Test: Linear Inequalities- 1 - Commerce MCQ

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.

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, ∞)

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)

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

(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)

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.

|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).

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)

|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,∞)

Let score be x in the fifth paper, then

Hence Rishi must score between 52 and 77 marks.

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.

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)

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. 

Let x degree Fahrenheit be the temperature of the solution. 

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

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.

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.

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

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

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.

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

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

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

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}.

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, ∞)