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Test: Linear Inequalities- 1 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Linear Inequalities- 1

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Test: Linear Inequalities- 1 - Question 1

By solving the inequality 3(a - 6) < 4 + a, the answer will be

Detailed Solution for Test: Linear Inequalities- 1 - Question 1

Given inequality:

3(a - 6) < 4 + a

Step 1: Expand the left-hand side

3a - 18 < 4 + a

Step 2: Move all terms involving aaa to one side and constants to the other side

Subtract aaa from both sides:

3a - a - 18 < 4

This simplifies to:

2a - 18 < 4

Step 3: Add 18 to both sides

2a < 22

Step 4: Divide both sides by 2

a < 11

The correct option is D: a < 11.

Test: Linear Inequalities- 1 - Question 2

What is the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 2

Given:

Eliminate the denominators by multiplying both sides of the inequality by 35 (the least common multiple of 5 and 7):

This simplifies to:
7 × 2(x − 1) ≤ 5 × 3(2 + x)
Which further simplifies to:
14(x − 1) ≤ 15(2 + x)
Distribute the constants:
14x − 14 ≤ 30 + 15x
Move all terms involving x to one side and constants to the other side:
14x − 15x ≤ 30 + 14
This simplifies to:
−x ≤ 44
Solve for x by dividing by -1 (remember to reverse the inequality):
x ≥ −44x
The solution set is (−44, ∞)

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Test: Linear Inequalities- 1 - Question 3

Identify the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 3

Step 1: Start with the given inequality:

Step 2: Simplify both sides:

Step 3: Combine the constants on both sides:

which simplifies to

Step 4: Multiply through by 15 to clear the denominators:

which simplifies to
5(x − 1) + 90 < 3(x − 5)

Step 5: Distribute and combine like terms:
5x − 5 + 90 < 3x − 15
which simplifies to
5x + 85 < 3x − 15
then subtract 3x from both sides:
2x + 85 < −15
then subtract 85 from both sides:
2x < −100
finally, divide by 2:
x < −50

Final Answer:

(B) (−∞, −50)

Test: Linear Inequalities- 1 - Question 4

What is the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 4

check for interval (7/3, ∞ ) the whole would be +ve
check for interval (-∞, 3/2 ) the whole would be +ve

Test: Linear Inequalities- 1 - Question 5

Identify the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 5

(7x-5)/(8x+3) > 4
(7x-5)/(8x+3) - 4 > 0
7x - 5 - 4 ( 8x + 3 ) / 8x + 3 > 0
- 25 x - 17 / 8x + 3 > 0
Now furthermore solving for general range:
x ∈ ( -17/ 25, - 3/8)

Test: Linear Inequalities- 1 - Question 6

What is the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 6

Given Inequality:

Step 1: Break down the inequality

We need to solve both parts of the compound inequality:

Step 2: Solve each part separately

Multiply both sides of the inequality by 2 to get rid of the denominator:
0 < -x
Now, multiply both sides by -1 (which reverses the inequality):
x < 0

Multiply both sides by 2 to eliminate the denominator:
-x < 6
Multiply both sides by -1 (which reverses the inequality):
x > -6
Step 3: Combine the results
From both inequalities, we now have:
-6 < x < 0
The solution set is x ∈ (−6, 0), which corresponds to Option C.

Test: Linear Inequalities- 1 - Question 7

Identify solution set for | 4 − x | + 1 < 3?

Detailed Solution for Test: Linear Inequalities- 1 - Question 7

|4 − x| + 1 < 3
⇒ 4 − x + 1 < 3
Add −4 and −1 on both sides, we get
4 − x + 1 − 4 − 1 < 3 − 4 − 1
⇒ − x < −2
Multiply both sides by −1, we get
x > 2
Also,|4 − x| + 1 < 3
⇒ −(4 − x) + 1 < 3
⇒ − 4 + x + 1 < 3
Add 4 and −1 on both sides, we get
− 4 + x + 1 + 4 − 1 < 3 + 4 − 1
⇒ x < 6
Thus, x ∈ (2, 6).

Test: Linear Inequalities- 1 - Question 8

What is the solution set for

Detailed Solution for Test: Linear Inequalities- 1 - Question 8

Test: Linear Inequalities- 1 - Question 9

Identify the solution set for  

Detailed Solution for Test: Linear Inequalities- 1 - Question 9

Given Inequality:

Step 1: Eliminate the denominators by multiplying both sides by 15

Multiply both sides of the inequality by 15, which is the least common multiple (LCM) of 3 and 5:

This simplifies to:
5(x − 1) + 60 < 3(x − 5) − 30
Step 2: Simplify both sides
Distribute the constants:
5x − 5 + 60 < 3x − 15 − 30
Simplifying further:
5x + 55 < 3x − 45
Step 3: Move all terms involving 
x to one side and constants to the other side
Subtract 3x and subtract 55 from both sides:
5x − 3x + 55 − 55 < 3x − 45 − 3x − 55
Simplifies to: 2x < −100
Step 4: Solve for x
Divide both sides by 2:
x < −50
The solution set is: x ∈ (−∞, −50)

Test: Linear Inequalities- 1 - Question 10

What is the solution set for 

Detailed Solution for Test: Linear Inequalities- 1 - Question 10


|x| > a
⇒ x > a
or x < -a
(2x-1)/(x-1) > 2 and (2x-1)/(x-1) < -2
(2x-2+1)/(x-1) > 2
⇒ (2(x-1) + 1)/(x-1) > 2
⇒ 2 + (1/(x-1)) > 2
1/(x-1) > 0
x-1 < 0
x < 1...........(1)
Now taking, (2x-1)/(x-1) < -2
2 + (1/(x-1) < -2
= 1/(x-1) < -4
x-1 > -1/4
x > -1/4 + 1
x > 3/4.......(2)
From (1) and (2)
x implies (3/4, 1)∪ (⁡1,∞)

Test: Linear Inequalities- 1 - Question 11

In the first four papers each of 100 marks, Rishi got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks, find the range of marks he should score in the fifth paper .

Detailed Solution for Test: Linear Inequalities- 1 - Question 11

Let score be x in the fifth paper, then

Hence Rishi must score between 52 and 77 marks.

Test: Linear Inequalities- 1 - Question 12

A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3 cm longer than the shortest and third length is to be twice as long as the shortest. What are the possible lengths for the shortest board if the third piece is to be at least 5 cm longer than the second?

Detailed Solution for Test: Linear Inequalities- 1 - Question 12

Let the length of the shortest piece be x cm. Then, length of the second piece and the third piece are (x + 3) cm and 2x cm respectively.

Since the three lengths are to be cut from a single piece of board of length 91 cm,

x cm + (x + 3) cm + 2x cm ≤ 91 cm

⇒ 4x + 3 ≤ 91

⇒ 4x ≤ 91 ­– 3

⇒ 4x ≤ 88

⇒ x ≤ 22   ...(1)

Also, the third piece is at least 5 cm longer than the second piece.

∴ 2x ≥ (x + 3) + 5

⇒ 2x ≥ x + 8

⇒ x ≥ 8 …(2)

From (1) and (2), we obtain

8 ≤ x ≤ 22

Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.

Test: Linear Inequalities- 1 - Question 13

Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

Detailed Solution for Test: Linear Inequalities- 1 - Question 13

Let x be the smaller of the two consecutive odd natural number, so that the other
one is x + 2
Given x > 10 and x + (x + 2) < 40
⇒ 2x < 38
⇒ x < 19
∴ 10 < x < 19
Since x is an odd number, x can take the values 11, 13, 15 and 17.
∴ Required possible pairs are (11, 13), (13, 15), (15, 17), (17, 19)

Test: Linear Inequalities- 1 - Question 14

The marks scored by Rohit in two tests were 65 and 70. Find the minimum marks he should score in the third test to have an average of atleast 65 marks.

Detailed Solution for Test: Linear Inequalities- 1 - Question 14

Let x be the minimum  marks he scores in the third test. 

⇒ 135 + x ≥ 195( Multiplying both the sides by 3)
⇒ x ≥ 195 − 135
⇒ x ≥ 60
Hence, the minimum marks Rohit should score in the third test should be 60. 

Test: Linear Inequalities- 1 - Question 15

A solution is to be kept between 30C and 35C What is the range of temperature in degree Fahrenheit?

Detailed Solution for Test: Linear Inequalities- 1 - Question 15

Let x degree Fahrenheit be the temperature of the solution. 

Hence, the range of the temperature in Fahrenheit is between  86° and 95°. 

Test: Linear Inequalities- 1 - Question 16

The longest side of a triangle is three times the shortest side and the third side is 2cm shorter than the longest side if the perimeter of the triangles at least 61cm, find the minimum length of the shortest side.

Detailed Solution for Test: Linear Inequalities- 1 - Question 16

Let the shortest side of the triangle be x cm.
Then, the longest side will be 3x and the third side will be 3x − 2. 
∴ Perimeter of the triangle ≥ 61
⇒ x + 3x + 3x − 2 ≥ 61
⇒ 7x ≥ 61 + 2
⇒ x ≥ 9( Dividing throughout by 7)
Hence, the minumum length of the shortest side is 9 cm.

Test: Linear Inequalities- 1 - Question 17

Which of the following is correct?

Detailed Solution for Test: Linear Inequalities- 1 - Question 17

We are given a set of inequalities and asked to identify which one is correct. Let's analyze each option one by one.
Option A: If 0 > -7 , then 0 < 7

  • The inequality 0 > -7 is true because 0 is indeed greater than -7.
  • The statement 0 < 7 is also true because 0 is less than 7.
  • Since both parts of this conditional are true, Option A is correct.

Option B: If 8 > 1, then -8 > -1

  • The inequality 8 > 1 is true because 8 is greater than 1.
  • However, the statement −8 > −1 is false because -8 is smaller than -1. As numbers become more negative, they become smaller.
  • Therefore, Option B is incorrect.

Option C: If -4 < 7 , then 4 < - 7

  • The inequality −4 < 7 is true because -4 is smaller than 7.
  • However, the statement 4 < −7 is false because 4 is greater than -7, not less.
  • Therefore, Option C is incorrect.

Option D: If -2 < 5 , then 2 > 5

  • The inequality −2 < 5 is true because -2 is smaller than 5.
  • However, the statement 2 > 5 is false because 2 is less than 5, not greater.
  • Therefore, Option D is incorrect.

Option A is the correct one, as both parts of the statement are true.

Test: Linear Inequalities- 1 - Question 18

Solve the inequality 3 − 2x ≤ 9

Detailed Solution for Test: Linear Inequalities- 1 - Question 18

3 − 2x ≤ 9
Add -3 both sides we get
​3 − 2x − 3 ≤ 9 − 3
⇒ −2x ≤ 6​
Divided by -2 both sides we get
x ≥ −3

Test: Linear Inequalities- 1 - Question 19

Given that x is an integer, find the values of x which satisfy both 2x + 3 > 7 and x + 4 < 10

Detailed Solution for Test: Linear Inequalities- 1 - Question 19

We are given two inequalities:

  1. 2x + 3 > 7
  2. x + 4 < 10

We need to find the integer values of x that satisfy both inequalities.
Step 1: Solve the first inequality 2x + 3 > 7
Subtract 3 from both sides:
2x > 7 - 3
Simplifying:
2x > 4
Divide both sides by 2:
x > 2
So, the first inequality gives us:
x > 2
Step 2: Solve the second inequality x + 4 < 10
Subtract 4 from both sides:
x < 10 - 4
Simplifying:
x < 6
Step 3: Combine the results
From the first inequality, we know x > 2.
From the second inequality, we know x < 6.
Thus, the solution for x must satisfy both:
2 < x < 6
The integer values of x that lie between 2 and 6 are:
x = 3, 4, 5
The correct option is: D: 3, 4, 5

Test: Linear Inequalities- 1 - Question 20

The solution of 4x - 2 > 6 is

Detailed Solution for Test: Linear Inequalities- 1 - Question 20

To solve the given linear equation, 4x - 2 = 6, for the variable x begin by adding 2 to both sides of the equation in order to begin isolating the variable x on one side, in this case the left side, as follows:
4x - 2 = 6 (Given)

4x - 2 + 2 = 6 + 2

4x + 0 = 8

4x = 8

Now, divide both sides of the equation by 4 in order to isolate x on the left side, thus finally solving the equation for x as follows:
(4x)/4 = 8/4

(4/4)x = 8/4

(1)x = 2

x = 2

Therefore, x = 2 is indeed the solution to the given equation.

Test: Linear Inequalities- 1 - Question 21

x = 4, 5 and 6 are the solutions for:

Detailed Solution for Test: Linear Inequalities- 1 - Question 21

To determine which statement matches the solutions x = 4, 5 and 6, let's analyze each option:

a) x > 4 and x < 7

  • This means 4 < x < 7.
  • The solutions x = 5, 6 do not satisfy this condition because it does not contain 4.

b) x ≥ 4 and x ≤ 7

  • This means 4 ≤ x ≤ 7.
  • The solution x = 4, 5, 6, 7 do not satisfy this condition as it contains 7 also.

c) x ≥ 4 and x < 7

  • This means 4 ≤ x < 7.
  • The solutions x = 4, 5, 6 satisfy this condition because they are all greater than or equal to 4 and strictly less than 7.

d) x > 4 and x > 7

  • This means x > 7.
  • The solutions x = 4, 5, 6 do not satisfy this condition because none of them are greater than 7.

Therefore, after analyzing each option, we conclude that the solutions x = 4, 5, 6 correspond to statement: (c) x ≥ 4 and x < 7

Test: Linear Inequalities- 1 - Question 22

What are the integer values of x which satisfy the inequalities x > − 2 and x ≤ 2?

Detailed Solution for Test: Linear Inequalities- 1 - Question 22

We are given two inequalities:

  1. x > − 2
  2. x ≤ 2

Step 1: Analyze the inequalities
For x > -2:
This means x must be greater than -2 but not equal to -2. So, x can be any value larger than -2.
For x ≤ 2:
This means x must be less than or equal to 2.
Step 2: Combine the results
We are looking for integer values of xxx that satisfy both conditions: x > -2 and x ≤ 2x.
So the integer values of x that lie between −2 (not included) and 2 (included) are:
x = -1, 0, 1, 2
The correct option is: B: -1, 0, 1, 2

Test: Linear Inequalities- 1 - Question 23

Given that x is an integer, find the values of x which satisfy the simultaneous linear inequalities 2 + x < 6 and 2 −3x < − 1.

Detailed Solution for Test: Linear Inequalities- 1 - Question 23

Given Inequalities:

  1. 2 + x < 6
  2. 2 −3x < − 1

Step 1: Solve the first inequality 2 + x < 6
Subtract 2 from both sides:
x < 6 - 2
Simplifying:
x < 4
Thus, from the first inequality, we have:
x < 4
Step 2: Solve the second inequality 2 - 3x < -1
Subtract 2 from both sides:
- 3x < -1 - 2
Simplifying:
- 3x < -3
Now, divide both sides by -3, and remember to reverse the inequality sign when dividing by a negative number:
x > 1
Step 3: Combine the results
From the first inequality, we know x < 4.
From the second inequality, we know x > 1.
Thus, the solution for x must satisfy both:
1 < x < 4
The integer values of xxx that lie between 1 and 4 are:
x = 2, 3
The correct option is: B: 2, 3

Test: Linear Inequalities- 1 - Question 24

Solve: 30x < 200, when x is a natural number:

Detailed Solution for Test: Linear Inequalities- 1 - Question 24

We are given 30x < 200

When x is a natural number, in this case the following values of x make the statement true.
1, 2, 3, 4, 5, 6.
The solution set of the inequality is {1, 2, 3, 4, 5, 6}.

Test: Linear Inequalities- 1 - Question 25

The solution set for: 3x − 7 > x + 3.

Detailed Solution for Test: Linear Inequalities- 1 - Question 25

Given Inequality:
3x − 7 > x + 3
Step 1: Move all terms involving x to one side and constants to the other side
Subtract x from both sides:
3x − x − 7 > 3
Simplifying:
2x - 7 > 3
Add 7 to both sides:
2x > 3 + 7
Simplifying:
2x > 10
Step 2: Solve for x
Divide both sides by 2:

Simplifying:
x > 5
Step 3: Solution Set
The solution is x > 5, which means the solution set is:
x ∈ (5, ∞)
The correct option is: C: (5, ∞)

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