Algebra (Άλγεβρα Α Λυκείου)
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Σημειώσεις Άλγεβρα Α Λυκείου Φροντιστήριο "Το Μάθημα" 0. Λογική 1. ΠιΘανότητες 2. Οι Πραγματικοί Αριθμοί 3. Εξισώσεις 4. Ανισώσεις 5. Πρόοδοι - Ακολουθίες 6. Βασικές Έννοιες Συναρτήσεων 7. Μελέτη Βασικών Συναρτήσεων 8. Τράπεζα Θεμάτων Άλγεβρας 2014
Transcript of Algebra (Άλγεβρα Α Λυκείου)
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PQ ( P) Q
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3 16 . . . . . . . . . . . . .
4 P (Q R) (P Q) (P R). . . . . . . . . . . . . . . . . . . . 5 ( P) P. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 :
2
= 9 = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 :
2 6= 4 6= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 : x(x 1) = 0 x = 0 x = 1 . . . . . . . . . . . . . . . . . . . . .
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0.2
Cantor
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1. .
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= {1 3 5 7 9} = {1 2 3 100}
=
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1
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0.2
1 2
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2 . . . . . . . . . . . . . . . . . . . .
3 {} = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A B , A B + . . . . . . . . . . . . . . . . . . . . . . . . . .
6 A = A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 A = A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 (A ) (A ) = A. . . . . . . . . . . . . . . . . . . . . . . . . 9 (A ) = A ( ). . . . . . . . . . . . . . . . . . . . . . . . 10 (A ) = A . . . . . . . . . . . . . . . . . . . . . . . 11 (A ) = A . . . . . . . . . . . . . . . . . . . . . . .
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2. .
3. .
4. 10
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5. 50
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(sample spae).
, {1
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1. B " " " "
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2. B " " " "
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3. B " " -
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2 . .
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4 . . .
5
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8 , (A B) (A B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 , A B A. . . . . . . . . . 10 , A A B. . . . . . . . .
1.1 -
1.1.1 4 , 3
2 .
;
1.1.1 4 , 3
2 . , 4 3 2 = 24 .
1.1.2 -
;
1.1.2 24 -
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24 24 24 = 13824 .
1.1.3
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1.1.3 , :
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2 2 2 = 23 = 8 . :
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+
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1.1.5
: 7.
: 3.
: .
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1.1.5 :
= {11 12 13 14 15 1621 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66}
) :
A = {61 52 43 34 25 16}
) :
B = {13 1623 26
31 32 33 34 35 36
43 46
53 56
61 62 63 64 65 66}
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" " 11
) :
= {12 13 14 15 1623 24 25 26
34 35 36
45 46
56}
= {61 43 34 16} = {34 25 16}
= {12 13 14 15 1623 24 25 26
34 35 36
45 46
56
61 52 43}
1.1.6 .
:
1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
8. .
1.1.6 :
1: A B
2: A B
3: A B 4: A B 5: (A B) (A ) (B )6: (A B ) (A B ) (A B )7: (A B ) (A B ) (A B )8: A
B
9: (A B )
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12 " "
1.1.7 ,
1,2 3. .
1.1.7 :
1 { | | }
1 { | | }
1 { | | } 1 2 { | | } 2 3 { | | } 3
2 { | | }
1 { | | } 4 2 { | | } 5 3 { | | } 6
3 { | | }
1 { | | } 7 2 { | | } 8 3 { | | } 9
2 { | | }
1 { | | }
1 { | | } 10 2 { | | } 11 3 { | | } 12
2 { | | }
1 { | | } 13 2 { | | } 14 3 { | | } 15
3 { | | }
1 { | | } 16 2 { | | } 17 3 { | | } 18
3 { | | }
1 { | | }
1 { | | } 19 2 { | | } 20 3 { | | } 21
2 { | | }
1 { | | } 22 2 { | | } 23 3 { | | } 24
3 { | | }
1 { | | } 25 2 { | | } 26 3 { | | } 27
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-
" " 13
1.1 -
1.1.8 . -
Venn
:
i) ,
ii)
iii)
iv)
v) ,
1.1.9 .
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1.1.10 (), ()
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ii) :
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1.1.11
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14 " "
1.1.12 . -
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i)
ii) A B Biii) iv) (A B) ( B)v) () ()vi) Bvii) A
viii)
1.1.13 , ,
(), (),
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1.1.14
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1.1.15 (),
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iv) .
1.1.16 , .
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1.1.17
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i) 30 , 20 15 . -
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ii) 30 , 40% 50%
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A :
B :
1.1.18 -
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() ().
1 () 2 ()
3 . .
1.1.19 , (),
() (). () (),
() () () .
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1.1.20 ()
() .
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i) = ii) = B iii) AB = iv)
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-
" " 17
1.2
-
. 25
. ()
() = 7 () = 14 () = 25
: .
25 ,
P(A) =
N(A)
N()
=
7
25
P(B) =
N(B)
N()
=
14
25
,
Laplae 1812.
,
, P(A) A
:
P(A) =
=
()
()
,
.
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.
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18 " "
1: P() =
N()
N()
2: P() = 0N()
= 0
3: P(A) =
N(A)
N()
0 P(A) 1
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1).
:
= {1
2
} .
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-
" " 19
{i
} , - P(
i
), :
0 P(i
) 1P(
1
) + P(
2
) + +P(
) = 1
P(
i
) {i
}. = {
1
2
} 6= P(1
) + P(
2
) + + P(
),
P() = 0.
.
1: B P(A) P(B)
2: P(A
) = 1P(A)
3: P(A B) = P(A) + P(B)P(A B)
3 P(AB) = P(A)+P(B), :
P(A B ) = P(A) + P(B) + P()P(A B)P(A )P(B ) + P(A B )
1.2
1 ,
P(A B) = P(A) + P(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ,
P(A B) + P(A B) = P(A) + P(B) . . . . . . . . . . . . . . . . . . . . . . . . 3 () + () = 1 . . . .
4 () = 1 = , . . .
5 () = 0 = , . . . . 6 () = 1 . . . . . .
.
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20 " "
7 . . P() = 0. . . 8 0 < P(A) < 1. . . .
9 : P(A) =
3
4
P(A B) = 12
. . . . . . .
10 , -
: P(A) =
()
()
. . . . . .
11 , :
P(AB) = P(A)P(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 , :
P(A B)P(A B) = P(A) + P(B) . . . . . . . . . . . . . . . . . . . . . . . . 13
: P(A) + P(A
) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 ,
P(A) = P(A ) P(A ) . . . . . . . . . . . . . . . . . . . . . . .
1.2
1.2.1 1 100.
9.
1.2.1
9.
A ={9 19 29 39 49 59 69 79 8990 91 92 93 94 95 96 97 98 99}
={1 2 3 4 100}
P(A) =
N(A)
N()
=
19
100
= 019
1.2.2
P(A) =
3
4
P(A B) = 910
P(A B) = 920
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-
" " 21
:
) .
) .
) .
1.2.2 P(A B) = P(A) + P(B)P(A B)
) P(B) = P(A B) + P(A B)P(A)
=
9
10
+
9
20
34
=
3
5
) P( B) = P(B)P(A B)
=
3
5
920
=
3
20
) P( B) = 1P(A B)
= 1 910
=
1
10
1.2.3
:
A : . ;
B : . ;
: .
;
: .
;
1.2.3
1.
1 52.
= {1 2 3 4 52}
4
P(A) =
4
52
=
1
13
2. 13
P() =
13
52
=
1
4
3. ' 52 51 .
.
.
.
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22 " "
i. ,
4 51 = 204 , ii.
, (52 4) 4 = 192 .
P() =
4 51 + (52 4) 452 51 =
204 + 192
52 51 =396
2652
015
4. , 5251 .
i. ,
13 51 = 663 , ii.
, (52 13) 13 = 507 .
P() =
13 51 + (52 13) 1352 51 =
663 + 507
52 51 =1170
2652
044
1.2.4 = {0 1 2 3 20} :
: 2.
: 5.
: 2 5.
1.2.4 :
A = {0 2 4 6 8 10 12 14 16 18 20} P(A) = N(A)N()
=
11
21
= {0 5 10 15 20} P() = N()N()
=
5
21
= {0 10 20} P() = N()N()
=
3
21
1.2.5
.
.
1.2.5 : ,
:
. :{P(A
) = 1P(A)
P(A
) = 2P(A)
1P(A) = 2P(A) 3P(A) = 1 P(A) = 13
1.2.6 .
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:
) P(A B) ) P(A B) ) P(A B)
1.2.6 :
= {11 12 13 14 15 1621 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66}
A = {12 14 1621 23 25
32 34 36
41 43 45
52 54 56
61 63 65 }
B = {11 12 13 14 15 1621 31 41 51 61}
:
) A B = {12 14 16 21 41 61}
P(A B) = 636
=
1
6
) A B = {11 12 13 14 15 1621 23 25
31 32 34 36
41 43 45
51 52 54 56
61 63 65}
P(A B) = 2336
.
.
.
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24 " "
) A B = {23 25 32 34 36 43}45 52 54 56 63 65}
P(A B) = 1236
=
1
3
1.2.7 ,
1,2 3. :
A: .
B: .
1.2.7
. ,
.
= {2 3 4 5 7 910 11 13 15 17 18
19 21 23 24 25 26}
P(A) =
18
27
:
B = {6 8 12 16 20 22}
P(B) =
6
27
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1.2
1.2.8 . -
:
:
:
: 4
1.2.9 52 .
:
:
:
:
:
1.2.10 4 , 10 6 .
, :
:
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1.2.11 3 1,2 3.
, . -
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i) N .
ii) :
: 2
: 2
1.2.12 .
(
) = () (
) = () R {1}
.
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26 " "
() = () =
1
+ 1
1.2.13 52 .
i) . -
:
:
:
: 2 3
:
ii) , .
.
:
A: , ,
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1.2.14 = {1
2
} (
) = x, = 1 2 , :i) x ,
ii) (
) .
1.2.15 30 , 25 , 5
DVD 4 DVD. . -
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A:
B: DVD
:
:
1.2.16 12 18 .
40 .
. :
: 40
: 40
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1.2.17 ( 2001) 400
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240 . .
.
i) N
ii) :
P() 35
iii) .
iv)
.
1.2.18 50
. :
0 1 2 3 4
5
6 14 13 9
5
3
50 .
:
A:
: 3
: 3
: 3 4
: 2 4
1.2.19 10000 , 90%.
, 90% .
:
A:
: ,
:
1.2.20 160 ,
, 84 (), 66 () 36 ().
, 12
x
.
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28 " "
i) N Venn.
ii)
iii) , :
A
1
:
A
2
:
A
3
:
A
4
:
A
5
:
A
6
:
1.2.21 50 1 50.
. :
: 6 4
: 4 6
1.2.22 , , .
:
i) P(A B) P(A)ii) P(A B) P(A B)
1.2.23 .
:
i)P(A B) P(A) + P(B)ii)P(A B ) P(A) + P(B) + P()
1.2.24 ( 1994) (
) 028 () 071. :
i) P(A B) 101P(A B)ii) .
1.2.25 () = 06
() = 08.
i) N .
ii) :
) ( ) 06 ) ( ) 08 ) ( ) 04
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" " 29
1.2.26 ( 2003) 55% -
, 40% 30% .
.
:
i)
ii)
iii)
iv)
1.2.27 , . -
:
i) P(A B) P(A) + P(B) 1ii) P(A B ) P(A) + P(B) + P() 2
.
.
.
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30 " "
2
2.1
,
. ,
,
,
.
(-
) .
+ = + = + ( + ) = ( + ) + ( ) = ( )
+ 0 = 1 = a - + () = 0 1
= 1 6= 0 (+) = +
0 , 0
.
:
= + ()
= 1
6= 0
, , :
1.
=
=
} + = +
2.
=
=
} =
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" " 31
3.
= + = +
4.
6= 0 = =
5. 0, -
0.
= 0 = 0 = 0
=
> 1
= 1
6= 0
0
= 1
=
1
=
=
, .
(2)2 = 22 2 6= 2
:
1
= +
2
=
3 (
)
=
4
= ( )
5
=
(
)
.
.
.
" "
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32 " "
-
,
:
( + )
2
=
2
+ 2 +
2
( )2 = 2 2 + 2
( + )
3
=
3
+ 3
2
+ 3
2
+
3
( )3 = 3 32 + 32 3
2 2 = ( )( + )
3 3 = ( )(2 + + 2)
3
+
3
= ( + )(
2 + 2)
( + + )
2
=
2
+
2
+
2
+ 2 + 2 + 2
1. :
.
2. :
.
" "
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.
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-
" " 33
2.1
1 12011 = 12013 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1000
0
= 10
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 ( )2 = ( )2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ( )3 = ( )3 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ,
2
. . . . . . . . . . . . . . . . . . . . .
6 . . . . . . . . . . . . .
2.1
2.1.1
= 13
=
2
3
:
(i) 322 (ii) 322 (iii)52
3
2.1.1 :
i) 3(13
)2
(2
3
)2
= 319
4
9
= 427
ii) 3(13
)2(2
3
)2
= 3(3)2(4
9
)= 3 9 4
9
= 12
iii)
5
(2
3
)2
(1
3
)3 = 5 4
9
(3)3 =20
33 32 = 20
3
5
= 20243
2.1.2
8 125 (27) 64 (64) (4) 16(18
)(27)
( 1125
)
.
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34 " "
2.1.2 :
) 8 125 = 23 53 = (2 5)3 = 103
) (27) 64 = (3)3 43 = (3 4)3 = (12)3
) (64) (4) 16 = (82)(22) 16 = 82 22 16 = (8 2)2 16 = 163
)
(18
)(27)
( 1125
)=
( 12
3
)(33)
( 15
3
)
=
(12
)3
(3)3(15
)3
=
( 310
)3
2.1.3
2
0
3
8
3 (
)
1
2.1.3 :
)
2
0
3
8 = 2+03+8
=
3
)
3 = 3
)
(
)
1
=
=
(
)
2.1.4
=
x
4y
2
(x
1y
2)
4
(x
2y)
1
(x
2
y)
2y
3
x = (10)5 y = 104.
2.1.4 :
=
x
4y
2
(x
1y
2)
4
(x
2y)
1
(x
2
y)
2y
3
=
x
4y
2
(x
4y
8)(x
2
y1)(x
4y
2)y
3
=
x
44+2y
281
x
4y
23
=
x
6y
7
x
4y
5 = x2y
2
x = (10)5 y = 104
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.: 210.81.37.280
.
.
.
-
" " 35
=
(
(10)5)
2(
104)
2
= (1)2(
(10)
5)
2(1)2
(
10
4
)
2
= 10
10 108 = 100
2.1.5 :
3 3x + 7y
2 x2 y2 + + + x y x2 1
2.1.5 .
:
( 3)2 = 2 2 3 + (3)2
=
2 6 + 92
(3x + 7y)
2
= (3x)
2
+ 2(3x)(7y) + (7y)
2
= 9x
2
+ 42xy + 49y
2
( )2 = ()2 2() + ()2
=
2
2 2 + 2
(
2 x2 y2)2 = (2)2 + (x2)2 + (y2)2 + 22(x2) + 22(y2) + 2(x2)(y2)=
4
+ x
4
+ y
4 22x2 22y2 + 2x2y2
( + + )
2
= ()
2
+ ()
2
+ ()
2
+ 2()() + 2()() + 2()()
=
2
2
+
2
2
+
2
2
+ 2
2
+ 2
2
+ 2
2
( + x y)2 = ( )2 + 2( )(x y) + (x y)2
=
2 2 + 2 + 2(x y x + y) + x2 2xy + y2
=
2 2 + 2 + 2x 2y 2x + 2y + x2 2xy + y2
=
2
+
2
+ x
2
+ y
2 2 + 2x 2y 2x + 2y 2xy
(x
2 1)2 = (x2)2 2(x2) + 12
= x
4 2x2 + 1
.
.
.
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.: 210.81.37.280
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36 " "
2.1.6 :
i) x y + x yii) x
3 x2 x + 1
2.1.6 i)
x y + x y= (x y) + (x y)= (x y) + (x y)= (x y)( + )
ii) :
x
3 x2 x + 1= (x
3 x2) (x 1)= x
2
(x 1) (x 1)= (x 1)(x2 1)= (x 1)(x + 1)(x 1)= (x 1)2(x + 1)
2.1.7 ,,
6= 0 1
+
1
+
1
= 0
( + + )
2
=
2
+
2
+
2
2.1.7
1
+
1
+
1
= 0(1
+
1
+
1
) = 0
+ + = 0
( + + )
2
=
2
+
2
+
2
+ 2( + + ) =
=
2
+
2
+
2
+ 2 0=
2
+
2
+
2
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.: 210.81.37.280
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.
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" " 37
2.1
2.1.8 :
A =
3
2
5
6
B =
3
7
4
=
5
2
9
=
2.1.9 :
1)
1
+
1
+
1
2)
3 xx 2
x
2
+ 4
2x
3 8x +x + 3
x + 2
xx
2 4
2.1.10
1)
x 2x
+
4
x 2 8
x
2 2x 2)3
x + 2y
2x 2y +
2x + 16y
x
2 4y2
3)
y
2 6y
2 5y + 6 2
y 2 +3
y 3 4)x
2
x y +y
2
x + y
2xy2
x
2 y2
2.1.11 :
A =
[(x 1)2(x 1)4
(x 1)3]2
B =
7x
3
y
1 x3y1y
3
x
5
2.1.12 :
1)
1
1 x +1
1 + x
+
2
1 + x
2
2)
x
1 +
x
2 +
x
3
2.1.13 :
+ x 2 +
x
3
+ 2
2 y2
.
.
.
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.: 210.81.37.280
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38 " "
2.1.14
= + 5 + 9 = = 5(() + ()
)+ 4()
.
2.1.15 :
i) (7 + 3)( 1)(2 + 1) = 0ii) ( + 1)(2 + 5)( 3) 6= 0
2.1.16 :
(
2 2)2 + (2)2 = (2 + 2)2
( + )
2 ( )2 = 4 Legendre( + )
3
( ) (4 4) = 2(2 2)( + )
3 3( + 1) 1 = ( + 1)(2 + 2 + + + 1)( )2 + ( )2 + ( )2 = 2( )( ) + 2( )( ) + 2( )( )
2.1.17 :
i) 9
2
+ 12 + 4
2
ii) x
2 x + 14
iii) 3x
2
+ 6xy + 3y
2
2.1.18 :
i)
2 2522 ii) x3 125 iii) y3 27 iv) x2
16
y2
25
2.1.19 :
( + )
4
=
4
+ 4
3
+ 6
2
2
+ 4
3
+
4
( + )
5
=
5
+ 5
4
+ 10
3
2
+ 10
2
3
+ 5
4
+
5
( + + )
3
=
3
+
3
+
3
+ 3( + )( + )( + )
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.: 210.81.37.280
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.
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-
" " 39
2.1.20
i) x
2 xy + 4y 4x ii) x3 + x2 + 3x + 3 iii) 1 x2 2xy y2
2.1.21
i)
4
+
4 1122 ii) 4 + 44 iii) 8 8
2.1.22
x +
1
x
= 2
i) x
2
+
1
x
2
ii) x
3
+
1
x
3
iii) x
4
+
1
x
4
2.1.23 (
) :
i)
x + y + x + y
x
2 y2 ii)36x
2 12x + 148x 8
iii)
16
2
3 8334
2
2
iv)
(x + h)
2
h
v)
x
2 1(1 x)2 vi)
x
2 9y22x
2 12xy + 18y2
2.1.24 Newton:
(x + )(x + )(x + ) = x
3
+ ( + + )x
2
+ ( + + )x +
(x + )(x + )(x + )(x + ) =
= x
4
+ ( + + + )x
3
+ ( + + + + + )x
2
+ ( + + + ) +
= = = ;
.
.
.
" "
3
.: 210.81.37.280
-
40 " "
2.2
, > ,
. < .
-
, :
( > 0 > 0) + > 0( < 0 < 0) + < 0 > 0
> 0
< 0
< 0
, :
1.
>
>
} + > +
2. ,
>
>
} =
3.
> + > +
4.
> 0 > >
5.
> 0 > <
6. ,
> >
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.: 210.81.37.280
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.
.
-
" " 41
2.2
1 < , < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 x > y, yx > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 > , + 3 > + 3 . . . . . . . . . . . . . . . . . . . . . . . . .
4 x < y, x 3 > y 3 . . . . . . . . . . . . . . . . . . . . . . . . . 5 > , 2 > 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
6 x > y,
x
2 >y
2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 < 1,
2
< . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 x
2
> 2x, x > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
< 1,
> 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 < 1 < (1 )(1 )( ) > 0 . . . . . . . . . . . . . . . .
2.2
2.2.1 > 0
+
1
2
2.2.1 , :( +
1
)2
22
2
+ 2
1
+
1
2
4 2 2 + 1
2
0
( 1
)2
0
2.2.2 ,, :
2
+
2
+
2 + +
.
.
.
" "
3
.: 210.81.37.280
-
42 " "
2.2.2
2
+
2
+
2 + + 22 + 22 + 22 2 + 2 + 2
2 2 + 2 + 2 2 + 2 + 2 2 + 2 0( )2 + ( )2 + ( )2 0
2.2.3 x y :
(x + y)(
1
x
+
1
y
) 4
2.2.3
(x + y)(
1
x
+
1
y
) 4 (x + y)x + yxy
4
(x + y)
2 4xy x2 + 2xy + y2 4xy x2 2xy + y2 0(x y)2 0
2.2.4 ,, 0 :
i) ( + )
2 4 ii) ( + )( + )( + ) 8
2.2.4 i)
( + )
2 4 2 + 2 + 2 4
2 2 + 2 0 ( )2 0
ii) i) ( + )
2 4( + )
2 4( + )
2 4 ( + )2( + )2( + )2 64222
( + )( + )( + ) 8
2.2.5 + 0 :
3
+
3 2 + 2
2.2.5 , :
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.: 210.81.37.280
.
.
.
-
" " 43
3
+
3 2 + 2 (3 2) + (3 2) 0 2( ) 2( ) 0 ( )(2 2) 0 ( )( + )( ) 0 ( )2( + ) 0
2.2.6 + = 2, :
i) 1 ii) 2 + 2 2
2.2.6 + = 2 = 2 , i :
1 (2 ) 1 2 2 1 2 2 + 1 0 ( 1)2 0
ii) :
2
+
2 2 2 + (2 )2 2 2 + 4 4 + 2 2 22 4 + 2 0 2 2 + 1 0 ( 1)2 0
.
.
.
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.: 210.81.37.280
-
44 " "
2.2
2.2.7 > 3, :
1) 6 + 2 > 3 + 2) 4 < 3 + 2
2.2.8 > 2, :
1) + 3 > 5 2) 2 + 4 > 8
3) 3 + 6 < 0 4) 2
1 > 0
2.2.9 < 4, :
1) 2 8 32
< 2)
32
2 96
>
23
2.2.10 < , :
1) 5 4 > 5 42)
3
7 < 3
7
3) 9 2
> 9 2
4) 2 > 2
2.2.11 > 1 > , :
+ > 1 +
2.2.12 < 2 < , :
2( + ) > 4 +
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.: 210.81.37.280
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.
.
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" " 45
2.2.13 1, :
3
+ 1 2 +
2.2.14 :
1) 3(
2 2) + 2 2( + 2)2
2) 2(
2
+
2
) (b2 a2) 2(3 )
2.2.15 < , :
1) 5. . . . . . . . . . . . . . . . . .
2.3
2.3.1
A =
3 3 | 3|
2.3.1
3 < 3
3 3 < 0 3 3 = 33
> 3 3 > 0 | 3| = 3
=
3 3 | 3| = 33 + 3 = 3
2.3.2 :
A =
4x
2 4x + 1 B = x6 6x3 + 13
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.: 210.81.37.280
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.
.
-
" " 49
2.3.2
4x2 4x + 1 = (2x 1)2 0 =
4x
2 4x + 1 = 4x2 4x + 1 x6 6x3 + 13 = (x6 6x3 + 9) + 4 = (x3 3)2 + 4 0
B =
x
6 6x3 + 13 = x6 6x3 + 13
2.3.3 :
A =
x4 x2 3 B = x2 + 2x 5
2.3.3
x4 x2 3 = (x4 + x2 + 3) < 0 =
x4 x2 3 = x4 + x2 + 3 x2 + 2x 5 = (x2 2x + 1 + 4) =
(
(x 1)2 + 4)
< 0
B =
x2 + 2x 5 = x2 2x + 5
2.3.4 :
A =
4 | x2 + 2x 1| |x2 + 4| + 4x B = 2x+3 2x + x + |x| |x| x
2.3.4
x2 + 2x 1 = (x 1)2 < 0 4 | x2 + 2x 1| = 4 + (x 1)2 = 4 + (x 1)2 = x2 2x + 5
|x2 + 4| + 4x = |x2 + 4 + 4x| = (x + 2)2 = (x + 2)2 = x
2 2x + 5 (x2 + 4x + 4) = 6x + 1
:
2x+3 2x = 2x(23 1) = 7 2x > 0 |x| x |x| + x 0 |x| x |x| x 0
= 7 2x + (|x| + x) (|x| x) = 7 2x + 2x
.
.
.
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.: 210.81.37.280
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50 " "
2.3.5 :
A = |x 1| + |x 3| 1 < x < 3
B = | | + |2 2| | + 2| + +
2
< <
2.3.5
x > 1 x 1 > 0 |x 1| = x 1 x < 3 x 3 < 0 |x 3| = 3 x
= (x 1) + (3 x) = 2 :
< < 0 | | = + > > 0 2 2 > 0 |2 2| = 2 2
{ >
>
+ > 2 + 2 > 0 | + 2| = + 2
< + 2
+ 2
< 0 +
2
= + 2
= ( + ) + (2 2) ( + 2) +( +
2
)
= + + 2 2 + 2 + + 2
= 2
32
+
2.3.6 :
A = 2 + x |x 1| x R B = 2x |x 2| + |x + 1| x R
2.3.6 , -
. :
x 1 x 1 0 |x 1| = x 1 x < 1 x 1 < 0 |x 1| = 1 x
=
{2 + x (x 1) x 12 + x (1 x) x < 1 =
{3 x 12x + 1 x < 1
,
:
x
-1 2
x+1
-
0
+ + +
x-2
- - -
0
+
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.: 210.81.37.280
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.
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" " 51
:
B =
2x (2 x) + (x 1) x < 12x (2 x) + (x + 1) 1 x < 22x (x 2) + (x + 1) x 2
=
2x 3 x < 14x 1 1 x < 22x + 3 x 2
2.3.7 :
A =
x4 8xx 2
+x7 + x5x
5
+ x
3
x > 0 x 6= 2
2.3.7 , :
x4 8xx 2
=x(x3 8)
x 2
=x(x 2)(x2 + 2x + 4)
x 2
= |x| |x2 + 2x + 4| = x(x2 + 2x + 4)
x7 + x5x
5
+ x
3
=5(x2 + 1)x
3
(x
2
+ 1)
= |x2| = x2
= x(x
2
+ 2x + 4) + x
2
= x
3
+ 3x
2
+ 4x x > 0 x 6= 2
2.3.8 :
| | | 5| + | 5|
2.3.8 |x y| |x| + |y| :
| | = |( 5) ( 5)| | 5| + | 5|
2.3.9 :
| 3|2 + |3 + |2 = 10(||2 + ||2)
2.3.9 :
| 3|2 + |3 + |2 = ( 3)2 + (3 + )2 = 2 6 + 92 + 92 + 6 + 2
= 10
2
+ 10
2
= 10||2 + 10||2 = 10(||2 + ||2)
2.3.10 6= 0 | + ||| = || + || 0.
.
.
.
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52 " "
2.3.10 , :
2
+ 2|| + ||2 = ||2 + 2|||| + ||2 2|| = 2|||| = || 0
2.3.11 6= 0 + 1
2
2.3.11 , :
+ 1
2
22 ( +
1
)2
4
2
+ 2
1
+
1
2
4 2 2 + 1
2
0
( 1
)2
0
2.3.12 d(2) > d(2) || < ||.
2.3.12
d(2) > d(2) | 2| > |2 | | 2|2 > |2 |2 ( 2)2 > (2 )2
2 4 + 42 > 42 4 + 2 32 > 32 ||2 > ||2 || > ||
2.3.13
|| || ||
2.3.13 :
|| || || || || || + || 0||(|| ) + (|| ) 0 (|| )(|| + ) 0 {|| || 0|| || + 0
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" " 53
2.3
2.3.14
) | 2| ) | 3| + | 1| ) |2 1| ) |
2 2|
) | 3| ) | 3| | 4| ) |23 4| )
2
2
)
(1)1001 ) 2 |12| ) 1
3
12
2.3.15
.
1) |x2 + 1| 2) | x2 + 4x 4|
3)
|x2 + 1| + 124) |x2 6x + 9|
5) | x2 3| 6) |1 2| 3
2.3.16 :
1) |9x2 6x + 1| 2) |x6 + 6x3 + 17|
3) | x4 x2 5| 4) | x2 + 2x 7|
2.3.17 x < 2 :
1) x + |x 2| 2) 3x |x 2| + |3 x|
3) |x 2| + |2x 4| |x 3| 4) |4 2x| |6 3x|
2.3.18 0 x 1 :
.
.
.
" "
3
.: 210.81.37.280
-
54 " "
1) 2x + |x 1| 2) x |x 1| + |1 x|
3) |x 1| + |2 2x| |x2 1| + |2x 6| 4) |x2 x| + |2x2 5| |x3 4|
2.3.19 x > 5 :
1) 2x + |x 5| 2) x |x 5| + |5 x|
3) |x 5| + |x 4| |x 3| + |2x 4|4) |x2 25| + |x2 5x| |2x 3|
2.3.20 < 2 < :
1) + | 1| + | 2| 2) |2 | + |2 | | 3|
3) | 1| + || |2 4| + | + 2 2| 4) + 2
2
+ + 2
2
|2 4|
2.3.21 , , :
1) |x2 6x + 9| 2) | x2 + 8x 16|
3) |(x 2)(x + 2) + 6| 4) |x|3
+ 2x
2
|x| + 2
5)
|x|3 + 5x22|x| + 10 6) |x
2 4x + 4|
2.3.22 2 < x < 3 :
A = |x + 2| + |9 3x| B = |x + 2| 5 |2x 6|
2.3.23 , , :
1)
x
2
+ 3|x||x| + 3 2)
x
2 6|x| + 5|x| 1
3)
x
2
+ 6|x| + 9|x| + 3 4)
x
2 4|x| + 2
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.
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" " 55
5) |x2 2x + 1| x2 x + |x| 6) |x2 + 4x + 4| x2 + x + |x|
2.3.24 ,
:
A = x 1 + d(x2) B = x + d(x 2) + d(x 1 2)
2.3.25 + 4 + 2
= 2 || = 2.
2.3.26 2 + 33 + 2
< 1 || > ||.
2.3.27 |x| 1 |y| 3
) |4x 5y| 19 ) |3x 2y + 7| < 2000
2.3.28 |x| < 1 |y| < 1 x + y1 + xy
< 1
2.3.29 x1 + |x|
+ 11 + |x| = 2
2.3.30 x :x + 1x
= |x| + 1|x|
.
.
.
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.: 210.81.37.280
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56 " "
2.3.31
=
| 1||1 |
| 1|| + 1| +
2.3.32 1 < < 1 :2 | 1| = + 1
2.3.33 N :
x1 + x
2
12
2.3.34 || > || :
|||| ||
|||| || = 1
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.: 210.81.37.280
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-
" " 57
2.4
H
, , .
, > 0,
x
2
= .
:
2
= || =
=
H -
, , .
, > 0,
x
= .
- - :
0 : ( ) = = 0 : = || 0 :
=
=
=
=
.
> 0, , :
=
, ,
0
= 0
.
.
.
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.: 210.81.37.280
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58 " "
2.4
1 0 0 + =
+
. . . . . . . . . . . . . . . . . .
2
2
= R. . . . . . . . . . . . . . . . . . . . . . . . . . .
3 + =
+
+ 2
. . . . . . . . . . . . . . . . . . . . . . . . .
2.4
2.4.1 :
)
3
216 )
4
625 )
3
125
512
)
00009 )
3
64x
6
y
9
125
2.4.1 :
)
3
216 =
3
6
3
= 6 )
4
625 =
4
5
4
= 5 )
3
125
512
=
3
(5
8
)3
=
5
8
)
00009
(3
100
)2
=
3
100
= 003 )
3
64x
6
y
9
125
=
4x
2
y
3
5
2.4.2 x R :
) =
x
2
x
) =
(x 1)2 +
(3 x)2
2.4.2 :
=
|x|x
=
{1 x > 0
1 x < 0 ) = |x 1| + |x 3| 13.
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.
.
-
" " 59
x 1 x 1 0 x 3 < 0 |x 1| = 1 x |x 3| = 3 x1 < x 3 x 1 > 0 x 3 0 |x 1| = x 1 |x 3| = 3 x
3 < x x 1 > 0 x 3 > 0 |x 1| = x 1 |x 3| = x 3
B =
(1 x) + (3 x) = 4 2x x 1(x 1) + (3 x) = 2 1 < x 3(x 1) + (x 3) = 2x 4 3 < x
2.4.3 :
)
36x
4
+ 12x
2
+ 1 )
x
4
4
+
3x
2
5
+
9
25
)
x
4
25y
2
+ 1 +
25y
2
4x
4
2.4.3 :
)
36x
4
+ 12x
2
+ 1 =
(6x
2
+ 1)
2
= 6x
2
+ 1
)
x
4
4
+
3x
2
5
+
9
25
=
(x
2
2
+
3
5
)2
=
x
2
2
+
3
5
)
x
4
25y
2
+ 1 +
25y
2
4x
4
=
(x
2
5y
+
5y
2x
2
)2
=
x25y
+
5y
2x
2
2.4.4 :
)
x + 3 =
2x 1 )
x 2 =
2x + 3
) 4x 2 = 0 )
3x + 5 =
x 7
2.4.4
) x + 3 0 2x 1 0 x 0, x + 3 =
2x 1 x + 3 = 2x 1 x = 4
) x 2 0 2x + 3 0 x 2, x 2 =
2x + 3 x 2 = 2x + 3 x = 5
) x 2 0 x 2,
4x 2 = 0 16 = x 2 x = 18
) 3x + 5 0 x 7 0 x 53
x 7. .
.
.
.
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60 " "
2.4.5 :
)
4
16 )
9
(
5
3)
3
)
8
(
5 2)4
2.4.5 :
)
4
16 =
4
2
4
==
2
)
9
(
5
3)
3
=
3
5
3
)
8
(
5 2)4 =
5 2
2.4.6 :
)
19600 )
3
27 64 343 ) 5
32 243 3125
2.4.6 :
)
19600 =
4 49 100 =
4 49
100 = 2 7 10 = 140
)
3
27 64 343 = 3
27 3
64 3
343 =
3
3
3 34
3 37
3
= 3 4 7 = 84)
5
32 243 3125 = 5
32 5
243 5
3125 =
5
2
5 53
5 55
5
= 2 3 5 = 30
2.4.7 :
)
4
16
4
8
)
108x
5
y
6
)
3
4
3
5
3 )
3
4
2
2.4.7 :
)
4
16
4
8
=
4
(2
2
)
4
= 2||2
)
108x
5
y
6
=
2
2
3
3
x
5
y
6
= 2 3x2|y3|3x = 6x
2|y|33x
)
3
4
3
5
3 =
4
3
5
5
3 =
4
5
3
26
=
40
3
26
=
20
3
13
)
3
4
2
=
3
4
4
2
=
24
4
2
=
12
2||
2.4.8 :
)
5
2 15
4
)
12
7 20
3 15
2
)
2 33 51
6
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.: 210.81.37.280
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.
.
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" " 61
2.4.8 :
)
5
2 15
4
=
15
6 15
4
=
15
10
=
3
2
)
12
7 20
3 15
2
=
60
35
9
8
=
60
52
=
15
13
)
2 33 51
6
=
30
2
15
3
10
1
6
6
=
30
2
15
3
10
2
6
3
6
=
30
2
9
3
4
2.4.9 :
)
12
5
4
)
9
8
6
5
)
15
3
10
10
3
3
2.4.9 :
)
12
5
4
=
12
5
12
3
=
12
53=
6
)
9
8
6
5
=
18
16
15
=
18
)
15
3
10
10
3
3
30
3
20
3
9
=
30
3
11
2.4.10 :
)
8 +
32
18 ) 3
16 +
3
375 3
54
) 3
32 2
50 ) 8
20 + 3
80 2
500
2.4.10 :
)
8 +
32
18 =
2
2
2 +
2
4
23
2
2 = 2
2 + 2
2
2 3
2 = 3
2
) 316 +
3
375 3
54 = 3
2
3
2 +
3
5
3
5 33
3
2 = 2 32 + 5
3
5 3 3
2 = 5 3
2 + 5
3
5
) 3
32 2
50 = 12
2 10
2 = 2
2
) 8
20 + 3
80 2
500 = 8
2
2
5 + 3
2
4
5 210
2
5 = (16 + 12 20)5 = 8
5
2.4.11 :
)
5 2
6 )
9 4
5 )
54 + 14
5
2.4.11 :
.
.
.
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.: 210.81.37.280
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62 " "
)
5 2
6 =
3 + 2 2
3
2 =
(
3
2)
2
= |3
2| =
3
2
)
9 4
5 =
5 + 4 2 2
5 =
(
5 2)2 = |
5 2| =
5 2
)
54 + 14
5 =
49 + 5 + 2 7
5 =
(7 +
5)
2
= |7 +5| = 7 +
5
2.4.12
)
1
3
5
)
3 13 + 1
)
xx
2
+ 1
x +
x
2
+ 1
)
12 +
3 +
5
2.4.12 :
)
1
3
5
=
3
5
3
5
3
5
2
=
3
5
2
5
)
3 13 + 1
=
(
3 1)(
3 1)
(
3 + 1)(
3 1)
=
(
3 1)2
(
3)
2 12=
3 + 1 23
3 1 = 23
)
xx
2
+ 1
x +
x
2
+ 1
=
(xx
2
+ 1)
2
(x +
x
2
+ 1)(xx
2
+ 1)
=
x
2
+ x
2
+ 1 2xx
2
+ 1
x
2 x2 1 =
= 2x2 + 2xx
2
+ 1 1
)
12 +
3 +
5
=
2 +
3
5
(
2 +
3 +
5)(
2 +
3
5)
=
2 +
3
5
(
2 +
3)
2 (5)
2
=
2 +
3
5
5 + 2
6 5
=
(
2 +
3
5)
6
2
6
6
=
(
2 +
3
5)
6
12
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" " 63
2.4
2.4.13 :
)
(
11
7
)
(
7
3
)
)
3
6
3
2.4.14 -
510
3
2
3
xx
2x
2
+ 34x
2
+ 6
2.4.15 -
37 2
2x + 1
x 1
3x
2x
2
+ 4
xx
2
+ 2x x
2.4.16 :
)
18 +
3
+
18
3
) (23)
3+ (2 +
3)
3
2.4.17 :
4 + 2
3
4 2
3
2.4.18
A =
x
2 6x + 9x
2 2x + 1 1 < x < 3
B =
4x
2 12x + 9 +1 + 2x + x
2
1 < x < 32
.
.
.
" "
3
.: 210.81.37.280
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64 " "
2.4.19 ,
:
A =
5
2
4
3
> 0
B =
4
3
3
6
5
> 0
=
3
4
3
3
4
3
5
4
> 0 > 0
2.4.20
A =
7 4
3
4 2
3 +
12 + 6
3
B =
6 + 4
2
11 6
2
3 2
2
2.4.21
A =
3
4
3
3
3
3
B =
5
3
2
=
3
16
4
32
3
2
2.4.22 :
)
10
3
2
5
+
5
2
13(
3 2)
=
6
5 + 5
6 5
2
13
)
3 +
2
4
3 3
2
+
2 +
5
5
2 2
5
=
26 +
6 + 12
10
30
" "
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.: 210.81.37.280
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.
.
-
" " 65
3
3.1 1
x + = 0
x
x + = 0 x =
:
1. 6= 0
x =
2. = 0
i. 6= 0 0 x = .i. = 0 0 x = 0 .
. ,
:
R .
(
2 1)x + 1 = 0
1
1
-
. -
, .
.
.
.
" "
3
.: 210.81.37.280
-
66 " "
. " ."
:
1. |f(x)| = |g(x)|.
|f(x)| = |g(x)| {f(x) = g(x)
f(x) = g(x)2. |f(x)| = g(x).
|f(x)| = g(x){f(x) = g(x) g(x) 0f(x) = g(x) g(x) 0
:
1.
.
2. -
.
3. ,
. -
.
3.1 1
1 xy = x
2 x = y xy R . . . . . . . . . . . . . . . . . . . . . . 2 3
x11
+ 1 = 2 . . . . . . . . . . . . . . . . . .
3 3 2 x = x + 2, = 3 = 2 .
4
2
x = x + 1 = 1 . . . . . . . . . . 5
2
x = 4x + 2 = 2 . . . . . . . . . . . .
" "
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.: 210.81.37.280
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.
.
-
" " 67
3.1 1
3.1.1 :
(2x + 5)
2 (3x 4)2 = 0
3.1.1 :
(2x + 5)
2 (3x 4)2 = 0 (2x + 5 + 3x 4)(2x + 5 3x + 4) = 0 (5x + 1)(x + 9) = 0
{5x + 1 = 0
x + 9 = 0 {x = 1
5
x = 9
3.1.2 :
)
x 13x 6 =
x 23x + 1
)
7 5x1 + x
=
2 5xx
)
x 7x 9
x 13x 15 =
x 9x 11
x 15x 17
3.1.2 :
)
x 13x 6 =
x 23x + 1
(x 1)(3x + 1) = (3x 6)(x 2) 3x2 + x 3x 1 = 3x2 6x 6x + 12 2x 1 = 12x + 12
12x 2x = 12 + 1 10x = 13 x = 1310
= 1
3
)
7 5x1 + x
=
2 5xx
x(7 5x) = (1 + x)(2 5x) 7x 5x2 = 2 5x + 2x 5x2
7x + 5x 2x = 2 10x = 2 x = 15
.
.
.
" "
3
.: 210.81.37.280
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68 " "
, ,
.
)
x 7x 9
x 13x 15 =
x 9x 11
x 15x 17
x 9 + 2x 9
x 15 + 2x 15 =
x 11 + 2x 11
x 17 + 2x 17
1 + 2x 9 1
2
x 15 = 1 +2
x 11 12
x 17 2
x 9 2
x 15 =2
x 11 2
x 17 2x 30 2x + 18
(x 9)(x 15) =2x 34 2x + 22(x 11)(x 17)
12(x 9)(x 15) =
12(x 11)(x 17)
(x 9)(x 15) = (x 11)(x 17) x2 15x 9x = +135 = x2 17x 11x + 187 24x + 135 = 28x + 187 4x = 52 x = 13
3.1.3 x:
)
2
(x ) + 2(x ) = x 2 + 2 6= 0)
2
( x) 2(x + ) = x 2 + + 2 6= 0
)
1
1x
=
1
x
1
+ 6= 0
3.1.3 :
)
2
(x ) + 2(x ) = x 2x 3 + 2x 3 = x x(2 + 2 ) = 3 + 3
x(2 + 2 ) = ( + )(2 + 2) x = +
)
2
( x) 2(x + ) = x 3 2x 2x 3 = x 3 3 = (2 + 2 + )x ( )(2 + + 2) = (2 + 2 + )x x =
" "
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.: 210.81.37.280
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.
.
-
" " 69
)
1
1x
=
1
x
1
1
+
1
=
2
x
+
=
2
x
x2
=
+
x = 2 +
3.1.4
.
1) (1 x) 2x = 32)
2
x 2 = 4 + x + 6
3.1.4 x + = 0. 1) :
(1 x) 2x = 3 x 2x = 3( 2)x = 2
:
6= 2, 1)
x =
2
2
= 2, 1) 0x = 4, . 2
:
2
x 2 = 4 + x + 6
2
x x = 2 + 4 + 6(
2 1)x = 6 + 6
x
2 1 = ( + 1)( 1), : 2 1 6= 0 6= 1 6= 1, 2)
x =
6( + 1)
( + 1)( 1) =6
1
.
.
.
" "
3
.: 210.81.37.280
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70 " "
= 1, 2) 0x = 12, . = 1, 2) 0x = 0, .
3.1.5
2
x 2 = 9x 6 + 9
1) , 2) .
3.1.5 x = .
2
x 2 = 9x 6 + 9
2
x 9x = 2 6 + 9(
2 9)x = ( 3)2
1. H x = = 0 6= 0. { = 0
6= 0 {
2 9 = 0( 3)2 6= 0
{ = 3 = 3 6= 3 = 3
2. H x = 6= 0. 2 9 6= 0 6= 3 6= 3.
x =
( 3)2(
2 9) =( 3)2
( 3)( + 3) = 3 + 3
3.1.6 :
( x) 3x = 5( x) 6
3.1.6 :
( x) 3x = 5( x) 6 2 x 3x = 5 5x 6 x 3x + 5x = 5 6 +2
(2 )x = 2 + 5 6 ( 2)x = ( 2)( 3)
) 2 6= 0 x = 3.) 2 = 0 0x = 0, -
x.
3.1.7 :
" "
3
.: 210.81.37.280
.
.
.
-
" " 71
) |2x 1| + 3 = 0) |x 2| = 3
3.1.7 :
) |2x 1| + 3 = 0 |2x 1| = 3
) |x 2| = 3{x 2 = 3x 2 = 3
{x = 5
x = 1
3.1.8 x :
) |2x 3| = |x 1|) |2x 3| = |x 2| + |2x 4|
3.1.8 :
) |2x 3| = |x 1| {2x 3 = x 12x 3 = x + 1
{x = 2
x =
4
3
) |2x 3| = |x 2| + |2x 4| |2x 3| = |x 2| + 2|x 2| |2x 3| = 3|x 2|
{2x 3 = 3(x 2)2x 3 = 3(x 2)
{2x 3 = 3x 62x 3 = 3x + 6
{x = 3
x =
9
5
3.1.9 :
|x + 1| + | x + 2| + |x + 3| 4 = 0
3.1.9
) x + 1 = 0 x = 1) x + 2 = 0 x = 2) x + 3 = 0 x = 3
:
x
-3
-1
2
x+1
- - -
0
+ + +
-x+2
+ + + + +
0
-
x+3
-
0
+ + + + +
.
.
.
" "
3
.: 210.81.37.280
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72 " "
:
(x 1) + (x + 2) + (x 3) 4 = 0 x ( 3)(x 1) + (x + 2) 4 = 0 x = 3(x 1) + (x + 2) + (x + 3) 4 = 0 x (3 1)(x + 2) + (x + 3) 4 = 0 x = 1(x + 1) + (x + 2) + (x + 3) 4 = 0 x (12)(x + 1) + (x + 3) 4 = 0 x = 2(x + 1) + (x 2) + (x + 3) 4 = 0 x (2)
x = 4 x ( 3) 0x = 1 x = 3 x = 2 x (3 1) 0x = 1 x = 1 x = 0 x (12) 0x = 2 x = 2
x = 4 x (2)
, |x + 1| + | x + 2| + |x + 3| 4 = 0 4,2,0 4.
" "
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.: 210.81.37.280
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.
-
" " 73
3.1 1
3.1.10 :
1) 9 7x = 2x + 34 2) 0 = 10 3x + 8 + 9x
3) 11x 3 8x 5 = 7x 4) 3 2(x + 1) = 7 4(x + 2)
5) 8 3(x + 3) (5 + x) = 2 6) 9 (x 4) = 11 2(5 x)
3.1.11 :
1) 5 (3 x) 3(4 + x) = (2x) 2) (5 y)4 2(y 3) = y 4 3(y + 2)
3) 2(y + 2) 8(y 3) = 5(5 y) 2(3 y) 4) 14(5 4x) 07(5 6x) = 0
5) 12 04(2 3x) = 02(4x 7) 6) 5x 375(x + 1) = 875 25(5 x)
3.1.12 :
1)
2(3y + 4)
7
=
5 + y
3
2)
3(y 1)8
= y 32
3) 3 2y 34
= 1 1 y8
4) y + 12
3y 14
= 1
5)
2x + 3
10
x 22
= x 35
6) 2x 3 2x6
= 1 5 x4
3.1.13 :
1) 7 2(x 1) = 2(x 2) 5 2) 4(x 1) 2(x 2) = 3 x 3(1 x)
3) 3(x 2) 2(1 + 3x) = 2(x 4) x 164) x 1 x
2
= 2x 2x 74
5)
x
3
x 22
=
x
4
5x 1212
6)
x + 2
6
5 x2
= 7 2x6
+
x 33
.
.
.
" "
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.: 210.81.37.280
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74 " "
3.1.14 :
1)
x + 6
21
+ 5 x 123
=
x + 1
2
5x + 928
2)
3
4
(x 1) 53
(x 4) = 85
(x 6) + 512
3)
4x 16
= 43
(1 9x + 1
18
)4)
3
5
(x 4) 2x 93
= 025(x 1) 2
5)
x +
1
2
3 14
=
2x + 3
6
5)
x
4
16
5
4
38
=
1 2x3
1 14
3.1.15 :
)
2
5
(x 5
3
(x + 4)
)=
2
3
(x 32
(x + 2))
) 2
((3 x) x 6
6
)= 3
(1
3
15 5x9
)
3.1.16 x:
)
x
= ( )
) ( x) + (x
+ 1
)2
=
( + x)
2
6= 0
)
x + x +
x x 2 =
x x +
x
x
3.1.17 x:
) (
2 9)x = 2 + 3) 3( + 1)x + 4 = 2x + 5( + 1)
) ( + 2)x + 4(2 + 1) =
2
+ 4(x 1)
3.1.18 x:
) (x 1) = x + 2 7) (3x + ) + 7 2 = 2 + 3(1 + x)) ( )x = 2 ( + )x
3.1.19 : ( + 1)x + 2 = 3(x + 2). ,
x =
1
3
.
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3.1.20 :
1) 3 x 42
=
3
4
2(x 1)2) (2 6)x 5 = 1 (4x 2)
, (1) (2) .
3.1.21 :
A =
x 15
+ 4 B = 1 9 x6
i) x, .
ii) x ,
.
x +
3
+
5 x9
=
5x 318
5x 136
+ 8
3.1.22 :
=
(5 +
8 +
5
8
)2
(517)
i) .
ii) (i) :
1 2x
3
4
+
3 + x
a
1
4
=
2x + 5
a
1
2
+
1 10x24
3.1.23 :
2 6 + 2 4 + 13 = 0
i) ,.
ii) ,, (i), :
4
x +3
x x2 =5
x
3.1.24 :
=
8
62
+
24
6 +
2
=
3
3
3
4 +
7
3
4
7
.
.
.
" "
3
.: 210.81.37.280
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76 " "
i) .
ii) :
1) (x
2 9)(x2 ) = (x2 x)(x2 12x)
2)
2x + 4
+
x + 2
2 x =x
2
x2
3.1.25 24 3
. .
3.1.26 , -
1
20
.
3.1.27 . 5
.
.
3.1.28 41 9 .
;
3.1.29 :
1) ( + 1)( 4)x = 2 16 2) ( 1)x = 1
3)
2
x 4 = 16x 2 4) 4 ( 2x) = 2x
3.1.30 , :
1)
2
(x + 1) = (1 x) 2) (x + 6) = 2 9(1 x)
3) (2x + 1) 4(1 + x) = 2(x 1) + 4) 2(2 + 2x) (4 + x) = 0
" "
3
.: 210.81.37.280
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.
.
-
" " 77
3.1.31 , :
1)
2
(x + 2) (x + 1) = 0 2) 22x 2(2x 1) = 2(x 1)
3)
3
(x 1) 6(x + ) = 3(x 3) 4) (2x 2)( 2) + x ( 1)2 = 2
3.1.32 :
(x + )
2
= 2( ) + (x + )2
.
3.1.33 :
2
(x + 4) 5(x + ) = 25
, :
i)
ii) .
3.1.34 :
3
(x 1) 3(3x 2) = 9
, :
i)
ii) .
3.1.35 :
(x + 2) 3(2 x 3) = 0
, :
i) 3.ii) 3.
3.1.36 :
(x 5) = 2( x 2)
.
.
.
" "
3
.: 210.81.37.280
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78 " "
, :
i)
ii) .
3.1.37 :
( 2)2 6(1 + x) = (2 2x)( 1)( + 1) 2
, :
2
(x 1) + (5x 1) = 2(1 + 3x)
.
3.1.38 :
1) (2 + 6)x =
2 42) ( + 3)x = 2 + + 4
, (1) (2) .
3.1.39 :
1)
2
x = 1 (x + )2) 2x = (x 1) + 2011 2012
3) (x + 1) ( 4x) = 1
(1) (2) , (3)
2012
2011
.
3.1.40 :
1)
2
(x + 1) = 2
(( 1)2 1 + 8x
)
2)
2
(x 1)( 10) = 5(5(3 + x) 2(5x + 6)
)
(1) (2) , :
i) ,
ii) :
3x +
1
2
x 110
+
x 2
3
2
=
x + 1
4
" "
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.: 210.81.37.280
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.
.
-
" " 79
3.1.41 :
1) ( 1)( + 1)x = 3(x + 1)
2) ( 1)2x = 2(1 x(5 )
)
i) , (1) (2) .
ii) (),
+
x
2 x +2
x
2 =x 1x
2
+ x
.
.
.
" "
3
.: 210.81.37.280
-
80 " "
1
.
3.1.42 :
1) 2x(x
2 12) 4(2x 1) = 4 2) 8 x(5x + 6) = (x + 1)3 x(x + 4)2
3) x
2
(x 4) + 2x(x 4) + x 4 = 0 4) x3 2x2 (2x 1)(x 2) = 0
5) (x
2
+ 3x)(x 1) = (2x + 6)(x2 1) 6) 3x(x 3) + (x 3)2 + 9 x2 = 0
7) x(x + 1)
2 (x + 5)2 + 16 = 0 8) (x2 + 2)2 = x((x + 1)
3 (3x2 x + 1))
3.1.43 :
1)
3x 1x + 3
3x 7x 3 = 0 2)
x + 3
x 3 = 34(x 6)x + 6
3) 2 x 1x + 1
= 1 xx + 1
4)
2x
3x 6 x
2x 4 = 1 +5x 1212 6x
5)
15
x 2 4
x + 2
=
5
x
2 4 6)x + 1
x
2 1 +2
x
2 2x + 1 = 0
3.1.44 :
1)
1
x
55x x2 =
1
x 5 2) 1x + 2
x 2 =x 10x
2 2x x + 2
x
3)
x
x + 2
+
x
2x x2 = 2
x
2 4 4)1
x + 5
2xx
2
+ 5x
=
1
25 x2
5)
4
2
x1 1= 2
3.1.45 :
1) |7x 13| + 21 = 0 2) 3|2x 5| 21 = 0
" "
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.: 210.81.37.280
.
.
.
-
" " 81
3)
5 |x 2|2
= 4
4)
2|5 3x| 19
= 1
3.1.46 :
1) 2 +
|3x 4| 13
= 3 |3x 4|3
2) 1 1 + 3|x 7|4
=
4 |7 x|10
+
1
2
3)
|x 3|2
+
|6 2x|3
= 8 |3 x|6
4) 2|x + 3| = |x 5|
5)
|2x 1|3
=
|3 x|2
6)
|x + 1|4
|3x 2|6
= 0
3.1.47 :
1) 2|x 4| = 2 x 2) |1 3x| 3 = 2x
3) x 2|x + 2| 4 = 0 4) |3x 6| = 6 3x
5) x |x 13| = 13
3.1.48 :
1) |x 2| |x + 2| = |7x 4|2) |x2 6x + 9| | x2 3| = 123) 8 |2x x2 1| = |4x x2 4|4) |x2 4| + |x2 + 4x + 4| = 05) |x2 2x 3| + |9 x2| = 06) |x 4| |x + 5| = |2x + 7| |4 x|
3.1.49 :
1) d(x3) = d(3x) = 02) x d(2x 6) = 43)
4x
2 4x + 1x
2 10x + 25 = 04)
x
2 10x + 25 = 1 2x
3.1.50 :
.
.
.
" "
3
.: 210.81.37.280
-
82 " "
1)
|5 x| 6 = 2 2) 5 |2x 1| = 43)
1 |3 2x| = 6 4) |x + 3| 2 = |x 5|
5)
x |2x 6| = |x 8| 6) |x| 3 = 2|x| 1
3.1.51 :
1) 2|x + 1| |5 x| = x2) |x 1| 2|x 2| = 3 x3)
x
2 6x + 9 + 2x
2
+ 2x + 1 = 4
4) d(x1) d(0x) 4 = 2x d(2x 3)
3.1.52 :
1) x
2 10|x| + 25 = 0 2) |x|3 6x2 + 9|x| = 0
3) x
2
+ 6x + 9 |x + 3| = 0 4) |3 3x| x2 + 2x 1 = 0
5) |x2 2x 9| = x2 6x + 9 6) x2 3|x| 2 = 6
3.1.53 :
1)
|x| + 2 = x + 2 2) 3 + |x| + | x| = 15 |2x|3) x
2 |x 2| + x + 2 = 0 4) |x| + 5|x| x = 4
5)
2x + |x| = 14 4|x|
" "
3
.: 210.81.37.280
.
.
.
-
" " 83
3.2 x
=
, x
= -
. , - . ,
,
. :
1. > 0
x
= {x =
x =
x
2
= 8{x =
8
x = 8
2. > 0
x
= x =
x
3
= 8 x = 38 = 2
3. < 0 x
= .
4. < 0
x
= x = ||
x
3
= 8 x = 3| 8| = 2
3.2 x
=
1 x
= , R . . . . . . . . . 2 x
= , > 0 , . . .
.
.
.
" "
3
.: 210.81.37.280
-
84 " "
3.2 x
=
3.2.1 :
1) x
7
+ 27x
4
= 0
2) 25x
5
= 16x
3
3) x
7 8 = x3 8x4
3.2.1 1)
x
7
+ 27x
4
= 0
x4(x3 + 27) = 0
{x
4
= 0
x
3
= 27 {x = 0
x = 3
2)
25x
5
= 16x
3
25x5 16x3 = 0 x3(25x2 16) = 0
{x
3
= 0
25x
2
= 16
{x
3
= 0
x
2
=
16
25
{x = 0
x = 45
3)
x
7 8 = x3 8x4
x7 8 x3 + 8x4 = 0 (x7 x3) + (8x4 8) = 0 x3(x4 1) + 8(x4 1) = 0 (x4 1)(x3 + 8) = 0
{x
4
= 1
x
3
= 8 {x = 1x = 2
3.2.2 :
1) (x
2 5)4 256 = 02) (3x 1)4 + 8 = 24x
3.2.2 x
2 5 = 1)
" "
3
.: 210.81.37.280
.
.
.
-
" " 85
4 256 = 0
4 = 256 = 4256
{ = 4
= 4
= 4 x2 5 = 4 x2 = 9 x = 3 = 4 x2 5 = 4 x2 = 1 x = 1 2)
(3x 1)4 + 8 = 24x (3x 1)4 24x + 8 = 0 (3x 1)4 8(3x 1) = 0
3x 1 =
4 8 = 0(3 8) = 0{ = 0
3
= 8
{ = 0
= 2
:
= 0 3x 1 = 0 x = 13
= 2 3x 1 = 2 x = 1
.
.
.
" "
3
.: 210.81.37.280
-
86 " "
3.2 x
=
3.2.3 :
1) 8x
3
= 27 2) 32x
5
+ 1 = 0
3) 2x
5
= 8x
3
4) 3x
4
+ 24x = 0
5) 5x
6
+ 4x
2
= 0 6) 32x
11
= 2x7
3.2.4 :
1) x
4 8x = 0 2) x6 8x = 0
3) 2x
5
+ 16x
2
= 0 4) 8x
5
+ 27x
2
= 0
5) 27x
4
+ x = 0 6) 8x
4
+ x
2
= 0
3.2.5 :
1) (x
3
+ 27)(x
4 54) = 0 2) (x4 81)(x5 + 210) = 0
3) (x
12 166)(x9 + 83) = 0 4) (3x10 331)(4x9 220) = 0
3.2.6 :
1) 2x
3
= 8x 2) x
6
= 81x
2
3) 2x
5
+ 5x
2
= x
5 3x2 4) x3(x3 + 30) = 3x3
5) 2x
2
(2x
2
+ 3) = 3x
4 2x2 6) 5x(x3 5) = 2x(2x3 + 1)
" "
3
.: 210.81.37.280
.
.
.
-
" " 87
3.3 2
x
2
+ x + = 0 6= 0
" ". :
x
2
+
x =
x
2
+
x +
(
2
)2
=
+
(
2
)2
(x +
2
)2
=
2 44
2
=
2 4 () (x +
2
)2
=
4
2
:
1. > 0 {x +
2
=
2
x +
2
=
2
{x =
2
+
2
x = 2
2
x12
=
2
2. = 0
(x +
2
)2
= 0 x = 2
3. < 0 .
, , Vieta
,
:
x
1
+ x
2
=
+
2
+
2
=
22
=
x
1
x2
=
+
2
2
=
()2 ()
2
4
2
=
2 (2 4)4
2
=
4
4
2
=
.
.
.
" "
3
.: 210.81.37.280
-
88 " "
S x
1
+ x
2
P x
1
x2
,
:
S =
P =
Vieta. x
2
+ x + = 0,
Vieta, :
x
2
+ x + = 0
x2 +
x +
= 0
x2 (x1
+ x
2
)x + (x
1
x2
) = 0
x2 Sx +P = 0 , -
,
. .
x
2
+x+ = 0
< 0
1
< 0 <
2
= 0
0
> 0, 0,
> 0
0 <
1
2
> 0, 0,
< 0
1
2
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 , .
3 > 0 x
2
+ x + = 0 . . . . . . . . . . .
4 ,, > 0 x
2
+ x + x. . . . . . .
5 x
2
+ x + = 0, 6= 0 , = 0. . . . 6 x
2
+x+ = 0, 6= 0 1
,
2
,
x
2
+ x + = (x +
1
)(x +
2
). . . . . . . . . . . . . . . . . . . . . . . . . .
" "
3
.: 210.81.37.280
.
.
.
-
" " 89
3.3 2
3.3.1
) x
2 sx 1 = 0)
2
x
2 2x + 2 = 0 6= 0)
2
x
2 2x + 1 + 22 = 0 6= 0
3.3.1
) = (s)2 4 1 (1) = s2 + 4 > 0. . .
) = (2)2 4 2 2 = 0.
=
2
2
2
=
) = (2)2 4 2 (1 + 22) = 442. 6= 0 < 0 . = 0 = 0
=
2
2
2
=
1
3.3.2
;
) x
2 ( 1)x + 2 2 = 0) x
2 2( 1)x + 2 2 + 1 = 0) x
2 3( 3)x (2 + 10) = 0 6= 0
3.3.2
) = 0 (1)2 4(22) = 0 72 +6+1 = 0.
= 36 4(7) = 64
12
=
6 814
{
1
= 17
2
= 1
) = 4( 1)2 4(2 2 + 1) = 0. .
.
.
.
" "
3
.: 210.81.37.280
-
90 " "
)
= 09( 3)2 4((2 + 10)) = 09(
2 6 + 9) + 4(2 + 10) = 09
2 54 + 81 + 82 + 40 = 017
2 14 + 81 = 0
= 14
2 4 17 81 = 5312 < 0
.
3.3.3
x
2 2x + + 2 = 0
) 2 , ) 2 , ) 2 .
3.3.3
=
+ 2
1
= + 2 = 4 4( + 2) = 4 4
= 2
)
< 0 + 2 < 0 < 2
)
> 0 > 0
= 2 > 0{ + 2 > 0
4 4 > 0 { > 2 < 1 2 < < 1
)
> 0 > 0
< 0.
= 2 > 0,
.
3.3.4
1
,
2
-
1
+
2
1
2
.
)
2
1
+
2
2
)
3
1
+
3
2
)
1
1
+
1
2
)
2
1
2
+
2
2
1
" "
3
.: 210.81.37.280
.
.
.
-
" " 91
3.3.4
)
2
1
+
2
2
= (
1
+
2
)
2 21
2
)
3
1
+
3
2
= (
1
+
2
)
3 321
2
31
2
2
= (
1
+
2
)
3 31
2
(
1
+
2
)
)
1
1
+
1
2
=
1
+
2
1
2
)
2
1
2
+
2
2
1
=
3
1
+
3
2
1
2
=
(
1
+
2
)
3 31
2
(
1
+
2
)
1
2
3.3.5 x y
, .
3.3.5 S = x + y P = xy .
x,y
2 S+P = 0
0S
2 4P 0
P S2
4
P P =
S
2
4
.
= S
2 4P = S2 4S2
4
= 0
2 S+P = 0
x = y =
S
2
.
.
.
" "
3
.: 210.81.37.280
-
92 " "
3.3 2
3.3.6
x
2 7|x| 18 = 0
3.3.7 :
) x
2 5x 50 = 0) 2x
2 8x = 6) 3x
2
+ 14x 5 = 0) (x
2 16)(x2 + 6x 7) = 0
3.3.8 :
1) x
2
+ 2x 3 = 0 2) x2 + 2x 8 = 0
3) x
2
+ 6x + 9 = 0 4) x
2
+ 5x + 7 = 0
5) 3x2 + 5x 2 = 0 6) 9x2 6x + 1 = 0
7) x2 + 5x 2 = 0 8) 2x2 + 7x + 6 = 0
3.3.9 :
1) 3x2 + 12x = 0 2) 2x2 + 8x = 0
3) 36 16x2 = 0 4) 4x2 16x = 0
5)
2x
2 8x = 0 6)
3x
2 27x = 0
3.3.10 :
" "
3
.: 210.81.37.280
.
.
.
-
" " 93
1) x
2
+
(
3 + 1
)
x +
3 = 0 2) 5x
2 (
2 10
)
x 22 = 0
3) x
2 8x
2 = 0
4) 03x
2
+ 09x 3 = 0
5) 01x2 + x 25 = 0 6) 01x2 + 05x 14 = 0
3.3.11 :
1)
1
6
x
2 x + 32
= 0 2) 13
x
2
+
1
2
x 13
= 0
3)
x
2
2
4x 58
= 0
4) (3x
2 48)(x2 4x + 32) = 0
5) (9x
2 6x + 1)(x2 x + 2) = 06) (x
2 2x + 4)(
x
2
+ x +
1
4
)
= 0
3.3.12 :
1) (x 1)2 = 4x 5(2x + 1) 2) (x 1)3 x(x 2)(x + 2) = 1
3) (x+2)
3 x(x3)2 = 15 (3x+1)(13x)4)
x 22
x(6 x)6
=
x(x 2)6
x2 23
5)
5
6
(x + 1)(x 1)2
=
2 x3
(x 4)2
6
6)
x(3x 2)3
(2x 1)2
+ 2
6
= 1 x 62
3.3.13 :
) |x2 + 2x 9| = 0) |x2 + 3x 5| = |2x2 4x + 5|) |x 3| = x2 x 6
2
, .
3.3.14 :
) 2x2 + (a 3)x + a 1 = 0) 2x
2
+ (a 2b)x a(a + 2b) = 0
.
.
.
" "
3
.: 210.81.37.280
-
94 " "
3.3.15 :
) ( 3)x2 + 2x + + 3 = 0) ( 2)x2 2( + 1)x + + 4 = 0
3.3.16 ,
o :
) x
2 3( + )x + 9 = 0 6= 0) (
2 2)x2 22x 22 = 0 2 6= 2
) x
2 ( + 1
)x + 1 = 0 6= 0) ( + )x
2
+ ( + + 2)x + + = 0 6=
3.3.17 -
:
)
x
2
2
+ ( + 1)x +
2
+ + 1 = 0
) x
2 (2 4)x (3 ) = 0) ( 3)x2 + 2( 1)x + + 3 = 0
3.3.18
x
2
+ 2x + 1 = 0
,
x
2
+ (2 + 1)x +
2
+
9
4
= 0
.
3.3.19
x
2
+ (2 + 3)x + +
9
4
= 0
:
i)
ii)
" "
3
.: 210.81.37.280
.
.
.
-
" " 95
iii)
3.3.20
x
2 2x + 2 + 2 = 0
:
i)
ii)
iii)
iv)
3.3.21 :
1) x
2 x 12 = 02) x
2
+ (2 9)x + 2 6 = 0
(1) (2). :
i) ,
ii) (2)
3.3.22
x
2
+ (4 2)x + (2 1)2 = 0
i) ,
ii) ,
(35).
3.3.23 :
1) x
2
+ ( + 3)x 4 + 2 = 02) x
2
+ (1 2)x 3 4 = 0
i) , .
ii) , .
3.3.24 :
(
3
+ 10)x
2
+ (2
3
+ 4)x +
2
+ 4 + 22 = 0
.
.
.
" "
3
.: 210.81.37.280
-
96 " "
3. .
3.3.25 :
1) x
2
+ (2 + 1)x + |6 3| = 0
i) , (1) 1.ii)
: x
2 x + 2 = 0 (2). N , (2) .
3.3.26 :
(
2 1)x2 + ( 1)x + 1 = 0
. :
i) ,
ii) .
3.3.27 :
2x
2
+ 2( + )x + ( 2)( + 4) 2 = 0
. o :
i) ,
ii) .
3.3.28
x
2
+
+ 3x + = 0
i) N
ii) :
=
2
+ 6 + 9 +
2 2 + 1
3.3.29
x
2
+ (
3 + 1)x + 2(
3 1) = 0
i) N = (
3 3)2.
" "
3
.: 210.81.37.280
.
.
.
-
" " 97
ii)
iii)
=
1
2
.
3.3.30
x
2
+ ( 5)x + 4 = 0
i) .
ii) .
iii) .
iv) .
v) .
3.3.31
x2 + ( 7)x + 6 = 0
i) .
ii) .
iii) .
iv) .
v) .
3.3.32
( + 2)x
2
+ 2x + 1 = 0
i) n
ii) x
1
x
2
:
x
2
1
x
2
+ x
1
x
2
2
= 23
3.3.33
2x
2 4x + 3 = 0
.
.
.
" "
3
.: 210.81.37.280
-
98 " "
i) N .
ii) x
1
x
2
:
) x
3
1
x
2
2
+ x
2
1
x
3
2
= 8
)
1
x
2
1
+
1
x
2
2
= 2
3.3.34
x
4
+ (
3
+ 8)x
3 10x2 + 5 2 = 0
i) .
ii) .
.
3.3.35 :
) x
2 7x + 4 = 0) x2 3x 1 = 0)
3x
2 27x +
12 = 0
)
6x
2 3
2
x2
3
= 0
3.3.36 x
2
+ x + 8 = 0 x
1
x
2
x
1
+ x
2
= 6 x
1
x2
= 4.
i) .
ii) .
3.3.37 x
1
x
2
x
2 3x + 1 = 0
:
) x
1
+ x
2
) x
1
x2
) x
2
1
+ x
2
2
) x
3
1
+ x
3
2
)
1
x
1
+
1
x
2
)
x
1
x
2
+
x
2
x
1
" "
3
.: 210.81.37.280
.
.
.
-
" " 99
3.3.38 x
1
x
2
2x
2
+ 3x 4 = 0
:
) x
1
+ x
2
) x
1
x2
)
x
2
1
+ x
2
2
) (2x
1
3)(2x2
3) ) (x1
+ 1)(x
2
+ 1) ) (x
2
1
x1
x
2
)(x
1
x
2
x22
)
3.3.39 x
1
x
2
x
2
+ 6x + 3 = 0
:
) x
1
+ x
2
) x
1
x2
) x
2
1
+ x
2
2
) (x
1
+ 2)
2
+ (x
2
+ 2)
2
)
1
x
1
3 +1
x
2
3 ) (x3
1
x
2
+ 2x
2
1
x
2
2
+ x
1
x
3
2
)
3.3.40 2
:
) 6 1 ) 12
2 )5 + 1
2
15
2
) 1 + 1
3.3.41 x
1
x
2
x
2 3x 1 = 0. 2
:
) x
2
1
x
2
2
)
1
x
1
1
x
2
3.3.42 x
1
x
2
x2 + x + 3 = 0. 2
:
)
x
1
x
2
x
2
x
1
)
x
1
x
1
+ 2
x
2
x
2
+ 2
2
, , .
.
.
.
" "
3
.: 210.81.37.280
-
100 " "
3.3.43 :
1) x
2 6|x| + 8 = 0 2) 3x2 + 10|x| 8 = 0
3) 3x
2
+ |x| 2 = 3(|x| + 1) 4) 3x2 + | 5x| 2 = 0
3.3.44 :
1) (x 2)2 = 7|x| + 1 x(x + 4)2) 5|x| = 1 (x 3)(x + 3)
2
3.3.45 :
1) x
4 5x2 + 4 = 0 2) x4 5x2 + 6 = 0
3) x
6 16x3 + 64 = 0 4) x8 17x4 + 16 = 0
5) x 4x + 3 = 0
6)
x
(
x 2
)
= 3
3.3.46 :
1) (3x 5)2 + 7(3x 5) 8 = 0 2) (2x 3)2 6(3 2x) 7 = 0
3) (x + 1)
2
+ |x + 1| 2 = 0 4) (2x 1)2 8|2x 1| + 15 = 0
5) (x 3)2 + 5|3 x| 6 = 0
3.3.47 :
1)
x 10x
2 4 x
2 x =2
x + 2
2)
x + 2
x 1 7
x
2 x =x + 3
x
2
+ 3x
3)
x
x + 2
5x 20x
2 4x = 14
x
2
+ 2x
4)
1
1 1x
+
4
x 1x
=
5
4
3.3.48 :
" "
3
.: 210.81.37.280
.
.
.
-
" " 101
1) (2x 1)2 44x
2 4x + 1 + 3 = 0 2)(x 2
x
)2
5(x 2
x
)+ 4 = 0
3) 6
(2x
x 3
)2
(10x
x 3
) 6 = 0 4) x
2 32x
+
2x
x
2 3 = 2
5) |x2 x| + |x2 11x + 10| = 0 6)x
2 4|x| + 3 + x4 10x2 + 9 = 0
.
.
.
" "
3
.: 210.81.37.280
-
102 " "
4
4.1 1
x + > 0 x + < 0. ,
, .
, , '
. ,
x + > 0, :
x + > 0 x >
:
> 0
x > x
>
x >
< 0
x > x
i. > 0 .
ii. 0 .
x + < 0 . ,
.
, -
.
" "
3
.: 210.81.37.280
.
.
.
-
" " 103
, r
,
|x | r
[ r + r. .
|x | r r x + r{ r x
x + r
|x | r x r x r{x r
x + r
|x | r r x + r d(x ) r x [ r; + r
4.1 1
1 : x > 0 x2 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 : (x 1)2 0 x = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 3 : (x 1)(x 5) > 0 x > 1 x > 5 . . . . . . . . . . . . . . . . . . 4 : x x 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 : |x 5| < 2 x (3 5) . . . . . . . . . . . . . . . . . . . . . . . . 6 : x > 4 x < 4 |x| > 4 . . . . . . . . . . . . . . . . . . . . . . 7 : d(x 1) < 2 |x 1| < 2 . . . . . . . . . . . . . . . . . . . . . . . 8 |x + 1| 0 x. . . . . . . . . . . 9
|x| 1 1 x = 0. . . . . . . . . . . .
.
.
.
" "
3
.: 210.81.37.280
-
104 " "
4.1 1
4.1.1 .
i) |x 1| < 3 ii) |x + 2| < 2 iii) |x + 7| 0
4.1.1
i) |x 1| < 3 3 < x 1 < 3 2 < x < 4ii) |x + 2| 0.iii) |x + 7| 0 x + 7 = 0 x = 7.
4.1.2
x
3
2x 14
>
x
12
4.1.2 .. 12,
x
3
2x 14
>
x
12
4x12
3(2x 1)12
>
x
12
4x 6x + 3 > x 3x < 3 x < 1
4.1.3
x 12
+
3x + 2
4
5x4
4.1.3 .. 4,
x 12
+
3x + 2
4
5x4
2(x 1) + 3x + 2 5x 2x 2 + 3x + 2 5x 0x 0 x R
" "
3
.: 210.81.37.280
.
.
.
-
" " 105
4.1.4
x
2
+ 1
+
x 1
2
+ 1
> 2 R
4.1.4
2
+ 1 > 0 .
x
2
+ 1
+
x 1
2
+ 1
> 2
x + x 1 > 2(2 + 1) 2x > 22 + 3
x > 2 + 32
4.1.5 x
1)
x + 2
4
x3
1 2) x5
x + 115
< 0
4.1.5 1)
x + 2
4
x3
1 3(x + 2) 4x 12 3x + 6 4x 12 x 6 x 6
2)
x
5
x + 115
< 0
3x x 1 < 0
2x < 1 x < 12
[6 12
).
.
.
.
" "
3
.: 210.81.37.280
-
106 " "
4.1.6 (x+1) 12x .
4.1.6 x + 0. :(x + 1) 1 2x
( + 2)x 1
+ 2 > 0 > 2
x 1 + 2
+ 2 < 0 < 2
x 1 + 2
+ 2 = 0 = 2 0x 3 .
4.1.7 x + 3 x ,.
4.1.7 x + 0. :x + 3 x
( + 1)x 3
( + 1) > 0 > 1
x 3 + 1
( + 1) < 0 < 1
x 3 + 1
( + 1) = 0 = 1 0x 3 (2), 1. 3 0 3. (2) x.2. 3 < 0 < 3. (2) .
4.1.8 |x 2| 6.
4.1.8 :
|x 2| 6 6 x 2 6 6 + 2 x 6 + 2 4 x 8 x [4 8
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3
.: 210.81.37.280
.
.
.
-
" " 107
4.1.9 |x| , .
4.1.9 :
|x| |x|2 2
|x|2 2 0 x2 2 0 (x )(x + ) 0
, (x - ) (x + ) .
x 0 x + 0 x
x 0 x + 0 x
4.1.10
i) |x + 1| 2 ii) |x 1||x 2| > 1
4.1.10
i) |x + 1| 2 x + 1 2 x + 1 2 x 1 x 3ii)
|x 1||x 2| > 1
|x 1| > |x 2| |x 1|2 > |x 2|2
(x 1)2 > (x 2)2
x2 2x + 1 > x2 4x + 4
2x > 3 x > 32
4.1.11
2 |x 1| < 3.
4.1.11 :
2 |x 1| < 3 3 < 2 |x 1| < 3 5 < |x 1| < 1 1 < |x 1| < 5
1. |x 1| > 1 x.2. |x 1| < 5 5 < x 1 < 5 4 < x < 6
(4 6).
.
.
.
" "
3
.: 210.81.37.280
-
108 " "
4.1.12 |x + 1| |x 2| > 2x.
4.1.12
x
-1 2
x+1
-
0
+ + +
x-2
- - -
0
+
:
1. x < 1 |x + 1| = x 1 |x 2| = x + 2.
x 1 (x + 2) > 2x 2x < 3 x < 32
x ( 32
)
2. 1 x < 2 |x + 1| = x + 1 |x 2| = x + 2.
x + 1 (x + 2) > 2x 0x < 1
3. x 2 |x + 1| = x + 1 |x 2| = x 2.
x + 1 (x 2) > 2x 2x < 3 x < 32
" "
3
.: 210.81.37.280
.
.
.
-
" " 109
4.1 1
4.1.13 :
1) 2x 1 > 5 2) 3 2x < 15
3) 2x 3 < 15 x 4) 5 4x 29 x
5) 7 + 2x 3 + 4x 6) 5 3x 2x + 5
4.1.14 :
1) 2(x + 1) 3x 4 2) 2(x 1) 3(x + 2) x 1
3) 4(x + 1) + 2(x 2) 5(x 1) 4) 2(x + 3) + 5(x 1) 7(x + 3)
5) 4x 9 3 2x + 3(x 5) 6) 6(2x + 7) < 15(x + 2)
4.1.15
:
1) 3(x 2) + 4 > 13 2) 10 2(x 1) < 4
3) 7 5(x 1) 12 4) 3 (2x 5) 2 x
5) 6 (7 + 2x) < (x + 1) 6) 1 (x 1) 3 5(2 x)
4.1.16
.
1) 3(2x + 7) 4(15 x) 29 + 12x 2) 2(4x + 5) 3(x + 3) 5x 9(1 x)
3) 6(x 2) (5 3x) < 9(x + 3) 2x 4) 3(7 + 3x) (8 + 7x) > x 11(x + 1)
5) (x + 2)
2 2(x 2)2 25 (x + 1)2 6) 4x (x 1)2 > 8 (x 3)(x + 3)
.
.
.
" "
3
.: 210.81.37.280
-
110 " "
4.1.17
.
1)
2x + 5
3
> 7
2)
7 3x2
8
3)
5 2x5
+ 9 0 4) 7 3x 42
< 0
5) 1 2x 73
3x + 2
4
4.1.18
.
1) 2 x 13
56
2)
2
3
5 7x6
> x 2
3) 3 x 25
43
4)
3x + 1
2
2x 43
< 1
5) 3 2x + 13
x 12
6)
x + 1
2
x + 34
< 2 x 23
4.1.19 :
1)
x + 4
3
>
7x 52
2)
7x 34
9x + 48
> 0
3)
5x 29
x + 76
> x 1 4) x 32
x + 43
3x 4(3 x) 4) 3(x + 4) 4(2x + 1) > 5(x 2)
5) 2(x + 1) 4 (x + 3) 3(2 x) 6) 13 3(x 2) < 4(x + 3) 7(x 3)
" "
3
.: 210.81.37.280
.
.
.
-
" " 111
4.1.21 :
1) 2(x + 2) +
1
6
x 33
3)
17
4
15 x2
x x 32
4)
2
5
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