Inequalities Test - 2

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 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 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 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
• 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 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, ∞)