E2 f6 bộ binh

CREATED BY HOÀNG MINH THI; [email protected] TRUNG ĐOÀN 2 – SƯ ĐOÀN 6 – QUÂN ĐOÀN BỘ BINH

1

CHUYÊN ĐỀ PHƯƠNG TRÌNH VÀ BẤT PHƯƠNG TRÌNH

BÀI TẬP SỬ DỤNG BIẾN ĐỔI TƯƠNG ĐƯƠNG – NÂNG CAO LŨY THỪA (PHẦN 2) --------------------------------------------------------------------------------------------------------------------------------------------- Bài 1. Giải các phương trình và bất phương trình sau trên tập hợp số thực

2 2

2 2

2 2

2 2 2

2 2

2 2

2 2

2 2

2

2

1, 1 1 1

2, 3 1 2 2 4 3

3, 1 3 1 2 3 4

4, 2 2 1 4 3 1

5, 2 2 1 1

6, 5 3 1 5 1 2

7,2 3 1 12 2 2

8, 1 2 3 4

9, 3 2 4 18 4

10, 2 1 1

11, 2 2

x x x x

x x x x

x x x x

x x x x

x x x x x x

x x x x

x x x x

x x x

x x x

x x x x x

x x

+ + − − + =

+ + + = + +

+ + + = − + +

+ − < + +

+ + ≥ + + + − +

+ − ≤ + − +

+ + > + +

+ = − + +

− + + − =

− − + − = +

− −

( ) ( )( )( )( )

( )

( )

2 3 2

2 3 2

2

2

2 2

2

2

2

2 2

2 2

2 1

12, 2 3 6 5 2 2 1

13, 1 4 5 7 3

14, 1 1 4 3 0

15, 5 4 11 30

12 1216,

4 6

17, 3 4 3 0

18, 1 6 5 0

4 419,

41 3

5 1 5 120,

4 6 4

21

x x x

x x x x x x

x x x x x x

x x x x

x x x x

x x x x

x x

x x x

x x x

x x

xx x

x x x x

x x x

+ − < +

− − + + ≥ +

− − ≤ − + − +

− − + − + =

− − + > +

− − − −≤

+ −

− − + ≤

− − + ≥

− −≤

+ −

− − − −≥

− − +2

2

2

2

2

2 5 3 1 4,

24 4 7 10 122,

2 3 2 4 323, 2 1

2 2 1 2 1

4 3 2 324, 3 3 2

3 3 2 3 2

5 4 4 1 425, 5 4

5 4 5 4

x x xx

x x

x x xx

x x

x x xx

x x

x x x xx

x x

x x x xx

x x

+ + +− >

− + + +− <

+ − + −≥ + −

− −

− − −< + −

− −

+ − + +≥ − +

− −


2

Bài 2. Giải các phương trình và bất phương trình sau trên tập hợp số thực

( ) ( )

2

2

2 3

2 3

2

2

2 2

2

2 2

1, 1 2 2 1

2, 3 1 3 1

3, 1 1 1 1

4, 2 1 1 2 1 1

5, 2 4 2 4 1 2 3 2

6, 2 2 2 6 3 3 5 6

7, 5 6 2 3 6 2 . 2 3 2 . 2

8, 4 3 15 3 1 5 3

9, 3 1 2 1 6

x x x x

x x x x

x x x x

x x x x

x x x x x

x x x x x

x x x x x x x x

x x x x x x x

x x x x x

+ + − < − − +

+ − ≥ − +

− + + + = − +

+ + − + ≤ − +

+ − + + + = + +

+ − + + + > + +

+ + − + − + − = +

+ + + = + + +

− + + − + >

( )

( )

( )

( )

( ) ( )

( ) ( )

4 2

2

2 2

2

2

2

2

2

2 2

2

2

1 1

10, 3 7 3 1 8 7 1

11, 2 9 2 8

112, 3 1 9

1

213, 4 4 1

2

914, 5 81 3

9

15, 3 1 1 2

16, 2 1 4 2

17, 6 5 5 4 1

18, 2 1 1 2 1

19, 2 4 4 2 1 2

x

x x x x

x x x x

xx x

x

xx x

x

xx x

x

x x x x x

x x x x x

x x x x x

x x x x

x x x

+ + +

+ − > − + − + − +

+ − ≤ − −

+− = −

−

−− = −

+

+− > −

−

+ − + =

− − − ≤

− + + − + ≤ −

+ + − = − + +

− + − = + +

( )

( )

2 2

2

2

2

2

2

2

2

2

2

20, 2 2 1 2 1 1 2 3 2

21, 9 3 11 1 6 1 2 1

22, 3 2 2 1 2

23, 2 1 3

1 2 124, 2 2

4 2

25, 3 2 4 3 8

26, 4 1 3 2 4 3 1

27, 5 3 6 3 2 2 1

28, 1 4 3 0

29

x

x x x x x x x

x x x x

x x x x x

x x x x x

x xx x x

x x x x x

x x x x

x x x x

x x x

+

− − + − + − < + + + − +

− − − + − ≥ +

− − + = − − −

+ ≤ − − − +

− −− + = +

− + ≥ − + −

− + − = − + +

− − > + −

− − + ≤

2

2 2

4, 3 7 6 0

3

30, 6 5 3 2

xx x

x

x x x x

− − − + ≥

+ + ≤ + +


3


( )( )

( )

( ) ( ) ( )

( )

( )( )

2

3 2

2 3

2 2

2

2

2

2

1, 3 4 1 6 9

2, 2 5 3 4

3, 1 2 3 4

4, 1 6 1 1

5, 3 8 4 3 1 0

6, 2 2 4 1 2

37, 1

4 6 2

8, 1 2 3 2 1 2

9, 12 1 3 3

10, 6 1 4 13 9

11, 1 2 5 4 2 1

4 512, 7

1

x x x x

x x x x

x x x x x

x x x x x

x x x

x x x

x x

x x

x x x

x x

x x x

x x x

x x

x

− − ≤ − +

− − ≤ − +

− + − + + >

+ − − − < −

+ + + − + =

+ + + = +

+> −

− +

+ = + − +

+ ≥ + +

+ = − −

≤ + + + +

+ −<

−

( )

2

2

2

2

2

2

2

2

2

13, 25 39 11 4 1 0

5 11 314, 1

4 1

215, 1

2

2 4 316, 1

5 3

8 217, 1

22 407

18, 25 8 2 1 34

6 619, 1

3 14

20, 9 1 2 7 19

10321, 3 1 13

41 11

22, 1 42 4

12 123, 1

11 12

24,

x

x x x

x x

x x

x

x

x x

x x

x x

x

x x x

x x

x

x x x

x x x

x x x

x x

x

− + + − =

− +≥ −

−

−<

−

+ −>

−

+ −≤

−

− + + =

+ +=

+

+ + + ≤

+ + − >

+ + − =

+ +=

+2

2

2 2

5 6 1 225, 1

11

x x xx

x x

x x xx

xx

+ − −− ≤

+ − −= − +

−−


4

Bài 4. Giải các phương trình và bất phương trình sau trên tập hợp số thực 2

2

2

2

1, 4 4 5 1 12

2, 13 6 12 8 6

2 23, 1

9 6 1

4, 4 2 1 8 3 13

5, 9 2 2 5 2 5 4

6, 1 6 4 5

7, 4 3 3

8, 4 2 1 5 2 3 2

9, 7 3 2 2 2 6 1

10, 5 18 2 2 4 4

4 6 2 1 411, 2

4 4

12, 9 2 1 5 1

13,

x x x x

x x x x

x x

x

x x x

x x x

x x x

x x x

x x x

x x x

x x x

x x x

x

x x x

+ + ≥ +

+ + = +

−<

− +

+ − ≥ −

+ + = +

+ ≤ + +

+ − = +

+ − = + −

= + − + −

= + − − −

− − − +≥

−

+ + = +

( )

2

2

2

1 2 3 2

25 3 114, 1

713

435

15, 2 6 2 24

2 2 516, 1

9 12 31 10

17, 16 1 4 4

18, 8 13 12 1 4 1

19, 1 2 1 2 2

120, 1 2 2

2

21, 11 39 12 2 30 1

3922, 2 3 3 1

43 2

23,2 1

x x x

x x

x

x x x

x x

x

x

x x

x x x

x x x x

x x x x

x x x

x x x

xx

x

+ + + = +

+ −≤

−

+ + ≥ + −

+ +=

− −−

≥− − +

+ = + − −

+ + − > + −

+ + + ≤ + −

+ < + − +

+ = + −

+ − +2

2 2 2

2 2 2

2 2 2

2

9 10 0

24, 6 5 3 2 2 4 3

25, 2 3 3 9

26, 2

27, 3 2 1 2 8

12 2 228, 1

10

x x

x x x x x x

x x x x x x

x x x x x

x x x

x x x

x

+ − ≤

+ + + + + ≤ + +

− + − ≥ −

+ − − ≤

+ + + ≥

− −≤

−


5

Bài 5. Giải các phương trình và bất phương trình sau trên tập hợp số thực 3 23 3

3 32 23 3

2

2

2

1, 1 2 1 3 2

2, 1

3, 3 2 1 2 4 3

44, 3 4

314

5, 5 35

46, 2 0

2

1 17, 1

38, 2 2

2

9, 4 1 2

16 510, 3

3 36

11, 3 9 53

12, 3 2 13 2

1

x x x x

x x x x x

x x x x x x

xx x

x

xx

x x

x x

x

x xx x

xx

x

x x

xx

x x

x x

x

xx x

x

+ + + = + + +

+ + = + +

+ + + = + + +

+ + =+−

− − =+ −

− + + =+

− + − =

− − <−

− − = −

−+ − >

− −

+ − = −−

− − > −−

2 2 3

2

2

2

2

2 2 2

2

2 2 2

3, 1 1 2 2

3614, 4 5

510

15, 15 11

4 3 3 516,

17, 2 1 6 1

18, 10 21 3 3 3 7 6

19, 2 2 2 2 1

20, 2

21, 8 15 3 3 2 5 6

22, 3 2 4 3 5 4

23

x x x x x

x x

x

x x

x

x x xx

x x

x x x

x x x x

x x x x

x x x x x

x x x x

x x x x x x

− − + + − = +

− + = −−

+ − − =−

− − −+ >

+ + + ≤ +

+ + = + + + −

+ − − = − + +

− + + =

+ + = + + + −

− + + − + = − +2 2 2

2

2 2

2

2 3

, 2 2

24, 2 3 1 2 2 1 2 1

25, 2 3 1 2 2 2 1

26, 1 2 1 3 2

27, 1 2 2 2 2 1

x x x x x

x x x x

x x x x x x

x x x x

x x x x x

− − + =

− + + = − + −

− + − + − + + = −

− + − = + − +

− + + + = + − +


6


( ) ( ) ( )

( ) ( )

3 2 4

32

2

22 33

2 2

3 2 2 2

2 2

1, 1 2 3

2, 1 1 1 1

13, 1 1 3

3

4, 2 3 9 4

5, 3 3

6, 2 3 9 2 2 3 2

7, 1 2 1 2 2 2

8, 3 3 2 3 2 2

9, 4 7 3 7 3 6 2 7 46 21

310, 2 2 3 2

11

x x x x x x

x x x x x

xx x x x

x

x x x

x x x

x x x x x

x x x

x x x x x x x

x x x x x x x

x xx

− + − = +

− + + + + = + −

++ + = − + + +

+

+ = + +

− = +

+ + = + +

+ − = −

+ + + + = + + +

+ + − = + + −

+ − = +

( )

( )

( )

6 5 4 33

3 2 2 2

2 2

3 244 4

2

2

2

, 1 1 2

312, 4 3 1 4

13, 4 4 8 2 4

14, 2 1 4 3 1 2 2 6

15, 1 1

16, 2 2 2 1 4 1

17, 2 2 1

18, 4 3 5 4 1 3

19, 9 1 3 2 3 2 0

20, 3 2 2 2 1 1

21, 4 5 24

x x x x x

x xx

x x x x x

x x x x x x

x x x x

x x x

x x x

x x x

x x x

x x x

x x

− + = + − +

+ = + +

+ + − = − −

− + + + = + +

+ + = + +

+ + + = + +

= + + +

− = + +

− + + − =

+ = + + +

− − =

( )( )

( )

( )

2

3 2

3 2

2

2 2

2 2

2 2

2

10

22, 4 19 25 2 1 0

23, 2 1 2 1 1

24, 2 6 1 3 1 8 1

25, 1 1 3 3 1 1

26, 2 2 5 6 4 3

27, 2 1 4 1 2

28, 1 1 1

29, 3 1 3 1 1

30, 1 2 1 2 1 1 1

31,

x

x x x

x x x x

x x x x

x x x x

x x x x x x

x x x x x

x x x x

x x x x x x

x x x x

x

+ + + − =

+ + = + + − +

+ + = − + + +

+ − + − − = −

+ − − + + = − − +

− − + = − +

− + − = +

+ + − = + − +

− + − = − + +

−( ) 3 2 21 2 1 1x x x− + ≤ −


7


2 2 2

2 2

2

2

3 2

2

2 2 2

1, 3 2 4 3 1 4

10 52, 5

2 3 3 23, 2

3 1 5 14, 4

5, 9 3 4 5

6, 8 1 4 1 3

7, 9 1 3 1

8, 2 1 3 3 1

9, 1 2 1

310, 3 7 18 0

2 1

11, 6 5 3 2

x x x

x x

x x

x x x

x x x

x x

x x

x x x x x x

x x x x

x x x

x x x x

x x x

xx x x

x

x x x x x x

+ − + > −

− −+ ≤

− + −− ≤ −

+ −+ =

+ − − ≤ −

+ + = + + +

+ + ≤ +

+ + + > +

+ + ≤ −

− − − − ≤ +

− + ≥ − + + −

( )2 2 2

6 5

12, 4 1 1 1 16

13, 2 1 4 3 2

1 214, 9

7 3 4

4 3 4 2 615, 7

2 2

4 1 3 416, 3

5 3 1 4

4 3 2 3 117,

43 2 3 2

7 1 318, 6

5 1

5 3 5 919,

26 3 8 2 1

7 6 3 320,

51 2 3

21, 3 1 2 1

x

x x x x x

x x x

x

x

x x

x

x x

x x

x x

x x

x

x x

x

x x

x

x x

x x

+

− ≤ − + + − −

+ = − −

− +>

− +

+ − − −=

− +

+ − +< −

− +

+ −= −

+ + −

− −=

+ + −

+ −<

+ − + +

− −≥

− − −

+ + +

( )

2

2

2

22, 5 1 3 4 1

5 1 8 923,

3 2 424, 2 1

1 1

25, 4 5 5 1

x

x x x

x x xx

x x

x xx

x x

x x x

≤

− + ≥ +

+ + −− ≤

− +≥ − +

− −

+ − = +


8


( )( )

( )( )

( )

2

2

2 3

2

2

1, 7 5 5 1 1 2 1

2, 6 10 1 1 3 1

3, 3 3 9 1

4, 2 2 3 3 1 6 6

5, 1 2 1 2 2 1

6, 4 3 2

7, 2 3 1 3 2 3 2 2 1

8, 1 2 2 1

3 3 19, 1

1 3

10, 2 2 2 1

11, 1

x x x x

x x x x

x x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x xx

x

x x x x x x x

x x

+ = − + − − +

= − + − − +

+ + − = − +

− + + = + − − +

+ − − − = −

− + − − =

− + − − − + = −

− + = + −

− + +− =

+ +

+ + − − = + −

−

( )

( )( )( )( )

2

2

2

2

2

2

2

1 3 2 1 2 2 2 2

12, 5 4 3 1 4 2 2 4

13, 2013 2012 1 2012 1

2 6 6 3 4 114, 1

4 3 13 12 7 34

15, 6 5 15 1

16, 4 3 1 7 12 1

17, 7 1 1 8 7 6

2 2 3 119, 3 1 1

20, 5

x x x x x

x x x x x

x x x x

x x

x x x

x x

x x

x x x x

x x x x

xx

x x

x

− + − + = − + − + −

− + + − + = + −

− + + = − + −

− + − +=

− + + −

+ + − =+ + +

+ + + − + + = −

+ − + + + + =

−− + = +

( ) ( )( )( )( )

( )( )

( ) ( )( )

2 2

2

2

2 2

2

2 2

2

2

2

2

7 6 1 7 5 6 1

21, 8 7 1 1

22, 3 9 7 2 1 1

23, 5 20 15 4

24, 3 1 4 1

25, 6 6 1 5 4 5

8 826, 4 3 0

7 1

16 127, 1

9 1 5

28, 3 1 3 2 2

29, 4 6 7 4 2 1

x x x x x

x x x x

x x x x

x x x x

x x x

x x x

x

x x

x

x

x x

x x x x

x x x x

+ + + = + + + + +

+ + = + +

+ + ≥ + +

+ + = +

= + + +

+ ≤ − +

−+ − =

− −

+<

− +

− ≤ + +

+ = + − −


9


( )

2

2

2

2

2 2

2

2

1, 2 7 1 5 1

2, 2 1

3, 5 11 4 1 3

4, 16 20 4 2 1 1 3

5, 4 2 3 4 2 2

6, 4 2 3 1 1 8 4 5

7, 5 3 3 1 2 4 1

8, 4 1 8 1 4 2

9, 3 1 9 5 6 6

10, 4 1 3 16 4

11, 3 4 3 9 22

12, 2 4 3 1

x x x

x x x

x x x

x x x x

x x x

x x x

x x x

x x x

x x x

x x x x x

x x x x

x x x

= + + +

+ = +

+ + = +

+ = + − +

+ + + = +

+ − + ≥ + −

= + − + −

+ ≤ − + −

− = − + −

+ + + ≤ +

+ − + =

− = +

( )

2

2

2 2

2 2

2 2

2 2

2 2

2

2 2

1 13

13, 1 3 4

14, 3 13 6 5 44

15, 15 8 7 4 2 1

16, 3 3 7 2 9 1

17, 4 4 3 4 21 7

18, 2 1 2 5 2 1

19, 4 2 3 4 1

20, 10 22 6 3 7

21, 12 1 2 12 20

22, 7 16 8 3 11 8

x

x x x

x x x x

x x x x

x x x x

x x x x x

x x x x

x x x x x

x x x x

x x x

x x x x x

+

− = +

+ ≥ − +

+ = + + −

+ + = + +

+ + − ≤ −

+ = + + −

+ + > + +

+ ≥ − +

− = + +

+ + − + <

( )

( )

( )

22

2

2 2

2

2 2

2 2

2 2

2 2

2 2

2 2

23, 3 2 1 5 2

24, 2 2 3 1 5

3 125, 5 6 1

4 2

26, 2 5 3 1

27, 6 6 1 2 1 3

28, 11 21 6 4 1 3 5

29, 4 11 4 3 4 5 6

30, 11 10 9 4 3 4

31, 4 20 3 1 19 1

32, 6 9 2 2 2

x x x

x x x x

x x x x x

x x x

x x x

x x x x x

x x x x x

x x x x x

x x x x

x x x

− + + = −

+ + + ≤ +

+ + = + +

+ + ≤ +

+ = + +

− ≥ + + + +

+ + + + >

+ + ≤ + +

= − + + +

− ≤ + ( )2 2

2 2

9

33, 3 6 2 2 2

34, 9 4 4 7 10

x

x x x x

x x x x x

+

− + + + =

+ = + + +


10


( )

( )

( )

( )

2 2

22

2

2

2

2 2

2

22

1, 4 84

2, 9 2 3 2 0

5 43, 4 1 1 0

3 2

4 54, 2

2

95, 7 81 2011 2

91

6, 3 14

97, 4 2 3 2

431

8, 3 13 14

129, 4 2 3 4 20 5

4 11 210, 2 3 1

1 2

11, 3 2 2 2

xx x

x x x

x xx

x

x x

x

xx x

x

x x x

x x x

x x x x

x x xx

x xx x

x

x x

+ − ≥ −

− − − ≥

− +− − ≤

−

− +≥

−

+− ≤ −

−

− = − +

− = − +

− + ≤ + +

+ + − ≤ +

− ++ + =

−

− −

( )

( )

2 2

2 2

2

2

3 2

2

2 2

22

2

4 1 22 2

12, 3 4 1 4 5 1 0

13, 10 7 2 2 1 5 1 4 1 2

14, 7 2 1 5 1 1 1

13 12 3 415, 1

3 1

16, 4 1 2 1 4

17, 5 4 1 3 1 2 4

14 9 118, 2

2

2 4 319, 5

20, 11 40 6 2 3

x x x

x x x x

x x x x

x x x

x x x x

x

x x x

x x x x

x xx x

x

xxx x

x x

− + ≥ −

+ + − + + =

+ + + > − + −

+ − = − + + +

− + − + −≤

− −

+ + + =

+ + + + = −

− −+ + >

++ + =

+ ≤ +

( ) ( )

2

2

2

2

2 2

1 4 321, 3

33 2

1 2 3 222,

3 1 4 5 123, 3 1

3 13 1

24, 4 3 4 2 3

x x

x x

x xx x

x x xx

x x

x x xx

xx

x x x

+

− ++ ≤

+ −+ +

− − −+ ≤

+ − ++ − ≤

−−

≤ + − +

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E2 f6 bộ binh