = = x 0 x = = pi pi.

:

. 0< | x | = . pi 2| x | 1, | x 3 | 5, | x | ( 1),= = = + .

. pi pi pi ;

. : | x | | | x x = = =

pi:

.

. 0 : | x | x

pi:

, pi pi :

2 2 2 2| x | x 0 (x )(x ) 0 +

pi pipi pi x ,x + - pi pi:

1 2x 0 x 0( ) ( )x 0 x 0

+ +

1 2 pi.

:

. 0< | x | . pi : 2| x | 1, | x 3 | 5, | x | ( 1),< + .

. pi pi pi ;

. 0 : | x | x x .

pi:

• pi: pi 18

pi .

:

. 0< | x | | x | 0 . pi : 2| x | 1, | x 3 | 5, | x | ( 1),> > + , x .

. pi pi pi ;

. pi 1 2 x ,x ,...,x :

| | | | ... | | ...1 2 1 20 0x x x x x x + + = = = = =+ + = = = = =+ + = = = = =+ + = = = = =

pi:

1 2| | | | ... | | 0vx x x+ + = . pi pi 0, pipi : 1 2 ... 0vx x x= = = = .

, 1 2 ... 0vx x x= = = = 1 2| | | | ... | | 0vx x x+ + = .

:

( , , , )

1. pi :

-

| x 1| 3+ =

| x 1| 3+ =

| x 1| | 1|,+ = +

pi

0 : | x | x x = = =

• pi: pi 19

| x 1| 3+ <

| x 1| 3+

| x 1| 3+ <

| x 1| 3+

2. : 1 |1 3x | 4< (1).

:

(1) pi pi : 1 |1 3x |< (1) |1 3x | 4 (2)

pi . :

(1) 1

2x1 3x 1 3x 2 3

|1 3x | 1 ( )1 3x 1 3x 0 x 0

> < <

> > >

• pi: pi 20

pipi, pi pi pi pi :

1 : x 3< (1) : (1) (x 3) (x 2) x 5 x x 5 + + = = = pi pipi x 3< .

2 : 3 x 0 (1) : 1(1) (x 3) (x 2) x 2x 1 x 3x 1 x3

+ + = + = = = 3 x 0 .

3 : 0 x 2< (1) : (1) (x 3) (x 2) x 2x 1 x x 1 + + = + = = pi pipi 0 x 2< .

4 : x 2> (1) : (1) (x 3) (x 2) x 5 x x 5 + = = = pi x 2> .

pi pi

1. pi : | |=| |:

. = . = -

. = = - . .

2. > : . > . > > > 0 . < . > 0 > > .

3. | x | | x 1| 0+ = :

. . . pi .

4. pi x - :

. | x 2 | | x 1| > + . | x 2 | | x 1| < + . | x 2 | | x 1|+ > . | x 2 | | x 1|+ <

10. | x | 1 | y | 2 pi | 2x y | :

. 0 . 4 . -1 .

pi

1. 0< < < : | | | | | 2 | + = .

2. : . | x 1| 6 = . 6 1| x 1| =

3. : . | x 1| 6 . 6 1| x 1|

4. - : 1 |1 3x | 5< .

5. : | x 2 | | x | | 2x 1| + = + .

6. . x : x | x | x | x |+ = x 0= -.

• pi: pi 21

5. | - 3 | + | 2 - 4 | + | -1| 0 :

. - 3,2 - 4, -1: . - 3,2 - 4, -1: . - 3,2 - 4, -1: . - 3 = 2 - 4 = -1= 0 .

6. | + |=| | + | | , :

. . . . .

7. | 2x 1| 0x

+< :

. x 0> . x 0<

. 1

x 02

< . 1

x2

< .

8. d(2x 1,0) 1 :

. x 0 x 1 . 0 x 1 . x 1 . x 0

9. ( )d 1, x 2 3 + > : . 6 x 0 < < . x 0>

. x 6< . x 0 x 6> <

. pi : 2 1 | 2 1| 2 1 | 2 1| + = .

7. . pi 0

: 1 1| | .

. pi- :

3 2 5| | | | .

8. 1 1| | , | |2 2

< < :

| | | 1 |+ < + .

9. pi , , . . pi :

( )1 1 1 | | | | | | 9| | | | | |

+ + + +

. | | | | | | 3+ + = - pi

1 1 1| | | | | |

+ +

.

10. .

2 2| 2 7 | | 2 2 | 5+ + = + + +

. :

2 22 | x 2x 7 | | x 2x 2 | 10+ + = + + +

11. . 2x x 1 0 + > x.

. : 2 2| x x 1| | x 2x 3 | 1 + + + = .

12. . :

2 | | ,=

. : (x 3) | 3 x | (3 x) | x 3 | 2 + = .

13. x, y, z pi , :

x y zy z = =

:

• pi: pi 22

| x | 3 | y z | .

pi & pi pi

1. 0, 0, 2 0 < < <

2. (). : . x 7 x 5= = . pi- x 1 .

3. (). : . 5 x 7 . pi x 1 .

4. () (). : x 1,1,2= .

5. pi 3 pi: 12

= .

6. . -. . , x 2-1= .

7. . x: | x | x | x | . . pi :.

8. pi pi : 1 1,2 2

< < .

.

9. . pi pi : 1| | 2| |+ . . pi.

10. .

2 2

2 2 2

( ||)

| 2 2 | 5 | ( 1) 1| 5

( 1) 1 5 2 7 | 2 7 |+

+

+ + + = + + + =

+ + + = + + = + +

.

. pi . .

11. . 24(x x 1) 0 + > . . -

2 2| x x 1| | (x 1) 2 | 1 + + + = x = -1.

12. A. pipi . . :

22 (x 3) | x 3 | 2 (x 3) 1 = = pi - x 4 x 2= = .

13. x y z 0y z = = = > -

.