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IIT ASHRAM
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UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda. Ph. 90330 63029, 90810 62221
ASSIGNMENT MATHEMATICS
TOPIC : LINEAR INEQUATION
DRILL -I
Find the values of x
2
( x 1)( x 2)
1. (x – 1)(3 – x) (x – 2)2 > 0. 2. 0.
1 x
1 3x 2 x 1
1.
3. 2
0. 4. 2
2 x 21x 40 ( x 1)
2 x 2 3x 459 x 7 3x 1
5. 1. 6. 0.
x 2 1 x 5 2
1 3 (2 x 2 )(x 3)3
7. . 8. 0.
x 2 x 3 (x 1)(x 2 3x 4)
x 4 3x 3 2x 2 2(x 3)
1
.
9. 0. 10.
x 2 x 30 x(x 6) x 1
1 1 1 2x 3
11. . 12. 0
x 2 x 1 x 3x 7
x 2 5 x 12 x 2 5x 6
13. >3 14. <0
x 2 4x 5 x2 x 1
( x 1) 2 ( x 1) 3 x 1 x 5
15. <0 16.
4 x 1 x 1
x (x 2)
2( x 4 )
1 x 2 4x 4
17. 18. >0
( x 1)( x 7 ) x 2 2x 2 x 1
DRILL - II
Solve the following inequalitites
1. |x + 2 | = 2(3 – x) 2. |x – 3| + 2 |x + 1| = 4
3. |x|-|x - 2|= 2 4. |x – 1| + |x – 2| + | x – 3 | 6
5. |5 – 2x| < 1 6. |x – 2| |x + 4|
7. |x2 – 4x| < 5 8. x2 - |x| - 2 0
9. |x2 + x| - 5 < 0 10. |x2 – 2x| < x
11. |x2 – 2x - 3| < 3x – 3
I I T AS H R A M Page # 1
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
SINGLE CORRECT TYPE QUESTIONS
Linear Inequation
1. Which of the following linear inequations represents the above graph?
(a) x>2 (b) x 2 (c) x < 2 (d) x 2
2. Graph of y 2 is _______.
(a) (b)
(c) (d)
3. Which of the following inequations best describes the following graph?
Y-axis
x=2
y=3
3
2
1
X-axis
0 1 2
(a) x 2, y 3 (b) x < 2 ; y < 3
(c) x 2; y 3; x 0, y 0 (d) x < 2; y < 3; x > 0 ; y > 0
4. If a and b are two real numbers such that a > b and k is a non-zero number, then for
what values of k is ka > kb?
(a) All real values of k (b) All positive real values of k.
(c) All negative values of k. (d) All positive integral values of k.
5. The solution set of which of the following inequations is an empty set?
(a) x + y 3, x + y 2 (b) x + y 5, x + y 5
I I T AS H R A M Page # 2
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
(c) x + y 5, x + y 3 (d) x + y 2, x + y>2
6. The solution set formed by the inequations x 0, y 0 and 3x – 4y 12 is a/an
(a) finite set (b) singleton set
(c) infinite set (d) null set
7. The region formed by the inequation 3x – 8y< –5 is ______.
(a) does not contain the origin (b) contains the origin
(c) can’t say (d) none of the above
8. The point (–2, –2) does not belong to which of the following regions?
(a) 3x – 6y – 7 0 (b) 7x + 4y + 28 0
(c) x 2 (d) 2x – y 4
9. A shop keeper sells not more than 45 bottles of pepsi and atleast 50 bottles of sprite.
The total number of both kinds cannot exceed 150. Frame the inequations for the
above data.
(a) y 50, x 45, x + y 150 (b) x + y 150, 45 x 50, y 50
(c) x 45, 45 y 50, x + y 150 (d) None of these
10. If x {0, 1, 2, 3, 5 7, 9, 11}, the solution set of 25 – 3(4x – 1) < 4 is
(a) {0, 1} (b) {0, 1, 2, 3, 5, 7, 9, 11}
(c) {3, 5, 7, 9, 11} (d) {2, 3, 5, 7, 9, 11}
11. The least integral value of x that satisfies 2x + 4 + 3(x – 5) > 7 is
(a) 3 (b) 4 (c) 5 (d) 6
12. The solution set of 2x – 3 < x + 2 3x + 5, x R is
3 3
(a) {x : x < 5, x R} (b) {x : < x < 5, x R}
2 2
3 3
(c) {x : x < 5, x R} (d) {x : < x < 5, x R}
2 2
4x 3
13. The least value of x which satisfies the inequation 3 3 is
5
(a) –2 (b) –3 (c) –4 (d) 0
15 1
14. The solution set of (3x – ) (3x – 30) is.
2 2
(a) {x/x < 5} (b) {x/x –5} (c) {x/x > 5} (d) {x/x –5}
15. The solution set of 3(–7x + 2) 5 [3x + 2 (x – 11)] is
58 58 58
(a) x / x (b) x / x (c) x / x (d) None of these
23 23 23
7 x
16. The solution set of 2x – 1 x + > 2 is
3
5 1
(a) {x : x , x R} (b) {x : x > , x R}
2 2
(c) {x : x 5 / 2, x R } (d) None of these
I I T AS H R A M Page # 3
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
17. The cardinal number of the integral solution set formed by the inequations x 0, 7x
+ 4y 28 and y 0 is
(a) 16 (b) 21 (c) 18 (d) 17
18. The solution set formed by the inequations x + y 5 and x + y 7 in the first
quadrant represents a
(a) triangle (b) rectangle (c) trapezium (d) rhombus
19. The common solution set of the inequations 3 4x + 3 5 and 5 3x + 7 8 is
1 1 2 1 2 1
(a) 0 x (b) 0 x (c) x (d) x
2 3 3 3 3 2
20. If x, y and z are three non-zero numbers such that x + y z – x, y + z x – y and
z + x y –z, the maximum value of x + y + z is
(a) 0 (b) –1 (c) 1 (d) None of these
21. The length of a rectangle is twice that of its breadth. The perimeter of the rectangle
cannot exceed 48m. Find the dimensions of the rectangle if its area is the maximum
(in metres).
(a) 24, 12 (b) 20, 10 (c) 16, 8 (d) 7, 9
Quadratic/Rational Inequality
1. The solution set of x2 – 5x + 6 0 is
(a) (2, 3) (b) [2, 3]
(c) (– , 2) (3, ) (d) (– , 2] [3, )
2. If 2 – 3x – 2x2 0 , then
1 1
(a) x 2 (b) 2 x (c) x 2 (d) x
2 2
3. The solutions of the inequation 2x 2 + 3x 9 0 are given by
3 3 3
(a) 3 x (b) 3 x (c) x3 (d) none of these
2 2 2
4. The solution of inequation –x2 + 6x – 8 > 0 is
(a) 4<x<2 (b) 2 < x < 4 (c) 2 > x > 4 (d) none of these
5. If a < b, then the solution of x 2 + a + b x + ab < 0 , is given by
(a) x < –b or x – a (b) a < x < b (c) x < a or x > b (d) –b < x < – a
6. Solve (x + 2)2 – (x2 + 3x + 3) < -(x + 3)2.
(a) (-5, -2) (b) (2, 5)
(c) (- , -5) (-2, ) (d) No solution
7. (x2 + 2x) < m (x + 1)2 for all x R, then the possible values of m can be
3 3
(a) (1, ) (b) , (c) [1, ) (d) ,
4 4
I I T AS H R A M Page # 4
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
x 2 36
8. 0 , find the value of x?
x 2 9x 18
(a) x (3,6) (b) x (6,3)
(c) x (, 6) (3, ) (d) x (, 3) (6, )
9. x2 < n, if n (- , 0), then the set of all possible values of x is
(a) any real number. (b) only positive number.
(c) no solution. (d) unconditional.
2 2
10. Solve (x + 1) + (x – 1) < 6.
(a) (- 2 , 2) (b) (-1, 1)
(c) (- , -2) (2, ) (d) (- , -1) (1, )
x2 1
11. 2
, find the solution set of x?
x 1 2
(a) x (3,1) (b) x (, 3) (1, )
(c) x (1,3) (d) x (, 1) (3, )
12. If mx2 < nx such that m and n have opposite signs then x can be (m being positive)
n n n
(a) x> (b) x < (c) <x<0 (d) None of these
m m m
13. The solution of the inequation, 15x2 – 31x + 14 < 0 is given by
7 2 2 7 7 2 7 2
(a) <x< (b) <x< (c) x > or x < (d) x > or x >
5 3 3 5 5 3 5 3
14. The sum of a number and its square is greater than 6, then the number can be
(a) (- , 2) (3, ) (b) (- , -3) (2, )
(c) (2, 3) (d) [2, 3]
15. If x2 – 1 0 and x2 – x – 2 0, then x line in the interval/set
(a) (–1, 2) (b) (–1, 1) (c) (1, 2) (d) {–1}
16. If x2 - 4x + 3 > 0 and x2 - 6x + 8 < 0, then
(a) x>3 (b) x < 4 (c) 3 < x < 4 (d) 1 < x < 2
17. Solve the simultaneous quadratic inequations x2 + 7x – 18 0 and x2 – 4x + 3 0.
(a) [–9, 1] [2, 3] (b) [–9, 2] [3, [
(c) [–9, 1] (d) None of these
18. The values of x for which -2x - 4 (x + 2)2 -2x - 1 satisfies is
(a) [-5, -1] (b) [-5, 0)
(c) [-5, -4] (-2, -1] (d) [-5, -4] [-2, -1]
19. Solve x2 – 7x + 3 < 2x + 25.
(a) (-2, 11) (b) (2, 1)
(c) (- , -1) (2, 11) (d) (-8, -2) [11, )
I I T AS H R A M Page # 5
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
x 2 2x 3
20. The solution of inequation > is
x+2 4x +1
1
i) x < –2 ii) x < 1 iii) x > 4 iv) x >
4
(a) Only (i) and (iii) (b) Only (ii) and (i)
(c) All (i), (ii), (iii) and (iv) (d) none of these
x 2 2x + 5 1
21. The range of values of x for which 2
>
3x 2x 5 2
5 5
(a) –5 < x < –1 or <x<3 (b) 5 < x < 1 or <x<3
3 3
5
(c) – 5 < x < 1 or <x<3 (d) none of these
3
x 1 x3
22. Find the range of real values of x for which <
4x + 5 4x 3
5 3 5 3 5 3
(a) <x< (b) x (c) x (d) none of these
4 4 4 4 4 4
x 3 x 1 x 2
23. The value of x for which x< , 2 x > 2x 8 is
4 2 3
10 10
(a) 3 , 1 (b) 1,
3
(c) R (d) none of these
x +1 1
24. The number of integral solutions of 2
> is
x +2 4
(a) 1 (b) 2
(c) 5 (d) none of these
x 3
25. The values of x for which expression is 0
2
x 3x 54
(a) (-6, -3] (9, ) (b) [-6, -3] [9, ]
(c) (-6, -3) (9, ) (d) (-6, )
x2 1
26. Solve < .
1 x
(a) xR (b) x R– – {-1} (c) x R+ (d) x R –{0} .
x 2 x 12
27. If 0, then x lies in
x 2 3x 2
(a) (-4, 3) (b) (-4, 2)
(c) [-4, 1) [2, 3] (d) (-4, 1) (2, 3)
I I T AS H R A M Page # 6
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
x
x2 1
28. For all real values of x, 2 5 is
x2 1 4
1
(a) equal to 1 (b) 0 (c) (d) 0
4
x 5
29. The least integral value of x such that 2 > 0 is
x 5x 14
(a) –4 (b) 2 (c) –6 (d) 5
x 2 + 4x + 3
30. The value of x is If >0
x 3 6x 2 +11x 6
(a) (–3, –2) (3, 31) (b) (–3, 3/2) (5, )
(c) (– , –4) (5/2, ) (d) (–3, –1) (1, 2) (3, )
2x 1
31. If S is the set of all real x such that is positive. Then S is
2x + 3x 2 + x
3
(a) (–3, –2) (3, 31) (b) (–3, 3/2) (5, )
(c) (– , –1) (-1/2, 0) (1/2, ) (d) (–3, –1) (1, 2) (3, )
x 2 + 2x + 7
32. The value of x is If < 6, x R .
2x + 3
(a) (–3, –2) (3, 31) (b) (– , -3/2) (-1, 11)
(c) (– , –1) (-1/2, 0) (1/2, ) (d) (–3, –1) (1, 2) (3, )
Modulus Equation & Inequation
1. If x is a real number and | x | < 3, then
(a) x 3 (b) – 3 < x < 3 (c) x – 3 (d) – 3 x 3
2. x and b are real numbers. If b > 0 and | x | > b, then
(a) x (– b, ) (b) x [– , b)
(c) x (– b, b) (d) x (– , – b) (b, )
3. If |x -1| > 5, then
(a) x (– 4, 6) (b) x [– 4, 6]
(c) x (– , – 4) (6, ) (d) x [– , – 4) [6, )
4. If |x + 2| 9, then
(a) x (– 7, 11) (b) x [– 11, 7]
(c) x (– , – 7) (11, ) (d) x (– , – 7) [11, )
5. If |3– 4x| 9 Then x lies in the interval
(a) x (– 13, 7] (b) x (– 13, 7)
(c) x (– , – 3/2] [3, ) (d) x [– , – 13]
6. The least positive integer x, which satisfies |x – 2| > 7?
(a) 9 (b) 10 (c) 7 (d) 5
I I T AS H R A M Page # 7
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
7. If 1 |x –2 | 3 Then x lies in the interval
(a) x (– 13, 7] (b) x (13, 7)
(c) x (– , – 13] [7, ) (d) x [– 1, 1] [3, 5]
8. If |x + 1|+|x|>3 Then x lies in the interval
(a) x (– 13, 7] (b) x (– 13, 7)
(c) x (– , – 2) (1, ) (d) x [– , – 13] [7, )
9. If |x – 1| < –2, then find the range of x.
(a) x < –3 (b) x < –1 (c) x R (d) None of these
10. The solution set of the inequation |5 – 2x| + x < 4 is ______.
(a) (1, 2) (b) (2, 3) (c) (1, 3) (d) (1, 4)
11. The solution set of the inequation |x + 1| < x + 2 is ________.
5 7 3
(a) x > –1 (b) x > (c) x > (d) x >
2 4 2
12. Which of the following is true?
(a) |x + y| > |x| + |y| (b) ||x| – |y|| |x – y|
x |x|
(c) ;y 0 (d) | x| 2
= –x2
y | y|
13. The roots of the equation |x|2 - |x| - 2 = 0 are
(a) {–2, –1, 1, 2} (b) {–2, 2} (c) {–1, 1} (d) {1, 2}
14. Solve the quadratic inequation, |x|2 – 2 | x | – 8 0.
(a) [–4, 4] (b) [0, 4] (c) [–4,0] (d) [–4, 2]
15. The number of integral values in the solution set of the inequation
|x – 1| + |x – 2| + |x – 3| < 0 is
(a) 0 (b) 1 (c) 2 (d) 3
x x2
16. If |x| then the value of x lies in the interval
x 1 |x 1|
(a) [0, ) (b) (0, ) (c) [1, ) (d) (1, ) {0}
I I T AS H R A M Page # 8
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
ANSWER KEY
DRILL - I
1. (1, 2) (2, 3) 2. (– ,–2) (–2, –1) (1, + ) 3. (5/2, 8)
4. (– , 0) (3, + ) 5. (– , –20) (23, + ) 6. [1, 3] (5, + )
7. (–9/2, –2) (3, + ) 8.
2, 1 1, 2 3, 4
9. (– , –5) (1, 2) (6, + ) 10. (– , 0) (1, 6)
11. (– 2 , 0) (1, 2 ) (2, + )
12. (– , 3/2) (7/3, + ) 13. (1/2, 3) 14. (2, 3)
15. (–1, 2) – {0, 1} 16. (– , – 1) 1, 3
17. (1, 2) (7, + ) 18. (– , – 2) (– 2, – 1/2 ) (1, )
DRILL - II
1. {4/3} 2. { – 1} 3. [2, + ) 4. (– , 0] [4, )
5. (2, 3) 6. [–1, + ) 7. (-1, 5) 8. (– , -2] [2, )
1 21
,
21 1
9. 2 2 10. (1, 3) 11. (2, 5)
SINGLE CORRECT TYPE QUESTIONS
Linear Inequation
1. B 2. C 3. C 4. B 5. D 6. C 7. A
8. D 9. A 10. C 11. B 12.A 13.B 14.D
15. C 16. C 17. B 18. C 19.B 20.A 21.C
Quadratic/Rational Inequality
1. B 2. B 3. B 4. B 5. D 6. A 7. A
8. B 9. C 10. A 11. A 12.C 13.B 14.B
15. D 16. C 17. C 18. D 19.A 20.A 21.A
22. B 23. B 24. C 25. C 26.B 27.D 28.D
29. C 30. D 31. C 32. B
Modulus Equation & Inequation
1. B 2. D 3. C 4. B 5. C 6. B 7. D
8. C 9. D 10. C 11. D 12.B 13.B 14.A
15. A 16. D
I I T AS H R A M Page # 9
UG–1 & 2, Concorde Complex, Above PNB . R.C. Dutt Road., Alkapuri Baroda.
