ΑΣΚΗΣΕΙΣ Α΄Λ ΕΠΑΝΑΛΗΠΤΙΚΟ.pdf
TRANSCRIPT
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31 :
2012-2013
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32 :
1. , . .
1. 2 = 9 = 3 ()
2. 2 = = 1 ()
3. 2 1 ()
4. 2 2 4 ()
5. > 2 2 > 4 ()
6. < 2 2 < 4 ()
7. 2 < 4 < 2 ()
8. 2 > 4 > 2 () 9. < 2 < 3 . < 6 () . //
2.
.
1. x(x 2) = 0 A x 0 x 2
2. x(x 2) 0 x = 2
3. 2x = 4 x = 2 x = 2
4. 2x = 4 x < 0 x = 0
5. x(x 2) = 0 x(x 1) = 0 x = 0 x = 2
6. 2x = 4 x > 0 x = 2 1 , 2, 3, 4, 5, 6 .//
-
33 :
10
2,3
13
5
-3,5
-5
0 20
5
100
NZ Q
R
3. 1. . 2. ; 3. Venn N,Z ,Q R .
.
4. .
1. = {xZx 16} = {xZx 24}, ) = {1, 2, 3, 4, 6, 8, 12, 24} ) = {1, 2, 4, 8}
2. = {xx } = {xx }. : ) = ) = ) = ) =
5. :
1. . : ) ) ) () ) ()
2. . :
3,5 0 10 13/5 2, 3 20/5 100 5
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34 :
) () ) )
6. . 1. , . : ) = ) = ) () = 2. . : ) = ) =
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35 :
1. . i) . ii) : 1: 2: 2 3: 2 4: 5: . iii) 45253 ,, AAAAA .
i) , : ,
},,,,,,,{ KKKKKKKKKKKK .
ii) :
},,,{1 A
},,{2 A
},,,{3 A ( 13 AA )
},{4 A
},,,{5 A .
iii) 3A 3A ,
2. , },,,{3 A .
25 AA 5A 2A , , . , },{25 AA .
45 AA
. , },,,,{45 AA .//
3 2 1
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36 :
2. . Venn : i) . ii) .
i) , , .
BA AB . , )()( ABBA
)()( BABA .
ii) , . BA , )( BA .//
3. . , .
2 3 1 = { , , , , , }.//
4. . : : 1 2 : : 5 )( , , ,
B A
( )A B
B A
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37 :
2 1
1 2 3 4 5 6
1 (1,1) (1,2) (1 ,3) (1,4) ( 1, 5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
= { (2,1) , (3,1) , (3,2) , (4,1) , (4,2) , (4,3) , (5,1) , (5,2) , (5, 3) , (5,4) , (6,1) , . (6,2), (6,3) , (6,4) , (6,5)} = {(1,1), (1,3) , (1,5) , (2,2) , (2,4) , (2,6) , (3,1) , (3,3) , (3 ,5) , (4,2), (4,4) , (4 , 6), ( 5 ,1) ,( 5, 3) , ( 5,5) , (6,2) , (6 ,4) , (6,6)}
={(1,1) , (1 ,2) , ( 1, 3) , ( 1,4) , ( 2,1) , ( 2 , 2) , (3 , 1) , (4, 1)}
= {(3 ,1) , ( 4, 2) , ( 5 , 1) , ( 5 , 3) , ( 6 , 2) , ( 6 , 4)}
{ ( 2 , 1) , (3 , 1) , ( 4 ,1)}
={ ( 1 , 1) , ( 1, 3) , ( 2 , 2) , ( 3 , 1) }
)( ={ ( 3,1) } .//
5. , :
. x x x x //
6. .
.
Venn )()()( .
.//
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38 :
7. . .
, , .
2 1
1
2
3
4
5
6
1 2 3 4 5 6
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)
(1,5) (2,5) (3,5) (4,5) (5,5) (6,5)
(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)
36 , 36)( N .
: ,
)}5,6)(6,5(),4,5(),5,4(),3,4(),4,3(),2,3(),3,2(),1,2(),2,1{(A 10)( AN
, 18
5
36
10
)(
)()(
N
ANAP .
, 28,018
5 , ,
28% .//
8. 5,0)( AP , 4,0)( BP 2,0)( BAP . :
i) . ii) .
i) )( BA .
)(1))(( BAPBAP
))()()((1 BAPBPAP
)2,04,05,0(1
7,01
3,0 .
( )A B
B A
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39 :
ii) )()( ABBA .
BA AB , :
)()())()(( ABPBAPABBAP
)()()()( BAPBPBAPAP
)(2)()( BAPBPAP
2,024,05,0 5,0 .//
9. 6,0)( AP 5,0)( BP .
i) . ii) 5,0)(1,0 BAP .
i) , : 1,15,06,0)()()( BPAPBAP
, , 1)( BAP , . , .
ii) BBA ABA ,
)()( BPBAP )()( APBAP ,
5,0)( BAP (1)
: )()()()( BAPBPAPBAP
)(5,06,0)( BAPBAP .
1)( BAP .
: 1)(5,06,0 BAP
)(15,06,0 BAP
)(1,0 BAP . (2) (1) (2) : 5,0)(1,0 BAP .//
10. ()
. 2 1
( ) ( )A B B A
B A
A B
B A
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40 :
= = ={ , , , } , () = 4 . : , = { } , () = 1.
4
1
)(
)()(
.//
11. : 10 (), 15 () , 5 () 10
(). .
: i) ii) iii)
10 + 15 + 5 + 10 = 40 , () = 40
i) : . () = 15
8
3
40
15)(
ii) : . () = 10
4
1
40
10)(
: , , .
8
5
4
1
8
3)()()(
iii) : , :
, 8
5 .//
12. 30 , .
4 11 9 3 2 1 0 1 2 3 4 5
, .
30 , ()=30 . 3 2 .
: .
() = 9
() =30
9
)(
)(
.//
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41 :
13. = { / 10 20} , = { / 3} = { / 4}. ,
i)
ii)
10 20 .
= { 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 } () = 11
3.
={ 12 , 15 ,18 } () = 3
4.
= {12 , 16 , 20 } () = 3
i) 11
3
)(
)()(
ii) .
() = 1 (). () = () = 111
8
11
3 .//
14. 30% ,
20% 40% .
i) ii)
: ,
() =100
30 , () =
100
20 () =
100
40
i) .
2
1
100
50
100
20
100
30)()()(
ii) )(
100
30
100
40
100
301)()(1)(1)(
( ) .//
11
3
)(
)(
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42 :
15. 30
17)(
7 2( ) ( ) . ( )
15 3
)()()()( 17 72 ( )3 30 15
17 7 2( )30 15 3
30
11)( .//
16.
12
1)(
3
2() ,
2
1)( . () .
() = 1 () = 3
2 () = 1 2
3 = 1
3
( )()()() 1 1 1( ) 2 3 12
= 6 4 112 12 12
= 9 312 4
.//
17.
)()()(
P(AB) 0 ( ) ( ) ( )
( 2 P(AB), ) .//
18. D V. 25% D , 55% V 15% . ,
;
D , D. (D) =100
25,
V , V. (V) =100
55 .
, (DV) .
15(D V)100
.
, , (DV) .
25 55 15(D V) (D) (V) (D V) (D V)
100 100 100 P(DV ) = 65
100 .//
3
2
)()()()(
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43 :
19. 10% , 6%
2% . ) )
: = , () =100
10
= , () =100
6
T : , ,
100
2)(
) : .
( ) ( ) ( ) ( ) = 10 6 2 14 100 100 100 100
) : ( ) ( ) , ,
P ( ) ( ) (1)
P( ) = () () = 100
10 2
100 = 8
100
( ) = () () = 6100
2100
= 4100
(1) P = 8
100 + 4
100 = 12
100 .//
20. 80% , 30% 20% . . ,
: , () =100
80
: , () =100
30
, 20( )100
( ) 1 ( )
= 1 [ ( ) ( ) ( ) ]
= 1 ( ) ( ) ( )
= 80 30 20 101 100 100 100 100
.//
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44 :
21. 15% , 40% 10% . . .
= () =100
15
= () =100
40
, ( )
100
10)(
, , . ( ).
Venn, () = ( )
( ) = ()
= 1 () = 1 [() + () - ()]
= 1 () () + ()
= 1 15 40 10 55 100 100 100 100
.//
22. 0 < () < 1 4)(
1
)(
1
() = 0 < < 1, ( ) = 1 > 0.
1
+ 11
4
1 + 4(1 )
1 4 4 2
4 2 4 + 1 0
(2 1 2) 0 .//
23. () = 0,6 () = 0,7 , 0,3 ( ) 0,6
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45 :
( ) 0,6
( ) () ( ) 0,6
0,3 ( )
( ) = ( ) ( ) ( ) ( ) = 0,6 + 0,7 ( )
( ) = 1,3 ( ) (1)
0,3 ( ) ( 1 )
0,3 1,3 ( )
1 ( )
( ) 0 .//
24. ,
( ) ( ) ( )
( ) ( ) ( ) ( ) (1 ( )) ( ) ( ) ( )
() 1 + () () + () ( )
1 + ( )
( ) 1 .//
25. = { 0 , 1 , 2 , 3 ,, 100}.
2
1)( ,
= 1 , 2 , 3 , , 100 . (0)
(0) + (1 ) + (2) + . . . + P(100) = 1
2 3 100
1 1 1 1(1) , (2) , (3) , . . . , (100)
2 2 2 2
(0) + 2 3 100
1 1 1 1 . . . 1
2 2 2 2 (1)
2 3 100
1 1 1 1 . . .
2 2 2 2 100
11 1
2 2
,
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46 :
100S = 100
1 ( 1)
1
100
100
1 11
2 2S
11
2
= 100
11
2
= 1 100
1
2
(1) (0) + 1 100
1
2 = 1 (0) =
100
1
2 .//
26. ,
. () 0,28 () 0,71 . i) ( ) 1,01 ( ) ii)
() 0,28 1 ( ) 0,28 ( ) 0,72 (1)
() 0,71 1 ( ) 0,71 ( ) 0,29 (2)
i) ( ) + ( ) 1,01
( ) ( ) ( ) ( )
( ) ( ) = ( ) ( )
( ) ( ) ( 1 ),( 2 )
0,72 + 0,29
( ) ( ) 1,01
ii) , ( 0)()
(i) 0 1,01 ( )
( ) 1,01 ,
1 , .//
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47 :
1. = [ 2
2 3x y
4
3xy ] :
33
1
x
y
i) = 9x 9y
ii) x = 2010 y = 12010
i) = [ 2
2 3x y
4
3xy ] :
33
1
x
y
= [ 4x 6y 4x 12y ] : 9
3
x
y
= 6y : 9 3
1
x y = 6y 9x 3y = 9x 9y
ii) = (x y 9) = 9
12010.
2010 = 91 = 1 . //
2. 3 2
2
2
3 2
2
2
=
2( 2 1)
( 1)
=
2( 1)
( 1)
= + 1 . //
3. 2
2
( ) 2 2
1
2
2
( ) 2 2
1
=
( 1) 2( 1)
( 1)( 1)
=
( 1)( 2)
( 1)( 1)
= 2
1
. //
4. 2
1
3 2
3( 1)
2
1
3 2
3( 1)
=
22 1
2
3
( 1)
( 1)
= 2
2
[( 1)( 1)]
2
3
( 1)
( 1)
= 2 2
2
( 1) ( 1)
2
2( 1)
= 2( 1) . //
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48 :
5. (x + y 2) . 2
1 1x y
(x + y 2) . 2
1 1x y
= (x + y 2)2
1 1
x y
= (x + y 2)2
y x
xy
= (x + y 2)2
xy
x y
= (x + y 2)
2 2
2
x y
x y = 2 2x y . //
6. 2 + 2 2 + 1 0. ;
2 + 2 2 + 1 = ( 2 2 +1) + 2 = ( 1 2) + 2 0 + 0 = 0
2 + 2 2 + 1 = 0 ( 1 2) + 2 = 0
1 = 0 = 0
= 1 = 0 . //
7. x y :
i) (x 2 2) + (y + 1 2) = 0 ii) 2x + 2y 2x + 4y + 5 = 0
i) (x 2 2) + (y + 1 2) = 0 x 2 = 0 y + 1 = 0
x = 2 y = 1
ii) 2x + 2y 2x + 4y + 5 = 0 2x + 2y 2x + 4y + 1 + 4 = 0
( 2x 2x + 1) + ( 2y + 4y + 4) = 0
(x 1 2) + (y + 2 2) = 0
x 1 = 0 y + 2 = 0
x = 1 y = 2 . //
8. 4,5 < x < 4,6 5,3 < y < 5,4,
i) x + y ii) x y iii) xy
iv) 2x + 2y
-
49 :
i) : 4,5 + 5,3 < x + y < 4,6 + 5,4 9,8 < x + y < 10 ii) 5,3 < y < 5,4 5,3 > y > 5,4 5,4 < y < 5,3 (1) 4,5 < x < 4,6 (2)
(2) + (1) : 0,9 < x y < 0,7
iii) 5,3 < y < 5,4 15,3
> 1y
> 15,4
1053
> 1y
> 1054
1054
< 1y
< 1053
(3)
(2).(3) : 4,5. 1054
< x. 1y
< 4,6. 1053
4554
< xy
< 4653
iv) 4,5 < x < 4,6 (4,5 2) < 2x < (4,6 2) 20,25 < 2x < 21,16 (4)
5,3 < y < 5,4 (5,3 2) < 2y < (5,4 2) 28,09 < 2y < 29,16 (5)
(4) + (5) : 48,34 < 2x + 2y < 50,32 . //
9. > 1 > , + > 1+.
1 > 0
1 1 > 0
1 1 > 0 (1)
> 1 > > 1 1 >
1 > 0 1 > 0
1 1 > 0 . //
10. , , ( + ) 1 1
4
4
(.. = > 0, , > 0)
4
2 2 4 0
2 22 0
2
0 . //
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50 :
11. :
i) 2 2 0 ii) 2 2 0
i) 2 2 0 2 22 2 2 0
2 + 2 2 22 0
2 + 2
+ 2 0
ii) 2 2 0 2 22 2 2 0
2 + 2 2 22 0
2 + 2
+ 2 0 . //
12. .
i) 3 ii) 4
iii) 3 + 4 iv) 2 3 3 2
i) 3 = 3
ii) 4 = ( 4) = 4
iii) 4 = (3 ) + 4 = 3 + + 4 = 1
iv) 2 3 3 2 = ( 2 3 ) ( 3 2 ) = 2 + 3 3 + 2 = 0 . //
13. 3 < x < 4, x 3 + x 4
3 < x x 3 > 0 x 3 = x 3
x < 4 x 4 < 0 x 4 = (x 4) = x + 4
x 3 + x 4 = x 3 x + 4 = 1 . //
14. x 0 y 0,
= x
x+
y
y
x, y : = xx
+ y
y = 1 + 1 = 2
x, y : = xx
+ y
y
= 1 1 = 2
x , y : = xx
+ y
y
= 1 1 = 0
x , y : = xx
+ y
y = 1 + 1 = 0 . //
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51 :
15. , .
x 4 2 d x,4 2 [2, 6]
x 3 4 d(x, 3) < 4 (7, 1)
x 4 2 d(x, 4) > 2 ( , 2) (6, + )
x 3 4 d(x, 3) 4 ( , 7] [1, + )
x 5 < 1 d(x, 5) < 1 (4, 6)
x 1 > 2 d(x, 1) > 2 ( , 3) (1, + )
x 5 1 d(x, 5) 1 ( , 4] [6, + )
x 1 2 d(x, 1) 2 [3, 1]
x < 2 d(x, 0) < 2 (2, 2)
x 2 3 d(x, 2) 3 [5, 1]
x 2 d(x, 0) 2 ( , 2] [2, + )
x 2 > 3 d(x, 2) > 3 ( , 5) (1, + )
16. +
=
= ( ) ( ) + ( ) . //
17. > , :
i) = 2
ii) =
2
> > 0 =
i) 2
=
2
= 2
2
=
ii) 2
=
( )
2
=
2
=
2
2
= . //
-
52 :
18. x y :
i) x + y = 0 ii) x + y > 0
i) x + y = 0 x = 0 y = 0
, x, y 0, ( x 0),
x > 0
x + y > 0,
ii) x + y > 0 x 0 y 0
, x = 0 y = 0
x + y = 0, . //
19. 0 < < .
i) 1,
,
ii)
1,
i) 0 < <
< 1 1 <
.
< 1 <
ii) (i)
1 < 0
1 > 0
1
= 1
1
=
1
1
< 1
1
<
1
+ 1 <
1
2 + < 2
0 < 2 + 2 2
0 < 2
. //
20.
i) 2( 4) ii) 2( 20) iii) 2(x 1) iv) 2x
4
-
53 :
i) 2( 4) = 4 = 4
ii) 2( 20) = 20 = 20
iii) 2(x 1) = x 1
iv) 2x
4 =
2
2
x
2 =
2x
2 = x
2 =
x
2 . //
21. 2
2 5 + 2
3 5 = 1
2
2 5 + 2
3 5 = 2 5 + 3 5 = 5 2 + 3 5 = 1 . //
22. ( x 5 x 3 )( x 5 + x 3 ) = 8
( x 5 x 3 )( x 5 + x 3 ) = ( x 5 2) ( x 3 2)
= x 5 (x + 3)
= x 5 x 3 = 8 . //
23. :
i) ( 8 18 )( 50 + 72 32 ) = 14
ii) 28 7 32 63 32 = 31
i) 8 18 50 72 32 2 2 2 2 2 42 .2 3 .2 5 .2 2 .3 .2 2 .2
22 2 3 2 5 2 2.3 2 2 2 2 5 2 6 2 4 2 2.7 2 14
ii). 28 7 32 63 32 = 2 4 2 42 .7 7 2 .2 3 .7 2 .2 = 2 7 7 4 2 3 7 4 2 = 3 7 4 2 3 7 4 2
= 2 2
3 7 4 2 = 63 32 = 31 . //
-
54 :
24. :
i) 2 . 2 2 . 2 2 = 2 ii) 3 2 . 3 3 5 . 3 3 5 = 2
i) 2 . 2 2 . 2 2 2 2 2 2 2 2 4 2 2.2 2
ii) 3 2 . 3 3 5 . 3 3 5 = 3 2 3 5 3 5 3 33 2 9 5 2.4 8 2 . //
25. :
i) 32 2 = 3 2 ii) 5 32 2 2 = 3 2
i) 32 23 123 4 32 .2 2 2
ii). 5 32 2 2 5 5 56 303 63 4 6 4 10 32 2 .2 2 2 2 .2 2 2 . //
26. :
i) 34 3 . 3 3 = 3 12 3 ii) 9 82 . 6 52 = 218 132
i) 34 3 . 3 3 = 3
431
33 = 3 1
4 33
= 13
123 = 1
1123
= 3. 1
123 = 3 12 3
ii) 9 82 . 6 52 = 8
925
62 = 8 5
9 62
= 31
182 = 13
1182
= 2. 13
182 = 2 18 132 . //
27.
:
i) 4
5 3 ii) 8
7 5 iii)
7 6
7 6
-
55 :
i) 4
5 3 =
4(5 3 )
(5 3 )(5 3 )
=
2
4(5 3 )
5 3
=
4(5 3 )
22
=
2(5 3 )
11
ii) 8
7 5 =
8( 7 5 )
( 7 5 )( 7 5 )
=
8( 7 5 )
7 5
= 4( 7 5 )
iii) 7 6
7 6
=
2( 7 6 )
( 7 6 )( 7 6 )
=
7 2 7 6 6
7 6
= 13 + 2 42 . //
28. :
i) 162 98
50 32
= 16 ii)
12 20
11 6
9 3
9 27
= 3,
i) 162 98
50 32
=
4 2
2 2
2 . 3 2 . 7
2 . 5 2 . 4
=
23 2 7 2
5 2 4 2
= 16 2
2 = 16
ii) 12 20
11 6
9 3
9 27
=
2 12 20
2 11 3 6
(3 ) 3
(3 ) (3 )
=
24 20
22 18
3 3
3 3
=
20 4
18 4
3 (3 1)
3 (3 1)
= 23 = 3 . //
29. i) 3 3 2 2
3 2
= 5 + 6
ii) , > 0
= ( ) +
i) 3 3 2 2
3 2
=
(3 3 2 2 )( 3 2 )
( 3 2 )( 3 2 )
=
3.3 3 6 2 6 4
3 2
= 5 + 6
ii)
=
( )( )
( )( )
= . .
= 2 2
= ( )( ) ( )
= ( ) + . //
-
56 :
30. i) 2
3 2 7 , 2
3 2 7
ii) 37 12 7 37 12 7 = 6
i) 2
3 2 7 = 9 + 12 7 + 28 = 37 + 12 7
2
3 2 7 = 9 12 7 + 28 = 37 12 7
ii) 2 2
37 12 7 37 12 7 3 2 7 3 2 7
= 3 2 7 3 2 7
= 3 2 7 3 2 7 = 6.
3 2 7 0 3 2 7 0 . //
31.
i) 35 3
+ 55 3
= 4 ii)
21
2 3
2
1
2 3 = 8 3
i) 35 3
+ 55 3
=
3 5 3 5 5 3
5 3 5 3
= 15 3 5 155 3
= 82
= 4
ii)
21
2 3
2
1
2 3 =
2 2
2 2
2 3 2 3
2 3 2 3
= 2
(4 4 3 3) (4 4 3 3)
(2 3 )(2 3 )
=
2
4 4 3 3 4 4 3 3
4 3
= 8 3
1 = 8 3 . //
-
57 :
1. 4x 3(2x 1) = 7x 42
4x 3(2x 1) = 7x 42 4x 6x + 3 = 7x 42
9x = 45
x = 5 . //
2. 1 4 1 4 5
5 4 20 4
x x x
1 4 1 4 5
5 4 20 4
x x x 4 1 4 5 1 4 5.5x x x
4 16 5 5 4 25x x x
16 5 4 5 4 25x x x
22 22x 1x . //
3. 2(3x 1) 3(2x 1) = 4
2(3x 1) 3(2x 1) = 4 6x 2 6x + 3 = 4
0x = 3 . //
4. 2x 5 x3
= 53
+ 7 x3
2x 5 x3
= 53
+ 7 x3
6x (5 x) = 5 + 7x
6x 5 + x = 5 + 7x
0x = 0 , x . //
-
58 :
5. ( 1)x = 1 , .
( 1) = 0, = 0 1 = 0
= 0 = 1.
) = 0, 0(0 1)x = 0 1
0x = 1
) = 1, 1(1 1)x = 1 1
0x = 0 ,
x
( 1) 0, 0 1 0
0 1
x = 1( 1)
= 1
. //
6. ( 1)x = 2 + , .
( 1) = 0, = 0 1 = 0
= 0 = 1.
) = 0, 0(0 1)x = 20 + 0
0x = 0 ,
x
) = 1, 1(1 1)x = 21 + 1
0x = 2
( 1) 0, 0 1 0
0 1
x = 2
( 1)
=
( 1)
( 1)
= 1
1
. //
7. :
i) 1R
= 1
1
R +
2
1
R ( 1R )
i) : R, 1R , 2R 0
1
R =
1
1
R +
2
1
R 1R 2R = R 2R + R 1R
-
59 :
1R 2R R 1R = R 2R
(2R R) 1R = R 2R (1)
2R R = 0, 2R = R
(1) 0 = R 2R R = 0 2R = 0
2R R 0, 2R R
(1) 1R = 2
2
RR
R R. //
8. ( 2x 4)(x 1) = ( 2x 1)(x 2)
( 2x 4)(x 1) = ( 2x 1)(x 2) (x 2)(x + 2)(x 1) (x 1)(x + 1)(x 2) = 0
(x 1)(x 2)[x + 2 (x + 1)] = 0
(x 1)(x 2)(x + 2 x 1) = 0
(x 1)(x 2) = 0
x 1 = 0 x 2 = 0
x = 1 x = 2 . //
9. 2
x 1
x 1
+
2
2
x 2x 1 = 0
: 2x 1 0 2x 2x + 1 0
(x 1)(x + 1) 0 (x 1 2) 0
x 1 0 x + 1 0
x 1 x 1
x 1(x 1)(x 1)
+
2
2
(x 1) = 0
1x 1
+ 2
2
(x 1) = 0
1 + 2x 1
= 0
x 1 + 2 = 0
x = 1 . //
-
60 :
10. 1x 1
+ 1x 1
= 2
2
x 1
: .. = (x 1)(x + 1) 0
x 1 0 x + 1 0
x 1 x 1
x + 1 + x 1 = 2
2x = 2
x = 1 . //
11. 2x 3 = 5
2x 3 = 5 2x 3 = 5 2x 3 = 5
2x = 8 2x = 2
x = 4 x = 1 . //
12. 2x 4 = x 1
2x 4 = x 1 2x 4 = x 1 2x 4 = (x 1)
x = 3 2x 4 = x + 1
x = 3 3x = 5
x = 3 x = 53
. //
13. x 2 = 2x 1
: x 2 0,
2x 1 0 2x 1 x 12
x 2 = 2x 1 x 2 = 2x 1 x 2 = (2x 1)
x = 1 x 2 = 2x + 1
x = 1 3x = 3
x = 1 x = 1
, x = 1 . //
-
61 :
14. x 4
3
x 4
5
= 2
3
x 4
3
x 4
5
= 2
3 5 x + 20 3 x 12 = 10
2 x = 2
x = 1
x = 1 x = 1 . //
15. 2 x 1
3
x 1
2
= 1
2
2 x 1
3
x 1
2
= 1
2 4 x + 2 3 x + 3 = 3
x = 2 . //
16. 3 x3 x
= 4
: 3 + x 0 x 3
3 x
3 x
= 4 3 x
3 x
= 4 3 x
3 x
= - 4
3 x = 12 + 4x 3 x = 12 4x
5x = 9 3x = 15
x = 95
x = 5 . //
17. x 1 x 2 = x 1
x 1 x 2 = x 1 x 1 x 2 x 1 = 0
x 1 ( x 2 1) = 0
x 1 = 0 x 2 1 = 0
x 1 = 0 x 2 = 1
x 1 = 0 x 2 = 1 x 2 = 1
x = 1 x = 3 x = 1
x = 1 x = 3 . //
-
62 :
18. (x + 2) (x 2) = 2( + ) ,
, .
(x + 2) (x 2) = 2( + ) 2x + 2x + 2 ( 2x 2x + 2 ) = 2 2 + 2
2x + 2x + 2 2x + 2x 2 = 2 2 + 2
2( + )x = 2 + 2 + 2
2( + )x = ( + 2) (1)
+ 0, (1) x = 2
,
+ = 0, (1) 0x = 0
. //
19. , ,
x
x
= 1;
0 x
x
= 1 x x =
( )x = (1)
0, , (1) x =
= 0, = ,
(1) 0x = 2 0, .
.
, 0 0 . //
20. xx
=
2
2 2
x
x .
: .. = (x )(x + ) 0 x 0 x + 0 x x
xx
=
2x
(x )(x )
-
63 :
x = 2x
x
( x 2) = 2x
2x + 2x + 2 = 2x
2x = 2 (1)
0, (1) x = 2
= 0, (1) 0x = 0 ,
x x x . //
21. 2 x 1= 3
2 x 1= 3 2 x 1 = 3 2 x 1 = 3
2 x = 4 2 x = 2
x = 2 x = 1
x = 2 x = 2 . //
22. 2x 2x 1 = 3x 5
: 2x 2x + 1 0
(x 1 2) 0, x
2(x 1) = 3x 5
x 1 = 3x 5
x 1 = 3x 5 x 1 = (3x 5)
2x = 4 x 1 = 3x + 5
x = 2 4x = 6
x = 2 x = 32
. //
23.
i) 3x 125 = 0 ii) 5x 243 = 0 iii) 7x 1 = 0
-
64 :
i) 3x 125 = 0 3x 35 = 0 3x = 35 x = 5
ii) 5x 243 = 0 5x 53 = 0 5x = 53 x = 3
iii) 7x 1 = 0 7x = 71 x = 1 . //
24.
i) 3x + 125 = 0 ii) 5x + 243 = 0 iii) 7x + 1 = 0
i) 3x + 125 = 0 3x + 35 = 0 3x = 35 3x = (5 3) x = 5
ii) 5x + 243 = 0 5x + 53 = 0 5x = 53 5x = (3 5) x = 3
iii) 7x + 1 = 0 7x = 71 7x = (1 7) x = 1 . //
25.
i) 2x 64 = 0 ii) 4x 81 = 0 iii) 6x 64 = 0
i) 2x 64 = 0 2x = 64 2x = 28 x = 8 x = 8
ii) 4x 81 = 0 4x = 81 4x = 43 x = 3 x = 3
iii) 6x 64 = 0 6x = 64 6x = 62 x = 2 x = 2 . //
26.
i) 5x 8 2x = 0 ii) 4x + x = 0 iii) 5x + 16x = 0
i) 5x 8 2x = 0 2x ( 3x 8) = 0
2x = 0 3x 8 = 0
x = 0 3x = 32 x = 0 x = 2
ii) 4x + x = 0 x ( 3x + 1) = 0
x = 0 3x + 1 = 0
x = 0 3x = 1
x = 0 3x = (1 3)
x = 0 x = 1
iii) 5x + 16x = 0 x ( 4x + 16) = 0
x = 0 4x + 16 = 0 x = 0 . //
-
65 :
27.
i) (x + 1 3) = 64 ii) 1 + 125 3x = 0 iii) (x 1 4) 27(x 1) = 0
i) (x + 1 3) = 64 (x + 1 3) = 34 x + 1 = 4 x = 3
ii) 1 + 125 3x = 0 125 3x = 1
35 3x = 1
(5x 3) = (1 3)
5x = 1 x = 15
iii) (x 1 4) - 27(x 1) = 0 (x 1) [(x 1 3) - 27] = 0
x 1 = 0 (x 1 3) 27 = 0
x = 1 (x 1 3) = 27
x = 1 (x 1 3) = 33
x = 1 x 1 = 3 x = 1 x = 4 . //
28.
i) 2 2x 5x + 3 = 0 ii) 2x 6x + 9 = 0 iii) 3 2x + 4x + 2= 0
i) = 25 24 = 1, x = 5 14
= 64
1 = 32
1
ii) = 36 36 = 0, x = 2
= 6
2.1
= 3 ( )
iii) = 16 24 = 8 < 0, . //
29.
i) 2x 1,69 = 0 ii) 0,5 2x x = 0 iii) 3 2x + 27 = 0
i) 2x 1,69 = 0 2x = 1,69 2x = (1,3 2) x = 1,3 x = 1,3
ii) 0,5 2x x = 0 x(0,5x 1) = 0 x = 0 0,5x 1 = 0 x = 0 0,5x = 1 x = 0 x = 2
iii) 3 2x + 0x + 27 = 0
= 20 4. 3 . 27 = 324 < 0, . //
-
66 :
30. 2x + ( + )x + = 0, 0 .
= ( + 2) 4 = 2 + 2 + 2 4
= 2 2 + 2
= ( 2) 0.
, ( = 0) ( < 0) . //
31. , 2x + 2x + = 0, 0 .
0, 2 . H = 0 4 4 = 0
4 4 2 = 0
2 = 1 = 1 = 1 . //
32. , ( 2 + 2 ) 2x + 2( + )x + 2 = 0. =
2 + 2 0, , 0,
2 ,
= 4 2( ) 8 ( 2 + 2 )
= 4( 2 + 2 + 2 ) 8 2 8 2
= 4 2 + 8 + 4 2 8 2 8 2
= 4 2 + 8 4 2
= 4( 2 2 + 2 )
= 4 2( ) < 0,
2 + 2 = 0, = 0 = 0,
2 = 0 . = = = 0, 2 = 0
= 0 ,
-
67 :
( 2 + 2 ) 2x + 2( + )x + 2 = 0
2 2 2x + 4 x + 2 = 0
2 2x + 2 x + 1 = 0, 2 0
= 4 2 4 2 = 0, . //
33. 2
i) 2 3 ii) 1 12
iii) 5 2 6 5 + 2 6
i) S = 2 + 3 = 5, P = 2. 3 = 6, 2x 5x + 6 = 0
ii) S = 1 + 12
= 32
, P = 1. 12
= 12
, 2x 32
x + 12
= 0
2 2x 3x + 1 = 0
iii) S = 5 2 6 + 5 + 2 6 = 10, P = (5 2 6 )(5 + 2 6 ) = 25 24 = 1
H 2x 10x + 1 = 0 . //
34. , , i) 2 15
ii) 9 10
i) 2x Sx + P = 0,
S = 2 P = 15 2x 2x 15 = 0
= 4 + 60 = 64, x = 2 82
= 5, 3
ii) 2x Sx + P = 0,
S = 9 P = 10 2x 9x + 10 = 0
= 81 40 = 41, x = 9 41
2
=
9 41
2
,
9 41
2
. //
35.
i) 2x ( 5 + 3 )x + 15 = 0 ii) 2x + ( 2 1)x 2 = 0
i) S = 5 + 3 P = 15 = 5 3
5 , 3
-
68 :
ii) = ( 2 1 2) 4 2 = 2
2 2 2 + 1 4 2
= 2
2 + 2 2 + 1
= ( 2 + 1 2)
x = ( 2 1) ( 2 1)
2
= 2 1 2 1
2
2 1 2 12
= 22
2 22
= 1 2 . //
36.
i) 2x 7 x + 12 = 0 ii) 2x + 2 x 35 = 0 iii) 2x 8 x + 12 = 0
i) 2x 7 x + 12 = 0 2
x 7 x + 12 = 0
= 49 48 = 1, x = 7 12
= 4 3
x = 4 4 3 3
ii) 2x + 2 x 35 = 0 2
x + 2 x 35 = 0
= 4 + 140 144, x = 2 122
= 5 7 x 0.
x = 5 5
iii) 2x 8 x + 12 = 0 2
x 8 x + 12 = 0
= 64 48 = 16, x = 8 42
= 6 2
x = 6 6 2 2 . //
37. (x 1 2) + 4 x 1 5 = 0
(x 1 2) + 4 x 1 5 = 0 2
x 1 + 4 x 1 5 = 0
= 16 + 20 = 36, x 1 = 4 62
= 1 5 x 1 0.
x 1 = 1 x 1 = 1 x 1 = 1
x = 2 x = 0 . //
-
69 :
38. 2
1x
x 5 1x x + 6 = 0
: x 0
1xx
= y (1), 2y 5y + 6 = 0
= 25 24 = 1, y = 5 12
= 3 2
y = 3, (1) 1xx
= 3
2x + 1 = 3x
2x 3x + 1 = 0
= 9 4 = 5, x = 3 5
2
y = 2, (1) 1xx
= 2
2x + 1 = 2x
2x 2x + 1 = 0
= 4 4 = 0, x = 2 02
= 1 . //
39. xx 1
+ x 1x
= 136
: x 0 x 1
x
x 1+ x 1
x
= 136
6 2x + 6(x + 1 2) = 13x(x + 1)
6 2x + 6( 2x + 2x + 1) = 13 2x + 13x
6 2x + 6 2x + 12x + 6 13 2x 13x = 0
2x x + 6 = 0
2x + x 6 = 0
= 1 + 24 = 25, x = 1 52
= 2 3 . //
-
70 :
40. 2x
+ 2x 3x 2
+
2
2
2 x
x 2x
= 0
: 2x 2x = x(x 2)
E.K. = x(x 2) 0 x 0 x 2
2
x+ 2x 3
x 2
+
2
2
2 x
x 2x
= 0 2(x 2) + x(2x 3) + 2 2x = 0
2x 4 + 2 2x 3x + 2 2x = 0
2x x 2 = 0
= 1 + 8 = 9, x = 1 32
= 1 2 , x = 1 . //
41. 4x + 6 2x 40 = 0
4x + 6 2x 40 = 0 ( 2x 2) + 6 2x 40 = 0
= 36 + 160 = 196, 2x = 6 196
2
= 6 14
2
= 4 10
2x = 4 x = 2 x = 2 . //
42. 2 4x + 7 2x + 3 = 0
2 4x + 7 2x + 3 = 0 2( 2x 2) + 7 2x + 3 = 0
= 49 24 = 25, 2x = 7 54
= 24
3 2x 0 . //
43. 2 2x 2 3 x + 4 1 = 0, 0.
i) = 4 2 .
ii) 2 1
2 1
i) = (2 3 2) 4 2 ( 4 1) = 4 6 4 6 + 4 2 = 4 2
ii) x = 3 2
2
2 4
2
=
3
2
2 2
2
=
2
2
2 ( 1)
2
=
2 1
. //
-
71 :
44. 2x (5 2 )x + 6 3 2 = 0
i) = (1 + 2 2)
ii) 3 2 2
i) = (5 2 2) 4(6 3 2 )
= 25 10 2 + 2 24 + 12 2
= 3 + 2 2 = 2 + 1 + 2 2 = 2
2 + 21 + 2 2 = (1 + 2 2)
ii) x = 5 2 (1 2
2
=
5 2 (1 2 )
2
5 2 (1 2 )
2
= 5 2 1 22
5 2 1 22
= 3 4 2 22
= 3 2 2 . //
45. 2x + x + = 0, . 0,
1
2x + x + = 0.
2x + x + = 0 2 + + = 0
( 2 ) + 1
+ 2
1
= 0
+ 1
+ 2
1
= 0
1
2 + + = 0, . //
46. x + 1
= + 1x
, 0
: x 0
x + 1
= + 1x
2x + x = 2 x +
2x + x 2 x = 0
2x + (1 2 ) x = 0
= (1 2 2) + 4 2 = 1 2 2 + 4 + 4 2 = 1 + 2 2 + 4 = (1 + 2 2)
= 2 21 (1 )
2
=
2 21 1
2
2 21 1
2
= 1
. //
-
72 :
47. 2x + 2x 8 = 0.
i) .
ii) ,
.
i) = 4 2 + 32 > 0 R,
R.
ii) 1x , 2x 2
1x = 2x (1)
1x + 2x = 2 (2) 1x 2x = 8 (3) Vieta
(3) ( 1 )
1x2
1x = 8 3
1x = 8 1x = 2
(1) (2 2) = 2x 2x = 4
(2) 2 + 4 = 2 2 = 2 = 1 . //
48. 4x 10 2x + = 0 1. .
. 41 10. 21 + = 0 1 10 + = 0 = 9
( 2x 2) 10 2x + 9 = 0
= 100 36 = 64, 2x = 10 82
= 9 1
x = 3 3 1 1 . //
-
73 :
3 xx 1
1. x 12 + 2x 3
4 < x
6
x 1
2 + 2x 3
4 < x
6 6(x 1) + 3(2x + 3) < 2x
6x 6 + 6x + 9 < 2x
10x < 3 x < 310
. //
2. x 122 + x
2 + 3
4 > x
x 12
2 + x
2 + 3
4 > x 2(x 12) + 2x + 3 > 4x
2x 24 + 2x + 3 > 4x
0. x > 21 0 > 21 . //
3. x 22 + 1 2x
5 < x
10 2
5
x 2
2 + 1 2x
5 < x
10 2
5 5(x 2) + 2(1 2x) < x 4
5x 10 + 2 4x < x 4
0. x < 4 xR . //
4. x
3x 1 < x + 5 2 x2
x + 12
3x 1 < x + 5 2x < 6 x < 3
2 x2
x + 12
4 x 2x + 1 3x 3 x 1
1 x < 3
-
74 :
3 xx -1
7
3-
1
7
xx
0x x < 0x < x < 0x +
x < < x <
5. :
x 12
> x2
+ 1 x 13 x
3 1
x 12
> x2
+ 1 2x 1 > x + 2 x > 3
x 13 x
3 1 3x 1 x 3 2x 2 x 1
6. xR :
2x x 18 > x x 4 + x 1
2 < 0
2x x 18 > x 16x x + 1 > 8x 7x > 1 x > 1
7
x 4 + x 12 < 0 2x 8 + x + 1 < 0 3x < 7 x < 7
3
17
< x < 73
1 7, 7 3 0, 1, 2 . //
7. :
i) x < 3 ii) x 1 4 iii) 2x 1 < 5
i) x < 3 3 < x < 3
ii) x 1 4 1 4 x 1 + 4
3 x 5
iii) 2x 1 < 5 5 < 2x + 1 < 5
5 1 < 2x < 5 1
6 < 2x < 4 3 < x < 2 . //
-
75 :
0x x > x < 0x x > 0x +
x > x < x >
0
x 1
8. :
i) x 3 ii) x 1 > 4 iii) 2x 1 5
i) x 3 x < 3 x > 3
ii) x 1 > 4 x 1 < 4 x 1 > 4
x < 3 x > 5
iii) 2x 1 5 2x + 1 5 2x + 1 5
2x 6 2x 4 x 3 x 2 . //
9. :
i) 2x 6 = 2x 6 ii) 3x 1 = 1 3x
i) 2x 6 = 2x 6 2x 6 0
2x 6
x 3
ii) 3x 1 = 1 3x 3x 1 = (3x 1)
3x 1 0
3x 1 x 13
. //
10. x 1 4
2
+ 5
3 <
x 1
3
x 1 4
2
+ 5
3 <
x 1
3
3( x 1 4) + 10 < 2 x 1
3 x 1 12 + 10 < 2 x 1
3 x 1 12 + 10 < 2 x 1
x 1 < 2
2 < x 1 < 2 1 < x < 3 . //
-
76 :
x
2 =
11. x 1
2
2 x
3 >
1 x
3
x 1
2
2 x
3 >
1 x
3
3( x + 1) 4 x > 2(1 x )
3 x + 3 4 x > 2 2 x
x > 1 xR . //
12. 2x 6x 9 5
2x 6x 9 5 2(x 3) 5
x 3 5
5 x 3 5 2 x 8 . //
13. 0x x < ,
(7, 3).
x(7, 3) 7 < x < 3 (1)
0x x < 0x < x < 0x + (2)
(1), (2) 0
0
x 7
x 3
0
( ) : 2x 4
( ) : 2 10
0
x 2
5
0x x < x ( 2) < 5 x 2 < 5 . //
14. x : 4 2 3x 2
4 2 3x 2 4 2 3x 2 2
6 3x 4
2 x 43
43
x 2 . //
-
77 :
x > x < x >
x < < x <
xx 4- 4 -2 2
xx 91 3 7
xx M(1) (7)K(x)
= d(, )
15. x :
i) 2 x 4 ii) 2 x 5 4
i) 2 x x 2
x 2 x 2 (1)
x 4 4 x 4 (2)
(1), (2)
4 x 2 2 x 4
ii) 2 2
x 5 2 x 5 2 x 3 x 7 (3)
x 5 4 4 x 5 4
4 + 5 x 4 + 5 1 x 9 (4) (3), (4)
1 x 3 7 x 9
16. 1 7 .
i) ;
ii) x 1 + x 7 = 6
. iii) , x 1 x 7.
i) 1 72 = 4
ii) (x)
x 1 + x 7 = 6 d(x, 1) + d(x, 7) = 6
() + () = 6 ( () = 7 1 = 6)
() + () = ()
1 x 7
x 5 x 5
-
78 :
iii) x < 1
x 1 + x 7 = 6 (x 1) + [ (x 7)] = 6
x + 1 x + 7 = 6 2x = 2 x = 1 , x < 1 1 x < 7
x 1 + x 7 = 6 (x 1) + [ (x 7)] = 6
x 1 x + 7 = 6 6 = 6 , 1 x < 7 x 7
x 1 + x 7 = 6 (x 1) + (x 7) = 6
x 1 + x 7 = 6 2x = 14 x = 7 , x = 7 1 x 7 . //
17. :
i) 2x 3x + 2 ii) 2 2x 3x 2
i) = (32) 4. 1 . 2 = 9 8 = 1 > 0
: x = ( 3) 1
2.1
= 3 1
2
= 3 12
3 12
= 2 1
2x 3x + 2 = 1.(x 2)(x 1) = (x 2)(x 1)
ii) = (32) 4. 2 . (2) = 9 + 16 = 25 > 0
: x = ( 3) 25
2.2
= 3 5
4
= 3 54
3 54
= 2
2 2x 3x 2 = 2(x 2) 1x 2 = (x 2)(2x + 1) . //
18. :
2 2x 3x 2 0
= 2x 3x + 2
= (32) 4. 1 . 2 = 9 8 = 1 > 0
1
2
2
2
x 3x 2
2x 3x 2
x 1 7 x 1 0 + + x 7 0 +
-
79 :
: x = ( 3) 1
2.1
= 3 1
2
= 3 12
3 12
= 2 1
2x 3x + 2 = 1.(x 2)(x 1) = (x 2)(x 1)
= 2 2x 3x 2
= (32) 4. 2 . (2) = 9 + 16 = 25 > 0
: x = ( 3) 25
2.2
= 3 5
4
= 3 54
3 54
= 2
2 2x 3x 2 = 2(x 2) 1x 2 = (x 2)(2x + 1) 2(x 2) 1x 2 0 x 2 0 x +
1
2 0
x 2 x 12
2
2
x 3x 2
2x 3x 2
=
(x 1)(x 2)
(x 2)(2x 1)
= x 1
2x 1
. //
19. : 2
2
4x 12x 9
2x 5x 3
2 2x 5x + 3 0
= 4 2x 12x + 9
= (122) 4. 4 . 9 = 144 144 = 0
x = 2
= 12
2. 4
= 32
4 2x 12x + 9 = 4 2
3x
2 = 22
23
x2
= (2x 32)
= 2 2x 5x + 3
= (52) 4. 2 . 3 = 25 24 = 1 > 0
: x = ( 5) 1
2. 2
= 5 1
4
= 64
44
= 32
1
2 2x 5x + 3 = 2 3x 2 (x 1) = (2x 3)(x 1)
2 3x 2 (x 1) 0 x 1 0 x 3
2 0
x 1 x 3
2
2
2
4x 12x 9
2x 5x 3
=
2(2x 3)
(2x 3)(x 1)
= 2x 3
x 1
. //
1
2
-
80 :
20. xR,
i) 2x 2x 15 ii) 4 2x 4x + 1 ii) 2x 4x + 13
i) = (22) 4. 1. (15) = 4 + 60 = 64 > 0
: x = ( 2) 64
2. 1
= 2 8
2
= 5 3
x 3 5 +
2x 2x 15 + 0 0 +
ii) = ( 42) 4. 4. 1 = 16 16 = 0
x = 2
= 4
2. 4
= 12
x 1/2 +
4 2x 4x + 1 + 0 +
iii) = ( 42) 4. 1. 13 = 16 52 = 36 < 0
x +
2x 4x + 13 +
21. xR,
i) 2x + 4x 3 ii) 9 2x + 6x 1 iii) 2x + 2x 2
i) = 24 4(1). (3) = 16 12 = 4 > 0
: x = 4 4
2( 1)
= 4 2
2
= 3 1
x 1 3 +
2x + 4x 3 0 + 0
ii) = 26 4(9). (1) = 36 36 = 0
x = 2
= 6
2( 9) = 1
3
-
81 :
x 1/3 +
9 2x + 6x 1 0
iii) = 22 4. (1)( 2) = 4 8 4 < 0
x +
+ 2x 2
22. : i) 5 2x 20x ii) 2x +3x 4
i) 5 20x 2x 4x 0
2x 4x = x(x 4) 0 4
x 0 4 +
2x 4x + 0 0 +
2x 4x 0 0 x 4 x[0, 4]
ii) 2x + 3x 4 2x + 3x 4 0
= 23 4. 1(4) = 9 + 16 = 25 > 0
2x +3x 4 : x = 3 25
2. 1
= 3 5
2
= 4 1
x 4 1 +
2x + 3x 4 + 0 0 +
2x + 3x 4 0 4 x 1 x[ 4, 1] . //
23. : i) 2x x 2 > 0 ii) 2 2x 3x 5 < 0
i) = 1 + 8 = 9 > 0 , 2x x 2 : x = 1 32
= 2 1
x 1 2 +
2x x 2 + 0 0 +
2x x 2 > 0 x < 1 x > 2 x( , 1) (2 , + )
2x
2x
-
82 :
ii) = 9 + 40 = 49 > 0, 2 2x 3x 5 : x = 3 74
= 52
1
x 1 5/2 +
2 2x 3x 5 + 0 0 +
2 2x 3x 5 < 0 1 < x < 52
x 51, 2 . //
24. :
i) 2x + 4 > 4x ii) 2x + 9 6x
i) 2x + 4 > 4x 2x + 4 - 4x > 0
(x 2 2) > 0 xR x 2
ii) 2x + 9 6x 2x + 9 6x 0
(x 3 2) 0
x 3 = 0 x = 3 . //
25. : i) 2x + 3x + 5 0 ii) 2 2x 3x + 20 > 0
i) = 9 20 = 11 < 0 = 1,
xR, . ii) = 9 160 = 151 < 0 = 2,
xR,
xR. . //
26. 14
( 2x 4x + 3) > 0
14
( 2x 4x + 3) > 0 4x + 3 < 0
= 16 12 = 4 > 0 , 2x 4x + 3 : x = = 3 1
x 1 3 +
2x 4x + 3 + 0 0 +
2x 4x + 3 < 0 1 < x < 3 x(1, 3) . //
2x
4 2
2
-
83 :
xx -4 4-1 3
xx 1 2 3 5
27. xR 2x 1 < 2x 4 < 12
2x 1 < 2x 4 < 12 2x 1 < 2x 4 2x 4 < 12
2.x 1 < 4 2x 3 > 0
= 4 + 12 = 16 > 0 , x = 2 42
= 3 1
x 1 3 +
2x 2x 3 + 0 0 +
2x 2x 3 > 0 x < 1 x > 3 (1)
4 < 12 2x < 16 x < 4 4 < x < 4 (2)
(1), (2)
4 < x < 1 3 < x < 4 . //
28. xR
2x 6x + 5 < 0 2x 5x + 6 > 0.
2x 6x + 5 < 0
= 36 20 = 16 > 0 , x = 6 42
= 5 1
x 1 5 +
2x 6x + 5 + 0 0 +
2x 6x + 5 < 0 1 < x < 5 (1)
2x 5x + 6 > 0
= 25 24 = 1 , x = = 3 2
x 2 3 +
2x 5x + 6 + 0 0 +
5x + 6 > 0 x < 2 x > 3 (2)
(1), (2)
1 < x < 2 3 < x < 5
2x 2x
2x
5 1
2
2x
-
84 :
29. i) :
2 + 2 2 6 2
ii)
i)
2 + 2 2
( x)
= 2 4. 1. (2 2 ) = 2 + 8 2 = 9 2 0
= 29
2
=
3
2
= 2
+ 2 2 = 1( )( + 2) (1)
2 6 2
( x)
= 2 4. 1. ( 6 2 ) = 2 + 24 2 = 25 2 0
= = = 3 2
2 6 = 1( 3)( + 2) (2)
ii) 6 2 0 3 2 2 2
2 2
2
6
( )( 2 )
( 3 )( 2 )
=
3
. //
30.
( )x -
= ( + 4
= 2 + 2 + 4
= 2 + 2 + 2 = ( + 2) 0
x = ( )
2
=
( )
2
( )
2
= 2
2
=
2x ( )x = (x )(x + ) (1)
2
2 2
2 2
2
6
2
225
2
5
2
2
2 (1 ), ( 2 )
2
2 2
x x x
x 3 x 2
2x2)
2
-
85 :
= 9 2 8 2 = 2 0
x = 32
= 2
3x + 2 2 = (x 2)(x ) (2)
2x 3x + 2 2 0 x 2 x 2
2 2
x x x
x 3 x 2
(1 ), ( 2 )
x
x 2
. //
31. 2x + 3x + + 5 = 0, R.
: i) ii) iii)
= 0 5 = 0 (1) 0 2 i) = 0
9 2 4( + 5) = 0
9 4 20 = 0
5 20 = 0
4 = 0 ( 4) = 0 4 = 0 0 = 4 ii) > 0
9 2 4( + 5) > 0
9 4 20 > 0
5 20 > 0
4 > 0
( 4) > 0
< 0 > 4
4 0 4 +
4 + 0 0 +
iii) < 0
9 2 4( + 5) < 0
9 4 20 < 0
5 20 < 0
4 < 0
( 4) < 0
0 < < 4 (2) (1), (2) 0 < 4 . //
2x
2 22
2
2 22
2
2
2
2 22
2
-
86 :
xx -3 -2 0
32. R 2x + 3x + > 0
xR
2x + 3x + > 0 xR
2x + 3x + = 1 xR
< 0
9 2 4 < 0 9 - 4
(9 4) < 0
0 < < 49
33. ( + 2) 2x 2x + 3, 2. i) < 0
ii) R
( + 2) 2x 2x + 3 < 0, 2
xR.
i) = 4 2 4( + 2). 3 = 4 2 12 2 24 = 8 2 24
< 0 8 2 24 < 0
2 + 3 > 0 + 3
( + 3) > 0
< 3 > 0
ii) ( + 2) 2x 2x + 3 < 0 xR
[ ( + 2) 2x 2x + 3
= + 2 x + 2 < 0 ]
< 0 + 2 < 0
[ < 3 > 0 ] < 2
< 3
34. i) 2 + 2 > 0 , R , 0.
ii) =
+
1
, 0.
2
2
0 4/9 +
9 2 - 4 + 0 0 +
0 4/9 +
9 2 4 + 0 0 +
-
87 :
i) ( x)
= 2 4 2 = 3 2 < 0 2 + 2 = 1, ( ) , 0.
ii) =
+
1 =
2 2
, . : , , > 0
, , < 0. . //
35. (x) = (2 3x)( 2x x 2)( 2x x + 1)
2 3x 0 3x 2 x 23
2x x 2
= 1 + 8 = 9 > 0 : 2 1
2x x 2 > 0 x < 1 x > 2
2x x + 1
= 1 4 = 3 < 0 2x x + 1 > 0 R
x 1 2/3 2 + 2 3x + + 0 0
2x x 2 + 0 0 + 2x x + 1 + + + +
P(x) + 0 0 + 0
36. (x) = ( 2x + 4)( 2x 3x + 2)( 2x + x + 1)
2x + 4
0 + 4 = 4 > 0 : 2 2
2x + 4 > 0 2 < x < 2
2x 3x + 2
= 9 8 = 1 > 0 : 1 2
2x 3x + 2 > 0 x < 1 x > 2
-
88 :
2x + x + 1
= 1 4 = 3 < 0 2x + x + 1 > 0 x
x 2 1 2 + 2 3x 0 + + 0
2x x 2 + + 0 0 + 2x x + 1 + + + +
P(x) 0 + 0 0 +
37. (x 1)( 2x + 2)( 2x 9) > 0 x 1 > 0 x > 1
2x + 2 > 0 R
2x 9 > 0 x < 3 x > 3
x 3 1 3 + x 1 0 + +
2x + 2 + + + +
2x 9 + 0 0 +
0 + 0 0 + 3 < x < 1 x > 3 . //
38. (3 x)(2 2x + 6x)( 2x + 3) > 0 3 x > 0 x < 3
2 2x + 6x > 0 2x(x + 3) > 0 x(x + 3) > 0 x < 3 x > 0
2x + 3 > 0 x
x 3 0 3 + 3 x + + + 0
2 2x + 6x + 0 0 + +
2x + 3 + + + +
+ 0 0 + 0 3 x 1 x 3 . //
-
89 :
39. (2 x 2x )( 2x + 2x + 1) 0
2 x 2x > 0 2x + x 2 < 0
= 1 + 8 = 9 > 0 : 2 1 2 < x < 1
2x + 2x + 1 > 0 (x + 1 2) > 0 x 1
x 2 1 1 +
2 x 2x 0 + + 0 2x + 2x + 1 + + 0 + +
0 + 0 + 0 x 2 x = 1 x 1 . //
40. (x 3)(2 2x + x 3)(x 1 2 2x ) > 0
x 3 > 0 x > 3
2 2x + x 3 > 0
= 1 + 24 = 25 > 0 1 54
= 1 32
2 2x + x 3 > 0 x < 32
x > 1
x 1 2 2x > 0 2 2x x + 1 < 0 = 1 8 = 7 < 0
x 1 2 2x < 0 R
x 3/2 1 3 + x 3 0 +
2 2x + x 3 + 0 0 + +
x 1 2 2x
+ 0 0 + 0
x < 32
1 < x < 3 . //
41. 2x 1x 3
0
: x 3 0 x 3 2x 1
x 3
0 (2x + 1)(x 3) 0 1
2 x < 3 . //
-
90 :
42. 2
2
x x 2
x x 2
0
: 2x + x 2 0 = 1 + 8 = 9 > 0 2 1 x 2 x 1
2
2
x x 2
x x 2
0 ( 2x x 2)( 2x + x 2) 0
2x x 2
= 1 + 8 = 9 > 0 1 2
2x x 2 > 0 x < 1 x > 2
2x + x 2
2x + x 2 > 0 x < 2 x > 1
x 2 1 1 2 + 2x x 2 + + 0 0 + 2x + x 2 + + +
+ 0 + 0 + 2 < x 1 1 < x 2 . //
43. 2x 3x 1
> 4
: x 1 0 x 1
2x 3
x 1
> 4 2x 3
x 1
4 > 0
2x 3 4x 4x 1
> 0
2x 7x 1
> 0
(2x + 7)(x 1) > 0 1 < x < 72
. //
44. 2x 3x 10
x 1
+ 2 0
: x 1 0 x 1
2x 3x 10
x 1
+ 2 0
2x 3x 10 2x 2
x 1
0
-
91 :
2x x 12
x 1
0 (x 1)( 2x - x 12) 0
x 1 > 0 x > 1
2x - x 12 > 0
= 1 + 48 = 49 > 0 1 72
= 3 , 4
2x x 12 > 0 x < 3 x > 4
x 3 1 4 + x 1 0 + 0 +
2x x 12 + +
0 + 0 + x 3 1 < x 4 . //
45. x3x 5
2x 1
: (3x 5 0 x 1 0) (x 53
x 1)
x
3x 5 2
x 1 x
3x 5 2
x 1 0
2x x 6x 10
(3x 5)( x 1)
0
2x 7 x 10
(3x 5)( x 1)
0
( x 2)( x 5)
(3x 5)( x 1)
0
(x 2)(x 5)(3x 5)(x 1) 0
x 1 5/3 2 5 + + + 0 0 +
1 < x < 53
2 x 5 . //
-
92 :
46. x 1x
> 2
: x 0
x 1
x
> 2 x 1x
< -2 x 1x
> 2
x 1x
+ 2 < 0 x 1x
- 2 > 0
x 1 2xx
< 0 x 1 2xx
> 0
3x 1x
< 0 x 1x
> 0
(3x + 1)x < 0 (x 1)x < 0
13
< x < 0 0 < x < 1 . //
-
93 :
1. :
i) = 2 , ii) = 2 1
1
i) 1 = 21=2, 2 = 22 = 4, 3 = 23 = 8, 4 = 24 = 16, 5 = 25 = 32
ii) 1 = 21 1
1 1
= 0 2 = 1, 3 = 2, 4 = 3, 5 = 4 . //
2.
i) 1 = 2, +1 = 1
ii) 1 = 0, +1=
2
+1 iii) 1 = 3, +1 = 2(1)
i) 1 = 2, 2 = 1
1
1
2 , 3 =
2
1
1
1
2
= 2 , 4 = 3
1
1
2 , 5 =
4
1
1
1
2
= 2
ii) 1 = 0, 2 = 2
1 +1 = 0 + 1 = 1, 3 = 2
2 +1= 1+ 1 = 2
4 = 2
3 +1= 4 + 1= 5, 5 = 2
4 +1 = 25 + 1 = 26
iii) 1 = 3, 2 = 2(11) = 22 = 4 , 3 = 2(21) = 23 = 6
4 = 2(31) = 25 = 10 , 5 = 2(41) = 29 = 18 . //
3.
i) = + 5, ii) = 2 ,
i) 1 = 6 +1 = ( + 1) + 5 ( + 5) 1 = 6 +1 = 1
1 = 6 +1= 1 + *
ii) 1 = 2 1
=
12
2
1 = 2 1
= 2
1 = 2 +1= 2 * . //
-
94 :
4. -
i) 1 = 1 +1 = + 2 ii) 1 = 3 +1 = 5
i) 1 = 1 2 = 1 + 2 3 = 2 + 2 . .. -1 = -2 + 2 = -1 + 2 = 1 + 2(1) = 1 + 22 =21 ii) 1 = 1 2 = 51
3 = 52
.
..
-1= 5-2
= 5-1
= 5-11 = 3 5-1 . //
5. - 7, 10, 13, . . . .
=
1 + ( 1) = 7 + ( 1) 3 = 7 + 3 3 = 3 + 4 . //
6. - 6, 9, 12, . . .
=
1 + ( 1) = 6 + ( 1) (3) = 6 3 + 3 = 3 3 . //
7. 15
2, 3, 8, . . .
15 =
1 + (15 1) = 2 + 14. 5 = -2 + 70 = 68 . //
8. , 6
= 12 10
= 16, 1
.
-
95 :
6
10
12
16
1
1
6 1 12
10 1 16
1
1
5 12
9 16
1
1
12 5
9 16
112 5
12 5 9 16
1
12 5
4 4
1
12 5.1
1
1
7
1
. //
9. 5 5 15 2. 50 .
5
15
5
2
1
1
5 1 5
15 1 2
1
1
4 5
14 2
1
1
5 4