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 = = = > -

    .

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