Inequalities Test

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Inequalities Test - 3

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Weekly Quiz Competition

Question 1
2 / -0.83

Find the pairs of consecutive even positive integers both of which are smaller than 10 and their sum of more than 11

A
(4, 8)

B
(6, 8)

C
(6, 8) and (4, 8)

D
(6, 4)(4, 2)

Solution

Let x be the smaller of the two consecutive even positive integers .
Then the other integer is x+2.
Since both the integers are smaller than 10,x <10 ....(1)
Also the sum of the two integers is more than 11.
x+(x+2)>11
⇒2x+2 >11
⇒2x >11 −2
⇒2x >9
⇒x >9/2
⇒x >4.5....(2)
From (1) and (2) we obtain 4.5 >x >11
Since x is an even number, x can take the values 6,8 and 10.
Thus the required possible pairs are (6,8).
Question 2
2 / -0.83

The solution to |3x –1| + 1 <3 is

A
2

B
-1/3

C
-1/3

D
-3

Solution

|3x - 1| + 1 <3
|3x -1| <2
Opening mod, we get
3x - 1 <2, -3x + 1 >2
3x <3, -3x >1
x <1, x >-1/3
-1/3
Question 3
2 / -0.83

Which of the following is not a linear inequality?

A
ax² + bx + c <0

B
ax + by + c ≥0

C
ax + b <0

D
ax + by + c ≤0

Solution

A is a quadratic equation not a linear equality because square of a function can't be negative
Question 4
2 / -0.83

For a student to qualify for a certain course, the average of his marks in the permitted 3 attempts must be more than 60. His first two attempts yielded only 45 and 62 marks respectively. What is the minimum score required in the third attempt to qualify?

A
73

B
91

C
83

D
89

Solution

No of attempts = 3
Average = (45+62+x)/3=60
x=73
Question 5
2 / -0.83

Which one of them is the solution for x, when x is integer and 12 x >30?

A
-1

B
3

C
1

D
0

Solution

12x >30
x >30/12
x >2.5
x is an integer. So, minimum value of x is 3.
Question 6
2 / -0.83

Find the value of x which satisfies 5x –3 <7, where x is a natural number.

A
{1, 2}

B
1

C
(1,∞)

D
[1,∞)
Question 7
2 / -0.83

If -5x+2 <7x -4, then x is

A
<3/2

B
<1/2

C
>1/2

D
>3/2

Solution

-5x + 2 <7x - 4
6 <12x
x >1/2
Question 8
2 / -0.83

The solution to 5x-3 <3x+1, when x is an integer, is

A
{x / x εZ, x <2}

B
{x / x εZ, x=2}

C
{x / x εZ, x >2}

D
{x / x εZ, x ≥2}

Solution

We have 5x −3 <3x+1
⇒5x −3+3 <3x+1+3
⇒5x <3x+4
⇒5x −3 ×<3x+4 −3x
⇒2x <4 ⇒x <2
When x is an integer the solutions of the given inequality are {.............,−4,−3,−2,−1,0,1}
Hence {x / x εZ, x <2}
Question 9
2 / -0.83

The inequations -4x+1 ≥0 and 3-4x <0 have the common solutions given by

A
{}

B
{0,2}

C
{-2,2}

D
None of these

Solution

- 4x+1 ≥0 and 3-4x <0
1 ≥4x and 3 <4x
1/4 ≥x and 3/4
so x ∈ (-∞,1/4] U (3/4 , ∞)
Question 10
2 / -0.83

Solve the following linear inequality for x:

A
x ≤34

B
x >34

C
x ≥34

D
x <34

Solution

(2-3x)/5 ≤(-x-6)/2
By cross multiply we get
4-6x ≤-5x-30
-x ≤-34
x ≥34
Question 11
2 / -0.83

A point P lies in the solution region of 3x –7 >x + 3. So the possible coordinates of P are

A
(6 , -1)

B
(-3 ,4)

C
(2, 5)

D
(0, 7)
Question 12
2 / -0.83

A connected planar graph having 6 vertices, 7 edges contains _____________ regions.

A
15

B
3

C
1

D
11

Solution

By euler ’s formula the relation between vertices(n), edges(q) and regions(r) is given by n-q+r=2.
Question 13
2 / -0.83

If 5x+6 <2x-3, then

A
x >-3

B
x <-3

C
x >3

D
x <3

Solution

5x+6 <2x-3
5x-2x <-3-6
3x <-9
x <-3
Question 14
2 / -0.83

The region x >-3 lies

A
To the left of the point (-3,0)

B
Totally above the point (-3,0)

C
Totally below the point (-3,0)

D
To the right of the point (-3,0)
Question 15
2 / -0.83

If a

A
–a <-b

B
–a ≤-b

C
–a ≥-b

D
-a >-b

Question 16
2 / -0.83

The solution of inequality 4x + 3 <5x + 7 when x is a real number is

A
[-4, ∞)

B
(-∞, 4)

C
(-4,∞)

D
(-∞, 4]

Solution

4x+3 <5x+7
4x+3 −7 <5x+7 −7
4x −4 <5x
4x −4 −4x <5x −4x
−4 Thus, the solution set of the given inequality is (−4,∞).

Question 17
2 / -0.83

Two less than 5 times a number is greater than the third multiple of the number. So the number must be

A
Greater than 0

B
Greater than 1

C
Less than 3

D
Less than 2

Question 18
2 / -0.83

What values of x satisfy -6x >24 and x is an integer?

A
all integers less than -4

B
all integers greater than -4

C
all integers greater than -6

D
all integers less than -6

Solution

- 6x <24
=>6x <-24
x <- 4

Question 19
2 / -0.83

The solution set of , where x is a real

A
(-6, 6)

B
(-∞, -6)

C
(-∞, -6]

D
(-6, ∞)

Solution

X/3-x/2 >1
-x/6 >1
x/6
x <-6
therefore x belongs to the range (-∞,-6)

Question 20
2 / -0.83

Find the value of x when x is a natural number and 24x <100.

A
{5,6,……..∞}

B
{1,2,3,4}

C
{1,2,3,4,5}

D
{0,1,2,3,4}

Solution

24x <100
⇒x <100/24
⇒x <25/6
It is evident that 1,2,3 and 4 are the only natural numbers less than 25/6,
Thus when x is a natural number ,the solutions of the given inequality are 1,2,3 and 4.
Hence, in this case, the solution set is {1,2,3,4}.

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Inequalities Test - 3

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Inequalities Test - 3

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Question 1

(4, 8)

(6, 8)

(6, 8) and (4, 8)

(6, 4)(4, 2)

Solution

Question 2

2

-1/3

-1/3

-3

Solution

Question 3

ax2 + bx + c <0

ax + by + c ≥0

ax + b <0

ax + by + c ≤0

Solution

Question 4

73

91

83

89

Solution

Question 5

-1

3

1

0

Solution

Question 6

{1, 2}

1

(1,∞)

[1,∞)

Question 7

<3/2

<1/2

>1/2

>3/2

Solution

Question 8

{x / x εZ, x <2}

{x / x εZ, x=2}

{x / x εZ, x >2}

{x / x εZ, x ≥2}

Solution

Question 9

{}

{0,2}

{-2,2}

None of these

Solution

Question 10

x ≤34

x >34

x ≥34

x <34

Solution

Question 11

(6 , -1)

(-3 ,4)

(2, 5)

(0, 7)

Question 12

15

3

1

11

Solution

Question 13

x >-3

ax² + bx + c <0