Solved Assignments Method of Interval or Wavy Curve Method Method
Pankaj Baluja Author, Trainer & Motivator
Chapter: Inequalities Website: www.mathematicsonline.co.in
Subj Su bjec ectt : Ma Math them emat atic ics s
Date Da te : 100-0 055-12 12
Category: I.I.T. Author : Pankaj Baluja
Meth Me thod od of in inte terv rva al (W (Wav avy y Cu Curv rve e Me Meth tho od)
Solved Assignment on Method of Interval
Contact: ba
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subj Su bjec ectt : Ma Math them emat atic ics s
Date Da te : 100-0 055-12 12
Category: I.I.T. Author : Pankaj Baluja
Meth Me thod od of in inte terv rva al (W (Wav avy y Cu Curv rve e Me Meth tho od)
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 x (1, 2) (2, 3)
2.
Sol.
6 x 5
+
+ 1
Sol.
6 x 5
x–
< 0,
4 x 1
2x 3
2x 3
4.
Sol.
1 4
+ –1/4
0. 5
x x 1 0. 5 2
x x 1 0 .5 x
2
Sol.
x 1
x 2 5x 6 x2 x 1 x 2 5x 6 x2 x 1
3/2
+ 5/6
–
+ 7/3
<0
<0
>0
0.5 > 0
Since
x
2
x
1 is always positive.
x (– , )
<0
<0
(x – 2)(x – 3) < 0
+
+
3
which is true for all values of x.
5.
3
7
3 7 , , 2 3
2
x
> 0,
3x 7
–
>0
3x 7
x
2
<0
4 x 1
1 5 x , 4 6 3.
–
Since
x
2
x
1 is always positive.
x (2, 3)
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
6.
Sol.
x 2 2x 3 x2 1 x 2 2x 3 x2 1
Date : 10-05-12
Sol.
<0
( x 1)( x 3) x2 1
( x 1)( x 2)2
1 x ( x 1)( x 2)2
1 x
8.
Sol.
<0
x (–3 , 1)
<0
< 0,
( x 1)( x 2)2 x 1
x –1
>0
–2
1
–1
x (–, –2) (–2, –1) (1, ) Ans.
x 2 4x 4 2x 2 x 1 x 2 4x 4 2x 2 x 1
Method of interval (Wavy Curve Method)
<0
( x 1)( x 3) < 0
7.
Category: I.I.T. Author : Pankaj Baluja
>0
>0
( x 2)2 ( 2 x 1)( x 1)
>0
–2
1
–1/2
1 x (– , –2) 2 , (1, ) Ans. 2 9. Sol.
10. Sol.
x4 – 5x 2 + 4 < 0 x4 – 5x 2 + 4 < 0 (x2)2 – 4x2 – x 2 + 4 < 0 (x2 – 4) (x2 – 1) < 0 (x – 2) (x + 2)(x – 1) (x + 1) < 0 x (–2, –1) (1, 2) x4 – 2x2 – 63 0 x4 – 2x2 – 63 0 x4 – 9x2 + 7x 2 – 63 < 0 (x2 + 7) (x2 – 9) 0 (x2 + 7) (x – 3) (x + 3) 0 (x – 3) (x + 3) 0 x (– , 3) (0, 3)
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–2
1
–1
–3
0
2
3
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 11.
Sol.
3 x2 3 x2
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
<1
x2
<1
3 x 2 <0 x2
5 x <0 x2
x5 >0 x2
2
5
1
+ 3/2
x (– , 2) (5, ) 12.
Sol.
1 x1 1 x1
2 2, 1
x 1
2x 3
x 1
4x 3 2x 5 4x 3 2x 5
+
0
<6 <6
4 x 3 6(2 x 5) 2x 5
<0
4 x 3 12 x 30 <0 2x 5
8 x 33 <0 2x 5
–
3 x (– , 1) , 2
Sol.
–20
3 2x 0 x 1
13.
x1
8 x 33 2x 5
x ,
>0
5/2
33/8
5
33 , 2 8
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 14.
Sol.
4x 3 2x 5 5x 6
Method of interval (Wavy Curve Method)
<6
x–6
< 1,
x6
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
5x 6 x 6
<0
x6 4 x 12 x6
<0
4( x 3 ) x6
+
<0
+ –6
–
3
x (– 6, 3) 15.
5x 8 <2 4x
Sol.
5x 8 <2 4x
x4
5x 8 8 2x 4x
<0
7x <0 4x 7x x4
+
>0
+ 0
–
4
x (– , 0) (4, ) 16.
Sol.
x 1 x3 x 1 x3
>2
x–3
>2
x 1 2( x 3 ) x3 x 1 2x 6 x3
>0
>0
x 7 >0 x3 x7 x3
<0
–7
–3
x (–7, –3)
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 17.
Sol.
7x 5
Method of interval (Wavy Curve Method)
–4>0
8x 3 7x 5
x
– 4 > 0,
8x 3
Date : 10-05-12
7 x 5 4(8 x 3 ) 8x 3
3 8
>0
25 x 17 >0 8x 3
25 x 17
8x 3
<0
17 3 , x 25 8
18.
Sol.
x
x
Sol.
x5
x5 2( x 5)
x2 3
1
–
3 8
>0
2
2( x 5)
5x 1
>
2x x 5
x2 3
17 25
x5
2
x
5x 1
1
>
x5
–
1 2
>
x5
19.
Category: I.I.T. Author : Pankaj Baluja
0
–5
5
0 x (– , –5] [5, )
<1 <1
5x 1 x 2 3 x2 3
x 2 5x 4 x2 3 x 2 5x 4 x2 3
x 3
<0
>0
( x 1)( x 4 ) 2
<0
+
+
>0
+ 0
1
–
4
x (– , 1) (4, ) Contact:
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 20.
Sol.
x2 x 1 x2 x 1
2 1 2
<–
2
x2 x2 1
Method of interval (Wavy Curve Method)
1
<–
2
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
1 <0 2
+
2x 4 x 2 1 ( x 2 1) 2
<0
( x 1)( x 3)
0
3
<0
( x 2 1) 2
x (–3, 1) 21.
Sol.
x1 ( x 1)2 x1 ( x 1)2
<1
<1
x 1 ( x 1)2 ( x 1)2 x1 ( x 1)2
<0
<1
x 1
–1<0
x 2 1 2x
x 1 x 2 1 2x ( x 1)2
x 2 3x ( x 1)2 x 2 3x ( x 1)2 x( x 3) ( x 1) 2
<0 0
1
3
>0
>0
x (– , 0) (3, )
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 22.
Sol.
x 2 7 x 12
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
>0
2x 2 4 x 5
3
x 2 7 x 12
4
>0
2x 2 4 x 5
x2 – 4x – 3x + 12 > 0 x (– , 3) (4, )
23.
Sol.
x 2 6x 7
x 2 6x 7
Sol.
2
x2 1
24.
2
x2 1
x 2 6x 7 2x 2 2 x2 1
x 2 6x 9 x2 1 x 2 6x 9 x2 1 ( x 3 )2 x2 1
0
0
>0
0
x (– , )
x 4 x2 1 x2 4x 5 x 4 x2 1 x2 4x 5
<0
<0
x2 x2 1 <0 x 5x x 5
x 4 x2 1 ( x 1)( x 5)
<0
x (–1, 5)
25.
1 3x 2 2 x 2 21x 40
<0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics Sol.
1 3x 2 2 x 21x 40 x 2 5x 7 2
2x 3 x 2
1
x
Sol.
x–
<0
2x 2 4 x x 2
2
2x 2 3 x 459 x2 1 2x 2 3 x 459 x2 1
x2 1
<0
2
–1/2
–1>0 –1>0
>0
>0
x2 1
x2 1
,2
x 23 x 20 x 460
( x 23 )( x 20 )
(2x 1)( x 2)
2
x2 1 x 2 3x 460
1
1
, 2
2x 2 3 x 459 x 2 1
Method of interval (Wavy Curve Method)
5 ,8 2
<0
( x 2 5x 7)
x
<0
2
26.
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
>0
–20
0
23
>0
x (– , – 20) (23, )
27.
x2 1 x2 x 1
<1
x2 1 x2 x 1 x2 x 1
<0
– x – 2 < 0 x+2<0 x (–2, )
28.
x 2
x 3x 4
>0
x 2
x 4x x 4
>0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
x
Method of interval (Wavy Curve Method)
>0
x( x 4 ) 1( x 4)
Category: I.I.T. Author : Pankaj Baluja
–1
0
–5
–2
4
x (–1, 0) (4, )
x 2 7 x 10 >0 x 2/3
29.
x 2 5 x 2x 10 >0 x 2/3 ( x 2)( x 5) x 2/3
–2/3
>0
2
x (– 5, –2)
3
,
3x 2 4x 6 <0 2x 5
30.
4 88
x=
6
x=2+
22 ,2 – 3
x (– , – 2 –
17 15 x 2x 2
31.
x3 2 D
=
b
2 22 3
=
22 3
22 ) (2 +
22 , )
<0
– 4ac = 225 + 136 = 361 Real roots
Roots are x =
14 24 , 34 34
=
7 12 , 17 17
d–3
12 17
7 17
7 12 x x 17 17 < 0 ( x 3)
12
x (– , –3)
7
17 17 ,
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics x2 9
32.
3 x x 2 24
( x 3)( x 3) x 2 3x 24
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
<0
>0
2
>0 x (– , –3) (3, ) (
33.
x
–
3
x7 x5
)
3x 1 0 2
+
x5
2 x 14 3 x 2 15 x x 5 2( x 5) 3 x 2 12 x 9
0
1
3
5
0
2( x 5)
x [1, 3] (5, )
34.
2x2 +
1 >0 x
x0
+ –
0
3
2x 1 x
>0
x (0, )
35.
x2 x 6 x 2 6x
0
x 2 2x 3 x 6 x( x 6 )
x 6, 0
–6
0
–2
3
0
x (–, –6) [–2, 0) [3, ) x (– 6, – 2] [3, )
36.
x 2 2x 3 x 6 2
x 6 x 5 x 30
( x 2)( x 3) ( x 6)( x 5)
<0
x 6, 5
+
+ 2
–
3
+ 5
–
6
<0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics x (2, 3) (5, 6)
x 2 8x 7
37.
x 2 7x x 7 4 x 2 2x 2x 1
1
2
x 2 9 x 18
( x 6)( x 6) 2
x 3 x 6x 18 3 –
<0
x (– 6, 3)
x 2 6x 9 5 4x x2
0
( x 3)( x 3)
( x 1)( x 5)
x 2 3x 3x x 2 5x x 5
<0
3
1
–5
40.
<0
39.
7
–6
( x 6)( x 3)
x3
1
<0
<0
( x 6)( x 6)
x6
1/2
(1, 7)
x 2 36
38.
Method of interval (Wavy Curve Method)
<0
4 x 2 4x 1
x ,
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
<0 =
( x 3) 2 ( x 1)( x 5)
<0
x (–5, 1) {3}
x 1 –x<0 x 1
–1
0
x 1 x2 x <0 x1
x2 1 <0 x1 x (–1, )
41.
1 x2
–
3 x3
<0
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x2 1 x 1
>0
–9/2
–2
3
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
x 3 3x 6 ( x 2)( x 3)
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
<0
2x 9 <0 ( x 2)( x 3) 2x 9 ( x 2)( x 3)
9
x
2
>0
, 2 (3, ) Ans.
14 x 9x 30 – <0 x1 x4
42.
x –1, 4
14 x( x 4 ) (9 x 30)( x 1) ( x 1)( x 4 )
<0
14 x 2 56 x 9 x 2 30 x 9 x 30 ( x 1)( x 4 ) 5 x 2 35 x 30
( x 1)( x 4 ) 5( x 2 7 x 6)
( x 1)( x 4 )
5( x 6)( x 1)
( x 1)( x 4)
<0
<0
–1
1
4
6
<
<0
x (–1, 1) (4, 6) 5x 2 2
43.
4x 2 x 3
<1
5 x 2 2 ( 4 x 2 x 3) 4x2 x 3
<0
2
– 2 – 4x 2 + x – 3 < 0 x2 + x – 5 < 0 D = b 2 – 4ac = 1 + 20 5
x
1 21 x 2
1 21 x 2
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
44.
Date : 10-05-12
Method of interval (Wavy Curve Method)
1 21 1 21 , x 2 2 Ans.
x 2 5 x 12
>3
x2 4x 5
x 2 5 x 12 3 x 2 12 x 15 x 2 4x 5
45.
Category: I.I.T. Author : Pankaj Baluja
– 2x 2 + 7x – 3 > 0 2x2 – 7x + 3 < 0 2x2 – 6x – x – 3 < 0 (2x – 1)(x – 3) < 0
x
1 2
x 2 3 x 24 x 2 3x 3 x 2 3 x 24 x 2 3x 3
>0
1/2
3
, 3
<4
–4<0
x 2 3 x 24 4 x 2 12 x 12 x 2 3x 3
3 x 2 9 x 12 x 2 3x 3
<0
4
–1
<0
3x2 – 9x – 12 > 0 3(x2 – 3x – 4) > 0 3(x – 4) (x + 1) > 0 x (– , –1) (4, )
46.
x2 1 –3<0 2x 5 x 2 1 3(2x 5) 2x 5
x–
<0
5 2
–5/2
–2
8
x 2 8 x 2x 16 <0 2x 5 ( x 2)( x 8) 2x 5
<0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
5
2
x ,
47.
x
x 2 1 8x 6 4x 3
>0
x 2 8x 7 4x 3
> 0
x 2 x 7x 7 >0 4x 3
x( x 1) 7( x 1)
4x 3 ( x 7)( x 1)
4x 3
3
x2 1 x2 2 x2 1
>0
<–2
+3<0
x 2 2 2x 2 2 x2 1 =
<0
( x 1)( x 1)
2
x 4x 5 3x 5 x 4x 5
>
–
1
<0
3x 2
3x 5
0
–1
x2 1
x2 1
>0
x 2 2 2( x 2 1)
3x 2
7
, 1 (7, )
4
x2 2
1
3/4
x
2
3 4
x 2 1 2( 4 x 3) >0 4x 3
49.
Method of interval (Wavy Curve Method)
(–2, 8)
x2 1 >2 4x 3
48.
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
<0
1 2 1 2
>0
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–5
1
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
2(3 x 5) ( x 2 4 x 5) 2( x 2 4 x 5)
50.
2x 3 2
x x 12
–
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
>0
1 0 2
2(2 x 3) x 2 x 12
0
( x 3)( x 4 )
–4
–3
6
3
x 2 3 x 18 0 ( x 3 )( x 4 )
x 2 3 x 18 ( x 3 )( x 4 )
0
x 2 6x 3 x 18 ( x 3 )( x 4 )
( x 3)( x 6) ( x 3)( x 4)
0
0
x (–, – 4) (–3, 3) (6, )
51.
5 2x 2
3 x 2x 16
<1
5 2 x 3 x 2 2 x 16 3 x 2 2 x 16
<0
3 x 2 21 3 x 2 8 x 6x 16
<0
3 x 2 21 x(3 x 8 ) 2(3 x 8) 3( x 2 7) ( x 2)(3 x 8)
>0
–2
0
8/3
>0
8 x (– , –2) , 3
52.
15 4 x x 2 x 12
<4
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Subject : Mathematics
15 4 x 4( x 2 x 12)
x 2 x 12 15 4 x 4 x 2 4 x 48
x 2 4 x 3 x 12
4 x 2 63
( x 3 )( x 4 )
2
x 5x 6
–
1 2
1 2
x 2x 3 x 6 2 ( x 2 5 x 6) ( x 3 )( x 2)
54.
< 12
<0
–3
0
4
>0
x (– , – 3) (4, )
1
Method of interval (Wavy Curve Method)
4 x 2 63 <0 ( x 3 )( x 4 )
53.
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
0
–
x 3, 2
1 0 2
0
x 2 5x 4 0 ( x 3)( x 2) x2 x 4x 4 ( x 3 )( x 2)
( x 4)( x 1) ( x 3)( x 2)
0
0
1
2
3
4
x [1, 2) (3, 4] [1, 2) (3, 4]
( 2 x 2 )( x 3)3 ( x 1)( x 2 3x 4)
0
( x 2 2)( x 3)3 ( x 1)( x 2 4 x x 4 ) ( x 2 2)( x 3 )3 ( x 1)( x 1)( x 4 )
( x 2 2)( x 3)3 ( x 1)2 ( x 4 )
<0
<0
0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
( x 2 )( x 2 )( x 3)3
3
2
–1
Method of interval (Wavy Curve Method)
0
( x 1)2 ( x 4)
– 2
Category: I.I.T. Author : Pankaj Baluja
4
x ( 2, 2 ) (3, 4) 55.
5 4x 3x 2 x 4
4x 5
<4
3x 2 x 4
–4<0
4 x 5 12 x 2 4 x 16 3x 2 x 4
<0 12 x 2 8 x 11
3x 2 x 4
12 x 2 8 x 11
<0
(3 x 4 ) 1(3 x 4 )
D = b 2 – 4ac = 64 + 528 = 526
56.
x=
8 529 24
19 33 x 7 x 2 11x 4 =
>2
19 33 x 14 x 2 22x 8 7 x 2 11x 4
<0
D = b 2 – 4ac
= 121 + 616
x=
11 737 11 737 x x 28 28 7x 2 7x 4x 4
11 737
57.
x 49 2
2(10 x x 2)
28
4 7
–11 + 737 28
–
1 2
x 49 10x 2 x 2 2(10 x 2 x 2)
x 11 737 x 11 737 28 28 7 x( x 1) 4( x 1)
11 + 737 28
1
<0
<0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
10 x 2 51
=
10 x 2 5 x 4 x 2
Method of interval (Wavy Curve Method)
<0
10 x 2 51
=
Category: I.I.T. Author : Pankaj Baluja
5x( 2x 1) 2(2 x 1)
>0 +
2
10 x 51
+ – –1/2
>0
(5 x 2)( 2 x 1)
2/5
–
51 10
51 1 2 , 10 , x 2 5
58.
( x 2)( x 2 2x 1) 4 3x x 2
0
( x 2)( x 2 x x 1)
0
x 2 4x x 4 ( x 2)( x 1)2
0
( x 1)( x 4 )
–2
1
–1
4
x (– , –2) (–1, 1) (1, 4) x (– , –2) (–1, 4)
59.
4 1 x
+
2 1 x
–1<0
4 4 x 2 2x (1 x )(1 x ) (1 x )(1 x )
<0
2x 6 [1 x x x 2 ] = <0 (1 x )(1 x )
–1
1
2x 6 1 x 2 = >0 ( x 1)( x 1) 1
( x 1)( x 1)
>0
x (– , –1) (1, )
60.
2+
3 x 1
–
2 x
>0
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
2 x 2 2x 3 x 2 x 2 x( x 1) 2x 2 3x 2
(2x 1)( x 2) x( x 1)
61.
>0
–2
2 x 1
–
6 x
1
2
x( x 1)
,
>0
x 2 2x 3 x 6
–1/2
>0
x( x 1)
x( x 1)
0
>0
x 2 x 2x 6 x 6
x 2 5x 6
–1
>0
x (– , –2) (–1, 0) 0 ,
1+
Method of interval (Wavy Curve Method)
>0
x( x 1)
Category: I.I.T. Author : Pankaj Baluja
0
>0
1
3
2
x (– , –5) (1, 2) (6, ) Ans.
62.
x1 x1 – <2 x x 1 ( x 1)2 ( x 1)x 2( x 2 x ) x( x 1)
0
x 2 2x 1 x 2 x 2x 2 2x
x( x 1)
2x 2 x 1 >0 x( x 1)
<0 2x 2 2x x 1 x( x 1)
>0
( 2x 1)( x 1)
x( x 1)
–1
0
1/2
1
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
1 x (– , –1) 0 , (1, ) 2
63.
2( x 3)
–
x( x 6)
1 x 1
0
2( x 3)( x 1) ( x 2 6 x )
x( x 6)( x 1) 2x 2 8 x 6 x 2 6x x( x 6)( x 1)
x 2 2x 6
0
0
x( x 6)( x 1)
+
+ – –
0
0
–
1
6
x (– , 0) (1, 6)
64.
2( x 4) ( x 1)( x 7 )
–
1
0
( x 2)
2( x 4 )( x 2) x 2 8 x 7 ( x 1)( x 2)( x 7 ) 2 x 2 12 x 16 x 2 8 x 7 ( x 1)( x 7 )( x 2)
1
2
0
0
x 2 4x 9
( x 1)( x 7 )( x 2)
0
7
x (1, 2) (7, )
65.
2x 2
x 9
–
1 x2
0
2x 2 4 x x 2 9 ( x 3)( x 3)( x 2) x 2 4x 9 ( x 3)( x 3)( x 2)
0
0
– 3
–2
3
x (– , –3) (–2, 3)
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 66.
1 x2
+
1 x 1
–
Date : 10-05-12
( x 2)( x 1) x x2 2
Method of interval (Wavy Curve Method)
1 >0 x
x 2 x x 2 2x x 2 3 x 2
Category: I.I.T. Author : Pankaj Baluja
>0
>0
x( x 2)( x 1)
x ( 2 , 0) (2, )
67.
7
+
( x 2)( x 3)
9 ( x 3)
+1<0
7 9( x 2) ( x 2)( x 3) ( x 2)( x 3)
<0
7 9 x 18 x 2 5 x 6
( x 2)( x 3 ) x2 4x 5 ( x 2)( x 3 )
( x 5)( x 1)
( x 2)( x 3)
<0
<0
<0
x (– 5 + 1) (2, 3)
68.
20 2
x 7 x 12
+
10 +1>0 x4
20 10( x 3 ) x 2 7 x 12 ( x 3)( x 4 ) x 2 x 2x 2 ( x 3 )( x 4 )
>0
–2
–4
3
4
>0
( x 2)( x 1) >0 ( x 3)( x 4)
69.
x (– , – 2) (–1, 3) (4, )
( x 2)( x 4 )( x 7 ) >1 ( x 2)( x 4)( x 7)
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
( x 2 6 x 8)( x 7 ) 1 [( x 2)( x 4 )( x 7 )] ( x 2)( x 4 )( x 7 )
x 3 13 x 2 50 x 56 x 3 13 x 2 50 x 56 ( x 2)( x 4 )( x 7)
26 x 2 112 >0 ( x 2)( x 4 )( x 7 ) 26 x 2 12 ( x 2)( x 4 )( x 7 )
70.
( x 1)( x 2)( x 3) ( x 1)( x 2)( x 3) ( x 2 3 x 2)( x 3 ) ( x 1)( x 2)( x 3)
<0
– 7
–4
–2
– 3
–2
–1
>1
(x3 + 6x 2 + 11x + 6)
>1
x 3 3 x 2 2x 3 x 2 9 x 6 ( x 1)( x 2)( x 3 )
–1>0
x 3 6 x 2 11x 6 x 3 6 x 2 11x 6 ( x 1)( x 2)( x 3 )
12 x 2 12 >0 ( x 1)( x 2)( x 3 )
12( x 2 1) >0 ( x 1)( x 2)( x 3 )
>0
12( x 2 1) ( x 1)( x 2)( x 3 )
>0
<0
x (–, –3) (–2, –1)
71.
2
(x – 2x) (2x – 2) –
9( 2 x 2) x 2 2x
0 ab –
b [a 2 9] 0 a
2x 2 (x 2 – 3x + x – 3) 0 x( x 2)
9b
a
0 2x 2
x( x 2)
a 2b 9b 0 a
(x 2 – 2x – 3) (x 2 – 2x + 3) 0
2( x 1) (x – 3) (x + 1) 0 x( x 2)
x (– , 1] (0, 1] (2, 3]
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
72.
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
2 x 3 0 2L2 2 x
(x 2 + 3x) (2x + 3) – 16
a 2b 16b 0 a b a
(a – 4) (a + 4) 0
2x 3 x( x 3) 2x 3 x( x 3) 2x 3 x( x 3)
(x 2 + 3x – 4) (x 2 + 3x + 4) 0
x(x + 4) – 1(x + 4) > 0
(x – 1) (x + 4) > 0
( 2x 3)( x 1)( x 4 )
0
x( x 3)
73.
x (– , –4] 3 ,
x 2 2x 3 x 6 | x | 7 .
3 2
[1, )
<0
– < x < 0
x( x 2) 3( x 2) x7 ( x 3)( x 2) x7 ( x 3)( x 2) >0 x7
.
x (2, 3) (7, ) 0x<
Rejected
( x 3)( x 2) <0 x7
–7
2
3
x (– , –7) (2, 3) Contact:
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
x (2, 3) Ans.
74.
x 2 6x 7 |x 4|
<0
x 2 7x x 7 |x 4| case- – < x < –4
( x 1)( x 7 ) <0 x4 ( x 1)( x 7 ) >0 x4 x (–7, –4) (1, ) x (–7, –4) case- – 4 x <
–7
–4
1
( x 1)( x 7 ) <0 x4 x (– , –7) (– 4, 1) (–4, 1) Ans.
75.
|x2| >0 x2 case- – < x < 2
x 2 >0 x2
x2 <0 x2
1 < 0 false case- 2x<
x2 >0 x2 1 > 0 True
76.
2 x4
>1
2 |x4|
>1
case– – < x < 4
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
Method of interval (Wavy Curve Method)
2 x 4 <0 x 4
2 –1<0 x4
x2 >0 x4
x (– , 2) (4, )
Intersection
Category: I.I.T. Author : Pankaj Baluja
x (– , 2)
case- 4
2 x4
–1<0
2 x 4 <0 2 x 6 x <0 2 x x6 >0 x2
77.
x (– , 2) (6, ) x (– , 2) (6, ) Ans.
x 2 5 | x | 2 | x | 10 x 2 6x 9
<0
| x |2 5 | x | 2 | x | 10 ( x 3) 2 | x | [| x | 5] 2 [| x | 5] ( x 3 )2
| x | 2 | x | 5 ( x 3 )2
78.
<0
<0
<0
|x3|x >1 x2 |x3|x x2 >0 x2 | x 3 | 2 >0 x2 – < x < – 3
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
Date : 10-05-12
Method of interval (Wavy Curve Method)
x 3 2 >0 x2
x32 >0 x2
x 5 > x2
x1 >0 x2
x5 <0 x2
–2
x (– 5, –2)
79.
Category: I.I.T. Author : Pankaj Baluja
–1
x (– , –2) (–1, ) x (–2, )
x (– 5, –2) (–1, ) Ans.
| x 1| x2
| x 1| x 2
<1
x2
<0
case-I – < x < 1
x 1 x 2 <0 x2 2x 1 >0 x2
2x 1 <0 x2
1
x (– , –2)
2
,
x (– , –2)
case- 1x<8
x 1 –1<0 x2 x 1 x 2 <0 x2
3 <0 x2 3 x2
x (–2, ) x (– , –2) (–2, ) Ans.
80.
>0
|x3| 2
x 2x 3 x 6
2
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
|x3| ( x 2)( x 3)
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
–2>0
| x 3 | 2( x 2 5 x 6) ( x 2)( x 3 )
>0
case- – < x < 3
2x 2 10 x x 12 3 >0 ( x 3 )( x 2) 2x 2 9 x 9 >0 ( x 3 )( x 2) 2x 2 9 x 9 ( x 3)( x 2)
<0
2x( x 3) 3( x 3) ( x 3)( x 2) ( 2x 3)( x 3) ( x 3)( x 2)
3
x
2
<0
<0
, 2
case- 3
>0
2 x 2 11x 15 >0 ( x 3 )( x 2) 2 x 2 11x 15 ( x 3)( x 2)
( 2x 5)( x 3) ( x 3)( x 2)
5
2
x 2 ,
<0
<0
2
5/2
3 5 , Ans. 2 2
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics 81.
x 2 | x | 12
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
2x
x3
x 2 4 | x | 3 | x | 12 2x x3 [| x | 3][| x | 4] x3
– 2x 0
[| x | 3][| x | 4] 2x( x 3) ( x 3) – < x < –3
0
–
( x 3)( x 4 ) 2x 2 6 x x3
0
x 2 x 12 2x 2 6 x 0 x3
x 2 5x 12 0 x3 x 2 5 x 12
0
x3
82.
1+
x (– , 3] 12 x
2
–
7 <0 x
x 2 12 7 x x2
<0
( x 4 )( x 3) x2
83.
x0
<0
x (3, 4)
x – 17 –
60 0 x
x 2 17 x 60 x
x0
0
x 2 20 x 3 x 60 x
0
( x 3)( x 20) 0 x
[email protected] x [–3, 0) [20, ) Ans. Contact:
–3
0
2
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Chapter: Inequalities Website: www.mathematicsonline.co.in
Subject : Mathematics
84.
x2 4x 5 x 2 5x 6
Date : 10-05-12
Category: I.I.T. Author : Pankaj Baluja
Method of interval (Wavy Curve Method)
0
1
0
2
x 2x 3 x 6 1 ( x 3)( x 2)
–3
1/2
0
x (– , –3) (– 2, )
85.
x
6 x5
x( x 5) 6
0
x5
x 2 5x 6 x5
0
( x 6)( x 1) ( x 5)
–1
5
6
–3
1
0
x (– , –1] (5, 6]
86.
x 1 2
x 4 x 3x 2
0
x 4, – 3
x 1 0 ( x 4)( x 3)
4
x (– , –3) [1, 4)
87.
30 x 9 25x + 50 x2 30 x 9 (25 x 50)( x 2) 0 x2 =
30 x 9 25x 2 50x 50x 100 x2
25x 2 30 x 91 0 x2
25x 2 30 x 91 0 x2
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