Method of Intervals-Wavy Curve Method

Solved AssignmentsMethod of Interval or Wavy Curve Method

Pankaj BalujaAuthor, Trainer & Motivator

Subject : Mathematics Date : 10-05-12 Method of interval (Wavy Curve Method)

Chapter: InequalitiesWebsite: www.mathematicsonline.co.in

Category: I.I.T.Author : Pankaj Baluja

Contact: [email protected] www.mathematicsonline.co.in

Solved Assignmenton

Method of Interval





Solve the following Inequalities for x:

1. (x – 1) (3 – x) (x – 2)2 > 0

Sol. (x – 1) (3 – x) (x – 2)2 > 0(x – 1) (x – 3) (x – 2)2 < 0 21

+ +3– –x (1, 2) (2, 3)

2.14

56

x

x < 0

Sol.14

56

x

x < 0, x –

41

–1/4+

+5/6

+

x

65,

41

3. 7x33x2

> 0

Sol. 7x33x2

> 0, x 37

3/2+

–7/3

+

x

2

3,

,37

4.1xx

5.02

< 0

Sol.1xx

5.02

< 0

1

5.02 xx

> 0 0.5 > 0 Since 12 xx is always positive.

which is true for all values of x. x (– , )

5.1xx6x5x

2

2

< 0

Sol.1xx6x5x

2

2

< 0

(x – 2)(x – 3) < 0 Since 12 xx is always positive.

x (2, 3)





6.1x

3x2x2

2

< 0

Sol.1x

3x2x2

2

< 0

1x

)3x)(1x(2

< 0

)3)(1( xx < 0 x (–3 , 1)

7.x1

)2x)(1x( 2

< 0

Sol.x1

)2x)(1x( 2

< 0, x –1

1x

)2x)(1x( 2

> 0–1–2 1

x (–, –2) (–2, –1) (1, ) Ans.

8.1xx24x4x

2

2

> 0

Sol.1xx24x4x

2

2

> 0

)1x)(1x2()2x( 2

> 0–1/2–2 1

x (– , –2)

21,2 (1, ) Ans.

9. x4 – 5x2 + 4 < 0Sol. x4 – 5x2 + 4 < 0

(x2)2 – 4x2 – x2 + 4 < 0 (x2 – 4) (x2 – 1) < 0 (x – 2) (x + 2)(x – 1) (x + 1) < 0

–1–2 1 2 x (–2, –1) (1, 2)

10. x4 – 2x2 – 63 0Sol. x4 – 2x2 – 63 0

x4 – 9x2 + 7x2 – 63 < 0(x2 + 7) (x2 – 9) 0(x2 + 7) (x – 3) (x + 3) 0(x – 3) (x + 3) 0

x (– , 3) (0, 3)0–3 3





11.2x

3

< 1

Sol.2x

3

< 1 x 2

2x

2x3

< 0

2xx5

< 0

2x5x

> 02 5

x (– , 2) (5, )

12.1x

1 2

Sol.1x

1 2, x 1

1x

1

– 2 0

1xx23

0

1x3x2 0 1 3/2

–

+ +

x (– , 1)

,23

13. 5x23x4

< 6

Sol. 5x23x4

< 6

5x2)5x2(63x4

< 0

5x230x123x4

< 0

5x233x8

< 0

5x233x8

> 0 5/2 33/8

x

25,

,8

33





14. 5x23x4

< 6

Sol. 6x6x5

< 1, x – 6

6x6x6x5

< 0

6x12x4

< 0

6x)3x(4

< 0 –6 3–

++

x (– 6, 3)

15.x48x5

< 2

Sol.x48x5

< 2 x 4

x4

x288x5

< 0

x4

x7

< 0

4x

x7

> 0 0 4–

++

x (– , 0) (4, )

16. 3x1x

> 2

Sol. 3x1x

> 2 x – 3

3x)3x(21x

> 0

3x6x21x

> 0

3x7x

> 0

3x7x

< 0–7 –3

x (–7, –3)





17. 3x85x7

– 4 > 0

Sol. 3x85x7

– 4 > 0, x 83

3x8)3x8(45x7

> 0

3x817x25

> 0

3x817x25

< 0

x

83,

2517

1725

– 38

–

18. 5xx

>21

Sol. 5xx

>21

x 5

5xx

>21

> 0

)5x(25xx2

0

–5 5

)5x(25x

0 x (– , –5] [5, )

19. 3x1x5

2

< 1

Sol. 3x1x5

2

< 1

3x

3x1x52

2

< 0

3x

4x5x2

2

< 0

3x

4x5x2

2

> 0

3x)4x)(1x(

2

> 0

10 4–

++ +

x (– , 1) (4, )





20.1x

2x2

< –

21

Sol.1x

2x2

< –

21

1x

2x2

+

21

< 0

2)1x(1x4x2

2

2

< 0

2)1x()3x)(1x(

2

< 0

x (–3, 1)

21. 2)1x(1x

< 1

Sol. 2)1x(1x

< 1

2

2

)1x()1x(1x

< 0

2)1x(1x

< 1

x21x

1x2

– 1 < 0

2

2

)1x(x21x1x

2

2

)1x(x3x

< 0

0 31

2

2

)1x(x3x

> 0

2)1x()3x(x

> 0

x (– , 0) (3, )

0 3





22.5x4x2

12x7x2

2

> 0

Sol.5x4x2

12x7x2

2

> 0

x2 – 4x – 3x + 12 > 0 x (– , 3) (4, )

23.1x

7x6x2

2

2

Sol.1x

7x6x2

2

2

1x

2x27x6x2

22

0

1x

9x6x2

2

0

1x

9x6x2

2

> 0

1x)3x(

2

2

0

x (– , )

24.5x4x1xx

2

24

< 0

Sol.5x4x1xx

2

24

< 0

5xx5x

1xx 22

< 0

)5x)(1x(1xx 24

< 0

x (–1, 5)

25.40x21x2

x312

2

< 0

3 4





Sol.40x21x2

x312

2

< 0 x

25

, 8

2x3x2

7x5x2

2

< 0 x –

21

, 2

2xx4x2

)7x5x(2

2

< 0 )2x)(1x2(

1 < 0

–1/2 2

x

2,21

26.1x

459x3x22

2

– 1 > 0

Sol.1x

459x3x22

2

– 1 > 0

1x

1x459x3x22

22

> 0

1x460x3x

2

2

> 0

1x

460x20x23x2

> 0

0 –20 23

1x)20x)(23x(

2

> 0

x (– , – 20) (23, )

27.1xx

1x2

2

< 1

1xx1xx1x

2

22

< 0

– x – 2 < 0x + 2 < 0x (–2, )

28. 4x3xx

2 > 0

4xx4xx

2 > 0





)4x(1)4x(xx

> 00–1 4

x (–1, 0) (4, )

29.3/2x10x7x2

> 0

3/2x10x2x5x2

> 0–2–5 –2/3

3/2x)5x)(2x(

> 0

x (– 5, –2)

,32

30.5x2

6x4x3 2

< 0

x =6

884 =

3222

x = 2 +322

, 2 –322

x (– , – 2 – 22 ) (2 + 22 , )

31.3x

x2x1517 2

< 0

D = b

2 – 4ac = 225 + 136 = 361 Real roots

Roots are x = 3414

, 3424

=177

,1712

d – 3 1217

717

)3x(1712x

177x

< 0

x (– , –3)

177,

1712





32.24xx3

9x2

2

< 0

24x3x)3x)(3x(

2

> 0

( x – 3 )

2 > 0x (– , –3) (3, )

33. 5x7x

+2

1x3 0 x 5

)5x(25xx15x314x2 2

0 1 3 5

)5x(29x12x3 2

0

x [1, 3] (5, )

34. 2x2 +x1

> 0 x 00–

+

x1x2 3

> 0

x (0, )

35.x6x6xx

2

2

0 x 6, 0

–2–6 0 3

)6x(x6x3x2x2

0

x (–, –6) [–2, 0) [3, )x (– 6, – 2] [3, )

36.30x5x6x6x3x2x

2

2

< 0 x 6, 5 32 5 6

++ +

– –

)5x)(6x()3x)(2x(

< 0





x (2, 3) (5, 6)

37.1x4x4

7x8x2

2

< 0 1/2 1 7

1x2x2x47xx7x

2

2

< 0

x

21, (1, 7)

38.18x9x

36x2

2

< 0 18x6x3x

)6x)(6x(2

–6 3–

)3x)(6x()6x)(6x(

< 0

3x6x

< 0

x (– 6, 3)

39. 2

2

xx459x6x

0

5xx5xx3x3x

2

2

< 0

–5 31

)5x)(1x()3x)(3x(

< 0 = )5x)(1x()3x( 2

< 0

x (–5, 1) {3}

40.1x1x

– x < 00–1

1xxx1x 2

< 0

1x

1x2

< 0 1x1x2

> 0

x (–1, )

41.2x

1

– 3x3

< 0–9/2 –2 3





)3x)(2x(6x33x

< 0

)3x)(2x(9x2

< 0

)3x)(2x(9x2

> 0

x

2,29

(3, ) Ans.

42.1xx14

–4x30x9

< 0 x –1, 4

)4x)(1x()1x)(30x9()4x(x14

< 0

)4x)(1x(30x9x30x9x56x14 22

< 0

)4x)(1x(30x35x5 2

< 01–1 4 6

)4x)(1x()6x7x(5 2

<

)4x)(1x()1x)(6x(5

< 0

x (–1, 1) (4, 6)

43.3xx4

2x52

2

< 1

3xx4)3xx4(2x5

2

22

< 0

5 x

2 – 2 – 4x2 + x – 3 < 0x2 + x – 5 < 0D = b2 – 4ac = 1 + 20

2211x

2211x





x

2

211,2

211Ans.

44.5x4x

12x5x2

2

> 3

5x4x15x12x312x5x

2

22

> 0 1/2 3

– 2x2 + 7x – 3 > 0 2x2 – 7x + 3 < 0 2x2 – 6x – x – 3 < 0 (2x – 1)(x – 3) < 0

x

3,21

45.3x3x24x3x

2

2

< 4

3x3x24x3x

2

2

– 4 < 0

3x3x12x12x424x3x

2

22

< 0

–1 4

3x3x12x9x3

2

2

< 0

3x2 – 9x – 12 > 03(x2 – 3x – 4) > 03(x – 4) (x + 1) > 0x (– , –1) (4, )

46.5x21x2

– 3 < 0 x –25

5x2)5x2(31x2

< 0–5/2 –2 8

5x216x2x8x2

< 0

5x2)8x)(2x(

< 0





x

25, (–2, 8)

47.3x41x2

> 2 x 43

3x4

)3x4(21x2

> 0

3x4

6x81x2

> 0 3x4

7x8x2

> 0

3x4

7x7xx2

> 03/4 1 7

3x4)1x(7)1x(x

> 0

3x4)1x)(7x(

> 0

x

1,43

(7, )

48.1x2x

2

2

< – 2

1x2x

2

2

+ 3 < 0

0 –1 1

1x)1x(22x

2

22

< 0

1x2x22x

2

22

< 0

1xx3

2

2

= )1x)(1x(

x3 2

< 0

49. 5x4x5x3

2

>

21

5x4x5x3

2

–

21

> 0–5 1





)5x4x(2)5x4x()5x3(2

2

2

> 0

50.12xx

3x22

–

21 0

)4x)(3x(12xx)3x2(2 2

0–3–4 3 6

)4x)(3x(18x3x2

0

)4x)(3x(18x3x2

0

)4x)(3x(18x3x6x2

0

)4x)(3x()6x)(3x(

0

x (–, – 4) (–3, 3) (6, )

51. 16x2x3x25

2

< 1

16x2x316x2x3x25

2

2

< 0

16x6x8x3

21x32

2

< 0

)8x3(2)8x3(x21x3 2

> 00 –2 8/3

)8x3)(2x()7x(3 2

> 0

x (– , –2)

,38

52.12xxx415

2

< 4





12xx

)12xx(4x4152

2

< 12

12x3x4x

48x4x4x4152

2

< 0

0 –3 4

)4x)(3x(63x4 2

< 0

)4x)(3x(63x4 2

> 0

x (– , – 3) (4, )

53. 6x5x1

2 –

21 0 x 3, 2

6x3x2x1

2 –

21 0

)2x)(3x()6x5x(2 2

0

)2x)(3x(4x5x2

0

)2x)(3x(4x4xx2

0

)2x)(3x()1x)(4x(

021 3 4

x [1, 2) (3, 4][1, 2) (3, 4]

54.)4x3x)(1x(

)3x)(x2(2

32

0

)4xx4x)(1x()3x)(2x(

2

32

< 0

)4x)(1x)(1x()3x)(2x( 32

< 0

)4x()1x()3x)(2x(

2

32

0





)4x()1x()3x)(2x)(2x(

2

3

0

–1 3 422–

x )2,2( (3, 4)

55. 4xx3x45

2

< 4 4xx3

5x42

– 4 < 0

4xx316x4x125x4

2

2

< 0

4xx311x8x12

2

2

< 0 )4x3(1)4x3(

11x8x12 2

D = b2 – 4ac = 64 + 528 = 526

x =24

5298

56.4x11x7

x33192

> 2

=4x11x7

8x22x14x33192

2

< 0

D = b2 – 4ac

= 121 + 616 4x4x7x7

2873711x

2873711x

2

x =28

73711

)1x(4)1x(x7

2873711x

2873711x

4 1–11 + 73728 7 11 + 737

28

57. )2xx10(249x

2

–

21

< 0

)2xx10(22xx1049x

2

2

< 0





=2x4x5x10

51x102

2

< 0

= )1x2(2)1x2(x551x10 2

> 0

)1x2)(2x5(51x10 2

> 0 2/5

++

–– –1/2 51

10

x

52,

21

,

1051

58. 2

2

xx34)1x2x)(2x(

0

4xx4x)1xxx)(2x(

2

2

0

)4x)(1x()1x)(2x( 2

0 1 –1 4 –2

x (– , –2) (–1, 1) (1, 4)x (– , –2) (–1, 4)

59.x1

4

+x1

2

– 1 < 0

)x1)(x1()x1)(x1(x22x44

< 0

= )x1)(x1(]xxx1[6x2 2

< 0–1 1

= )1x)(1x(x16x2 2

> 0

)1x)(1x(1 > 0

x (– , –1) (1, )

60. 2 +1x

3

–x2

> 0





)1x(x2x2x3x2x2 2

> 0–1/2–1–2 0

)1x(x2x3x2 2

> 0

)1x(x)2x)(1x2(

> 0

x (– , –2) (–1, 0)

,21,0

61. 1 +1x

2

–x6

> 0

)1x(x6x6x2xx2

> 0

)1x(x6x5x2

> 0

)1x(x6x3x2x2

> 0

310 2

x (– , –5) (1, 2) (6, ) Ans.

62.x

1x –

1x1x

< 2

)1x(x)xx(2x)1x()1x( 22

0

)1x(xx2x2xx1x2x 222

< 0

)1x(x1xx2 2

> 0 )1x(x1xx2x2 2

> 0

)1x(x)1x)(1x2(

10–1 1/2





x (– , –1)

21,0 (1, )

63. )6x(x)3x(2

–1x

1 0

)1x)(6x(x)x6x()1x)(3x(2 2

0

)1x)(6x(xx6x6x8x2 22

0

)1x)(6x(x6x2x2

0

1+ +

–– 0 – 6

x (– , 0) (1, 6)

64. )7x)(1x()4x(2

– )2x(

1 0

)7x)(2x)(1x(7x8x)2x)(4x(2 2

0

)2x)(7x)(1x(7x8x16x12x2 22

0 )2x)(7x)(1x(

9x4x2

0

21 7

x (1, 2) (7, )

65. 9xx2

2 –

2x1 0

)2x)(3x)(3x(9xx4x2 22

0

)2x)(3x)(3x(9x4x2

0 –2– 3 3

x (– , –3) (–2, 3)





66.2x

1

+1x

1

–x1

> 0

x)1x)(2x(2x3xx2xxx 222

> 0

)1x)(2x(x2x2

> 0

x )0,2( (2, )

67. )3x)(2x(7 + )3x(

9 + 1 < 0

)3x)(2x()3x)(2x()2x(97

< 0

)3x)(2x(6x5x18x97 2

< 0

)3x)(2x(5x4x2

< 0

)3x)(2x()1x)(5x(

< 0

x (– 5 + 1) (2, 3)

68.12x7x

202

+4x

10

+ 1 > 0

)4x)(3x(12x7x)3x(1020 2

> 04–4–2 3

)4x)(3x(2x2xx2

> 0

)4x)(3x()1x)(2x(

> 0

x (– , – 2) (–1, 3) (4, )

69. )7x)(4x)(2x()7x)(4x)(2x(

> 1





)7x)(4x)(2x()]7x)(4x)(2x[(1)7x)(8x6x( 2

)7x)(4x)(2x(56x50x13x56x50x13x 2323

> 0

)7x)(4x)(2x(112x26 2

> 0

)7x)(4x)(2x(12x26 2

< 0 –4– 7 –2

70. )3x)(2x)(1x()3x)(2x)(1x(

> 1

)3x)(2x)(1x()3x)(2x3x( 2

> 1 (x3 + 6x2 + 11x + 6)

)3x)(2x)(1x(6x9x3x2x3x 223

– 1 > 0

)3x)(2x)(1x(6x11x6x6x11x6x 2323

> 0

)3x)(2x)(1x(12x12 2

> 0

)3x)(2x)(1x()1x(12 2

> 0 –2– 3 –1

)3x)(2x)(1x()1x(12 2

< 0

x (–, –3) (–2, –1)

71. (x2 – 2x) (2x – 2) –x2x)2x2(9

2

0 ab – a

b9 0

ab9ba2 0

a

]9a[b 2 0 )2x(x

2x2

(x2 – 2x – 3) (x2 – 2x + 3) 0

)2x(x2x2

(x2 – 3x + x – 3) 0 )2x(x)1x(2

(x – 3) (x + 1) 0

x (– , 1] (0, 1] (2, 3]





72. (x2 + 3x) (2x + 3) – 16

x2L23x2

2 0

ab16ba2 0

ab

(a – 4) (a + 4) 0

)3x(x3x2

(x2 + 3x – 4) (x2 + 3x + 4) 0

)3x(x3x2

x(x + 4) – 1(x + 4) > 0

)3x(x3x2

(x – 1) (x + 4) > 0

)3x(x)4x)(1x)(3x2(

0

x (– , – 4]

23,3 [1, )

73. 7|x|6x3x2x2

< 0

. – < x < 0

7x)2x(3)2x(x

7x)2x)(3x(

7x)2x)(3x(

> 0

x (2, 3) (7, ) Rejected. 0 x <

7x)2x)(3x(

< 0

–7 2 3

x (– , –7) (2, 3)





x (2, 3) Ans.

74. |4x|7x6x2

< 0

|4x|7xx7x2

case-– < x < –4

4x)7x)(1x(

< 0

4x)7x)(1x(

> 0

x (–7, –4) (1, )x (–7, –4)

case-– 4 x < –7 –4 1

4x)7x)(1x(

< 0

x (– , –7) (– 4, 1)(–4, 1) Ans.

75.2x

|2x|

> 0

case-– < x < 2

2x2x

> 0 2x2x

< 0

1 < 0 falsecase-

2 x <

2x2x

> 0

1 > 0 True

76. 4x2 > 1 |4x|

2 > 1

case–– < x < 4





4x

2

– 1 < 0 4x4x2

< 0

4x2x

> 0 x (– , 2) (4, )

Intersection x (– , 2)

case-4 < x <

4x2

– 1 < 0

x24x2

< 0

x2x6

< 0

2x6x

> 0

x (– , 2) (6, ) x (– , 2) (6, ) Ans.

77.9x6x

10|x|2|x|5x2

2

< 0

2

2

)3x(10|x|2|x|5|x|

< 0

2)3x(]5|x[|2]5|x[||x|

< 0

2)3x(

5|x|2|x|

< 0

78.2x

x|3x|

> 1

2x2xx|3x|

> 0

2x2|3x|

> 0

– < x < – 3 – 3 x <





2x23x

> 02x

23x

> 0

2x5x

>2x1x

> 0

2x5x

< 0 –2 –1

x (– 5, –2) x (– , –2) (–1, )x (–2, )

x (– 5, –2) (–1, ) Ans.

79.2x

|1x|

< 12x

2x|1x|

< 0

case-I– < x < 1

2x2x1x

< 0

2x1x2

> 0

2x1x2

< 0 x (– , –2)

,21

x (– , –2)case-

1 x < 8

2x1x

– 1 < 0

2x2x1x

< 0

2x3

< 0

2x3

> 0

x (–2, ) x (– , –2) (–2, ) Ans.

80. 6x3x2x|3x|

2

2





)3x)(2x(|3x|

– 2 > 0

)3x)(2x()6x5x(2|3x| 2

> 0

case-– < x < 3

)2x)(3x(312xx10x2 2

> 0

)2x)(3x(9x9x2 2

> 0

)2x)(3x(9x9x2 2

< 0

)2x)(3x()3x(3)3x(x2

< 0

)2x)(3x()3x)(3x2(

< 0

x

2,23

case-3 < x <

)2x)(3x(12x10x23x 2

> 0

)2x)(3x(15x11x2 2

> 0

)2x)(3x(15x11x2 2

< 0

)2x)(3x()3x)(5x2(

< 0 2 5/2

x

25,2

25,

23

Ans.





81.3x

12|x|x2

2x

3x12|x|3|x|4x2

2x

3x]4|x][|3|x[|

– 2x 0

)3x()3x(x2]4|x][|3|x[|

0

– < x < –3 – < x < 4

3xx6x2)4x)(3x( 2

0

3xx6x212xx 22

0

3x12x5x2

0

3x12x5x2

0

x (– , 3]

82. 1 + 2x12

–x7

< 0 x 0

2

2

xx712x

< 0

2x)3x)(4x(

< 0

x (3, 4)

83. x – 17 –x

60 0 x 0

x60x17x2 0

x60x3x20x2 0

x)20x)(3x( 0 –3 0 2

x [–3, 0) [20, ) Ans.





84.6x5x5x4x

2

2

0

6x3x2x1

2 0 –3 1/2

)2x)(3x(1 0

x (– , –3) (– 2, )

85. x 5x6

5x6)5x(x

0

5x6x5x2

0 –1 5 6

)5x()1x)(6x(

0

x (– , –1] (5, 6]

86. 2x3x4x1x

2

0 x 4, – 3

)3x)(4x(1x

0 –3 1 4

x (– , –3) [1, 4)

87.2x9x30

25x + 50

2x)2x)(50x25(9x30

0

=2x

100x50x50x259x30 2

0

2x

91x30x25 2

0

2x

91x30x25 2

0





88. f (x) = 0 – 3 + 5x – x2

g(x) = 3x2 + 5x – 3 3x2 – 5x + 6 0x2 – 2x – 3x + 6 0(x – 2)(x – 3) 0 x (2, 3)

89. f (x) = 30x2 – 26x + 7g (x) = 33x2 – 30x30x2 – 26x + 7 33x2 – 30x– 3x2 + 4x + 7 03x2 – 4x – 7 03x2 + 3x – 7x – 7 0(3x – 7) (x +1) 0

x (– , –1)

,37

–1 7/3

90.2x

4

> 3 – x

2x

4

– (3 – x) > 0

4 – )2x()2x)(x3(

> 0

2x

6xx4 2

> 0 –2 –1 2

2x

2xx2

> 0

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

> 0

x (–2, –1) (2, )

Documents

Method of Intervals-Wavy Curve Method