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All questions of Inequalities for Year 10 Exam

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Can you explain the answer of this question below:

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,∞ )

The answer is b.

Krishna Iyer answered
The given inequality is 5x– 3 < 7
=> 5x – 3 + 3 < 7 + 3                             [3 is added both sides]
=> 5x < 10
=> x < 10/5
=> x < 2
When x is a real number, the solutions of the given inequality are given by x < 2, i.e., all real numbers x which are less than 2.

The solution to 5x-3<3x+1, when x is an integer, is
  • a)
    {x / xεZ, x<2}
  • b)
    {x / xεZ, x=2}
  • c)
    {x / xεZ, x>2}
  • d)
    {x / xεZ, x≥2}
Correct answer is option 'A'. Can you explain this answer?

Aadhar Academy answered
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}

 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}
Correct answer is option 'B'. Can you explain this answer?

Shreya Gupta answered
We are given: 24x < 100
24x < 100
=> 24x /24 < 100/24 [Dividing both sides by positive number.]
=> x < 25/6

When x is a natural number, in this case, the following values of x make the statement true

x = 1, 2, 3, 4.

The solution set of the inequality is {1, 2, 3, 4}.

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)
Correct answer is option 'B'. Can you explain this answer?

Gaurav Kumar answered
Let x be the smaller of the two consecutive even positive integers, then the other even integer is x + 2. 
 Given x < 10  and x + (x + 2) > 11.  
⇒ x < 10, and 2x + 2 > 11.  
⇒ x < 10, 2x > 9 
⇒ x < 10, x > 9/2 
⇒ 10 < x > 9/2   
∴ the required parity even integers is (6, 8)

The solution to |3x – 1| + 1 < 3 is
  • a)
    2 < x < 3/4
  • b)
    -1/3 < x < 1
  • c)
    -1/3 < x < 1/4
  • d)
    -3 < x < 3
Correct answer is option 'B'. Can you explain this answer?

Neha Joshi answered
|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 < x < 1

Solve : 30x < 200, when x is a natural number :
  • a)
    {2,3,4,5,6}
  • b)
    {1,2,3,4,5,6}
  • c)
    {4,5,6,7,8,9}
  • d)
    {1,2,3,4,5,6,7}
Correct answer is option 'B'. Can you explain this answer?

Mysterio Man answered
Given that 30x less than 200,
x is less than 200/30,
or x is less than 6.6,
but as it mentioned above x is a natural number
the possible values of x are
{1,2,3,4,5,6}

The solution of inequality 4x + 3 < 5x + 7 when x is a real number is
  • a)
    [4, ∞)
  • b)
    (-∞, 4)
  • c)
    (-4,∞)
  • d)
    None of the these
Correct answer is option 'C'. Can you explain this answer?

Sanchita Reddy answered
4x + 3 < 5x + 7
subtract 4 both sides,
4x + 3 - 3 < 5x + 7 - 3
⇒ 4x < 5x + 4
subtract ' 5x ' both sides ,
[ equal number may be subtracted from both sides of an inequality without affecting the sign of inequality]
4x - 5x < 5x + 4 - 5
-x < 4
now, multiple with (-1) then, sign of inequality change .
-x.(-1) > 4(-1)
x > -4
hence, x€ ( -4 , ∞)

 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}
Correct answer is option 'B'. Can you explain this answer?

Preeti Iyer answered
24x < 100

=> x < 100/24

x < 25/6

i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than.

Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3, and 4.

ii) The integers less than  are …–3, –2, –1, 0, 1, 2, 3, 4.

Thus, when x is an integer, the solutions of the given inequality are

…–3, –2, –1, 0, 1, 2, 3, 4.

Hence, in this case, the solution set is {…–3, –2, –1, 0, 1, 2, 3, 4}.

Hence, in this case, the solution set is {1, 2, 3, 4}.

Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.
  • a)
    (11 , 13) , (13 , 15) , (15 , 17) , (17 , 21)
  • b)
    (9 , 11) , (13 , 15) , (15 , 17) , (17 , 19)
  • c)
    (11 , 13) , (13 , 15) , (17 , 19) , (19 , 21)
  • d)
    (11 , 13) , (13 , 15) , (15 , 17) , (17 , 19)
Correct answer is option 'D'. Can you explain this answer?

Dipika Patel answered
**Explanation:**

To find the pairs of consecutive odd natural numbers that satisfy the given conditions, we need to consider the following:

1. The numbers should be consecutive odd natural numbers.
2. Both numbers should be larger than 10.
3. The sum of the two numbers should be less than 40.

Let's analyze each option to see if it satisfies these conditions:

**Option A: (11, 13), (13, 15), (15, 17), (17, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 21): The sum is 38, which is less than 40. This pair satisfies all conditions.

**Option B: (9, 11), (13, 15), (15, 17), (17, 19)**
- (9, 11): The numbers are not larger than 10. This pair does not satisfy the second condition.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.

**Option C: (11, 13), (13, 15), (17, 19), (19, 21)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.
- (19, 21): The sum is 40, which is not less than 40. This pair does not satisfy the third condition.

**Option D: (11, 13), (13, 15), (15, 17), (17, 19)**
- (11, 13): The sum is 24, which is less than 40. This pair satisfies all conditions.
- (13, 15): The sum is 28, which is less than 40. This pair satisfies all conditions.
- (15, 17): The sum is 32, which is less than 40. This pair satisfies all conditions.
- (17, 19): The sum is 36, which is not less than 40. This pair does not satisfy the third condition.

Therefore, the correct answer is **Option D: (11, 13), (13, 15), (15, 17), (17, 19)** as it is the only option where all the pairs satisfy all the given conditions.

Identify solution set for [| 4 −− x | + 1 < 3?
  • a)
    (2 , 6)
  • b)
    (3 , 6)
  • c)
    (2 , 4)
  • d)
    (2 , 3)
Correct answer is option 'A'. Can you explain this answer?

Neha Joshi answered
|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).

A solution is to be kept between 30∘C and 35∘C What is the range of temperature in degree Fahrenheit ?
  • a)
    Between 40∘F and 60∘F
  • b)
    Between 30∘Fand 35∘F
  • c)
    Between 86∘F and 95∘F
  • d)
    Between 76∘F and 105∘F
Correct answer is option 'C'. Can you explain this answer?

Manoj Patel answered
Explanation:

To convert Celsius to Fahrenheit, we use the formula:

F = (9/5)C + 32

where F is the temperature in Fahrenheit and C is the temperature in Celsius.

Let's find the Fahrenheit equivalents of the given temperature range:

Lower limit:

F = (9/5)30 + 32 = 86F

Upper limit:

F = (9/5)35 + 32 = 95F

Therefore, the range of temperature in degree Fahrenheit is between 86F and 95F.

Answer:

Option (c) is correct. The range of temperature in degree Fahrenheit is between 86F and 95F.

What is the solution set for 
  • a)
    (2 , ∞)
  • b)
    (0 , 2)
  • c)
    (-∞ , - 2)
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?

Mani Kandan answered
X=-2,2
x+2 ×-2 x+2/x-2
(-infinity,-2) (+) (-) ( - )

(-2,2) ( + ) ( - ) ( - )

(2,infinity ) ( + ) ( + ) ( +)


so answer is A=(2,infinity )

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,∞)
Correct answer is option 'B'. Can you explain this answer?

Vikas Saini answered
From inequality it comes out to be x less than 2 but it is also a natural no. so 1,2 is the. only answer in this case

By solving the inequality 6x - 7 > 5, the answer will be
  • a)
    x > 6
  • b)
    x < 5
  • c)
    x < 7
  • d)
    x > 2
Correct answer is option 'D'. Can you explain this answer?

Meera Nambiar answered
To solve the inequality 6x - 7, we need to isolate x on one side of the inequality symbol. Here's the step-by-step solution:

1. Add 7 to both sides of the inequality: 6x - 7 + 7 > 0 + 7
This simplifies to: 6x > 7

2. Divide both sides of the inequality by 6: (6x)/6 > 7/6
This simplifies to: x > 7/6

Therefore, the solution to the inequality 6x - 7 is x > 7/6.

If a , b , c are real numbers such that a ≥ b , c > 0, then
  • a)
    ac < bc
  • b)
    ac > bc
  • c)
    ac ≥ bc
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Palak Joshi answered
We know that $a$, $b$, and $c$ are real numbers such that $a< />

We can use the fact that $a< />

Now we can rewrite $ac+bc$ as $c(a+b)=c(a+\frac{a+c}{2})=\frac{3}{2}ac+\frac{1}{2}c^2$. Since $a+c=4$, we can substitute $c=4-a$ to get $ac+bc=\frac{3}{2}a(4-a)+\frac{1}{2}(4-a)^2=\frac{1}{2}(a-2)^2+\frac{7}{2}$.

The minimum value of $(a-2)^2$ is 0, which occurs when $a=2$. Therefore, the minimum value of $ac+bc$ is $\frac{7}{2}$, which occurs when $a=2$, $b=3$, and $c=4$.

If a < b then -a ______ - b
  • a)
    –a < -b
  • b)
    –a ≤ -b
  • c)
    –a ≥ -b
  • d)
    -a > -b
Correct answer is option 'D'. Can you explain this answer?

Bhargavi Sen answered
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However, it is important to note that a balanced diet should include a variety of food groups to ensure all essential nutrients are obtained. It is recommended to consult with a healthcare professional or registered dietitian before making any significant changes to your diet.

The solution of inequality 4x + 3 < 5x + 7 when x is a real number is
  • a)
    [-4, ∞)
  • b)
    (-∞, 4)
  • c)
    (-4,∞)
  • d)
    (-∞, 4]
Correct answer is option 'C'. Can you explain this answer?

Suheal Mohammed answered
Yes

given that 4x + 3 < 5x="" +="" />

then -x-4< />

there fore

x+4>0

because when sign changes the inequality also changes

so x>-4

therefore solution of x lies on (-4, infinity)

thanq.
all the best.

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
Correct answer is option 'B'. Can you explain this answer?

Sai Gavali answered
2 < 5(x)="" /> 8 
8 is the third multiple of the number i.e. 2 
put the number is greater than 1 in the value of xu will get that condition true. Hence the option b is correct.

By solving the inequality 6x - 7 > 5, the answer will be
  • a)
    x > 6
  • b)
    x < 5
  • c)
    x < 7
  • d)
    x > 2
Correct answer is option 'D'. Can you explain this answer?

Shraddha Deshpande answered
To solve the inequality 6x - 7 < 0,="" we="" need="" to="" isolate="" the="" variable="" />

Adding 7 to both sides of the inequality, we get:

6x - 7 + 7 < 0="" +="" />

Simplifying, we have:

6x < />

Dividing both sides of the inequality by 6 (since the coefficient of x is 6), we get:

x < />

Therefore, the solution to the inequality 6x - 7 < 0="" is="" x="" />< 7/6.="" />

Chapter doubts & questions for Inequalities - Mathematics for Year 10 2024 is part of Year 10 exam preparation. The chapters have been prepared according to the Year 10 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Year 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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