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All questions of Linear Inequalities for Grade 11 Exam
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?
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}
|
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}.
If -5x+2<7x -4, then x is- a)<3/2
- b)<1/2
- c)>1/2
- d)>3/2
Correct answer is option 'C'. Can you explain this answer?
If -5x+2<7x -4, then x is
a)
<3/2
b)
<1/2
c)
>1/2
d)
>3/2
|
Pooja Shah answered |
-5x + 2 < 7x - 4
6 < 12x
x > 1/2
6 < 12x
x > 1/2
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?
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}
|
|
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}
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 4x-2 > 6 is- a)x > 2
- b)x < 6
- c)x = 4
- d)x > 4
Correct answer is option 'A'. Can you explain this answer?
The solution of 4x-2 > 6 is
a)
x > 2
b)
x < 6
c)
x = 4
d)
x > 4
|
|
Arjun Kumar answered |
According to the laws of inequality, answer should be x is less than 2.
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?
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)
|
|
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.
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.
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?
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
|
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.
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.
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?
Identify solution set for [| 4 −− x | + 1 < 3?
a)
(2 , 6)
b)
(3 , 6)
c)
(2 , 4)
d)
(2 , 3)
|
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).
⇒ 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 connected planar graph having 6 vertices, 7 edges contains _____________ regions.- a)15
- b)2
- c)1
- d)11
Correct answer is option 'B'. Can you explain this answer?
A connected planar graph having 6 vertices, 7 edges contains _____________ regions.
a)
15
b)
2
c)
1
d)
11
|
Lavanya Menon answered |
By euler’s formula the relation between vertices(n), edges(q) and regions(r) is given by n-q+r=2.
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?
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
|
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 , ∞)
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 , ∞)
Given that x is an integer, find the values of x which satisfy the simultaneous linear inequalities 2 + x < 6 and 2 −3x < − 1.- a)1 , 2 , 3
- b)2 , 3
- c)2 , 3, 4,
- d)1 , 2, 3, 4.
Correct answer is option 'B'. Can you explain this answer?
Given that x is an integer, find the values of x which satisfy the simultaneous linear inequalities 2 + x < 6 and 2 −3x < − 1.
a)
1 , 2 , 3
b)
2 , 3
c)
2 , 3, 4,
d)
1 , 2, 3, 4.
|
EduRev JEE answered |
Given Inequalities:
- 2 + x < 6
- 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
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
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
Correct answer is option 'B'. Can you explain this answer?
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
|
Gaurav Kumar answered |
12x > 30
x > 30/12
x > 2.5
x is an integer. So, minimum value of x is 3.
x > 30/12
x > 2.5
x is an integer. So, minimum value of x is 3.
Solve the inequality 3 −− 2x ≤ 9- a)x ≥ − 6
- b)x ≥ − 3
- c)x ≤ − 3
- d)none of these.
Correct answer is option 'B'. Can you explain this answer?
Solve the inequality 3 −− 2x ≤ 9
a)
x ≥ − 6
b)
x ≥ − 3
c)
x ≤ − 3
d)
none of these.
|
Raghav Bansal answered |
3 − 2x ≤ 9
-2x ≤ 6
-x ≤ 3
-3 ≤ x
-2x ≤ 6
-x ≤ 3
-3 ≤ x
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?
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
|
|
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
|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
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
Correct answer is option 'A'. Can you explain this answer?
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
|
Tejas Verma answered |
No of attempts = 3
Average = (45+62+x)/3=60
x=73
Average = (45+62+x)/3=60
x=73
If 5x+6<2x-3, then- a)x>-3
- b)x<-3
- c)x>3
- d)x<3
Correct answer is option 'B'. Can you explain this answer?
If 5x+6<2x-3, then
a)
x>-3
b)
x<-3
c)
x>3
d)
x<3
|
|
Rohit Jain answered |
5x+6<2x-3
5x-2x < -3-6
3x < -9
x<-3
5x-2x < -3-6
3x < -9
x<-3
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?
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,∞)
|
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
Which of the following is correct ?- a)If 0 > -7 , then 0 < 7
- b)If 8 > 1 , then -8 > -1
- c)If -4 < 7 , then 4 < - 7
- d)If -2 < 5 , then 2 > 5
Correct answer is option 'A'. Can you explain this answer?
Which of the following is correct ?
a)
If 0 > -7 , then 0 < 7
b)
If 8 > 1 , then -8 > -1
c)
If -4 < 7 , then 4 < - 7
d)
If -2 < 5 , then 2 > 5
|
Aditi Nambiar answered |
The statement you provided is incomplete. Please provide the complete statement for accurate evaluation.
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
Correct answer is option 'A'. Can you explain this answer?
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
|
Shubham Rajput answered |
A is a quadratic equation not a linear equality because square of a function can't be negative
A pack of coffee powder contains a mixture of x grams of coffee and y grams of choco. The amount of coffee powder is greater than that of choco and each pack is atleast 10 grams. Which of the following inequations describe the conditions given ?
- a)x + y < 12 ; x > y
- b)x + y ≥ 10 ; x > y
- c)x + y < 10 ; x < y
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
A pack of coffee powder contains a mixture of x grams of coffee and y grams of choco. The amount of coffee powder is greater than that of choco and each pack is atleast 10 grams. Which of the following inequations describe the conditions given ?
a)
x + y < 12 ; x > y
b)
x + y ≥ 10 ; x > y
c)
x + y < 10 ; x < y
d)
none of these
|
Nabanita Nambiar answered |
The coffee powder is greater than choco
hence, x>y
each pack is at least 10 gm
Hence, x+y≥10
hence, x>y
each pack is at least 10 gm
Hence, x+y≥10
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
Correct answer is option 'D'. Can you explain this answer?
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
|
Abhishek Dasgupta answered |
-4x+1≥0 and 3-4x<0
1 ≥ 4x and 3 < 4x
1/4 ≥ x and 3/4 < x
so x ∈ (-∞,1/4] U (3/4 , ∞ )
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 .
- a)52 ≤ × < 77
- b)73 < × < 100
- c)25 < × < 75
- d)75 < × < 80
Correct answer is option 'A'. Can you explain this answer?
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 .
a)
52 ≤ × < 77
b)
73 < × < 100
c)
25 < × < 75
d)
75 < × < 80
|
Ayush Joshi answered |
Suppose Richard scores x in the fifth paper. Then,75
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.- a)16 cm
- b)11 cm
- c)61 cm
- d)9 cm
Correct answer is option 'D'. Can you explain this answer?
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.
a)
16 cm
b)
11 cm
c)
61 cm
d)
9 cm
|
|
Rajeev Choudhary answered |
Solution:
Let's assume the shortest side of the triangle is x cm.
According to the given information:
- The longest side is three times the shortest side, so it would be 3x cm.
- The third side is 2cm shorter than the longest side, so it would be (3x - 2) cm.
The perimeter of a triangle is the sum of all its side lengths. So, the perimeter of this triangle would be x + 3x + (3x - 2) cm, which simplifies to 7x - 2 cm.
Since the perimeter of the triangle is at least 61 cm, we can set up the following inequality:
7x - 2 ≥ 61
Now, let's solve this inequality to find the minimum length of the shortest side:
7x - 2 ≥ 61
7x ≥ 63
x ≥ 9
Therefore, the minimum length of the shortest side is 9 cm.
So, option D, 9 cm, is the correct answer.
Let's assume the shortest side of the triangle is x cm.
According to the given information:
- The longest side is three times the shortest side, so it would be 3x cm.
- The third side is 2cm shorter than the longest side, so it would be (3x - 2) cm.
The perimeter of a triangle is the sum of all its side lengths. So, the perimeter of this triangle would be x + 3x + (3x - 2) cm, which simplifies to 7x - 2 cm.
Since the perimeter of the triangle is at least 61 cm, we can set up the following inequality:
7x - 2 ≥ 61
Now, let's solve this inequality to find the minimum length of the shortest side:
7x - 2 ≥ 61
7x ≥ 63
x ≥ 9
Therefore, the minimum length of the shortest side is 9 cm.
So, option D, 9 cm, is the correct answer.
Solution of 2x + 2|x| ≥ 2√2- a)(-∞, log2(√2 + 1))
- b)(0, 8)
- c)(1/2, log2(√2 - 1))
- d)(-∞, log2(√2 - 1)) ∪ [1/2, ∞)
Correct answer is option 'D'. Can you explain this answer?
Solution of 2x + 2|x| ≥ 2√2
a)
(-∞, log2(√2 + 1))
b)
(0, 8)
c)
(1/2, log2(√2 - 1))
d)
(-∞, log2(√2 - 1)) ∪ [1/2, ∞)
|
Ruchi Basak answered |
Understanding the Inequality
To solve the inequality 2x + 2|x| ≥ 2√2, we need to consider two cases based on the definition of the absolute value.
Case 1: x ≥ 0
- Here, |x| = x. Thus, the inequality becomes:
- 2x + 2x ≥ 2√2
- 4x ≥ 2√2
- x ≥ √2/2
Case 2: x < />
- In this case, |x| = -x. The inequality becomes:
- 2x - 2x ≥ 2√2
- 0 ≥ 2√2, which is never true.
Hence, for x < 0,="" there="" are="" no="" solutions.="" />Combining Solutions
From Case 1, we have that x ≥ √2/2.
Now, we need to identify the range of values that this corresponds to in terms of logarithmic form. We can express √2/2 as log2(√2 - 1) to find the boundaries of the solution.
Final Solution Set
Thus, the solution to the inequality is:
- x is in the range of (-∞, log2(√2 - 1)) ∪ [√2/2, ∞).
This matches option 'D': (-∞, log2(√2 - 1)) ∪ [1/2, ∞).
Conclusion
- The solution involves understanding how to break the absolute value and analyze each case.
- The final answer confirms that there are no solutions for negative x, while positive x provides a clear range starting from √2/2.
This detailed breakdown explains why option 'D' is correct.
To solve the inequality 2x + 2|x| ≥ 2√2, we need to consider two cases based on the definition of the absolute value.
Case 1: x ≥ 0
- Here, |x| = x. Thus, the inequality becomes:
- 2x + 2x ≥ 2√2
- 4x ≥ 2√2
- x ≥ √2/2
Case 2: x < />
- In this case, |x| = -x. The inequality becomes:
- 2x - 2x ≥ 2√2
- 0 ≥ 2√2, which is never true.
Hence, for x < 0,="" there="" are="" no="" solutions.="" />Combining Solutions
From Case 1, we have that x ≥ √2/2.
Now, we need to identify the range of values that this corresponds to in terms of logarithmic form. We can express √2/2 as log2(√2 - 1) to find the boundaries of the solution.
Final Solution Set
Thus, the solution to the inequality is:
- x is in the range of (-∞, log2(√2 - 1)) ∪ [√2/2, ∞).
This matches option 'D': (-∞, log2(√2 - 1)) ∪ [1/2, ∞).
Conclusion
- The solution involves understanding how to break the absolute value and analyze each case.
- The final answer confirms that there are no solutions for negative x, while positive x provides a clear range starting from √2/2.
This detailed breakdown explains why option 'D' is correct.
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?
If a < b then -a ______ - b
a)
–a < -b
b)
–a ≤ -b
c)
–a ≥ -b
d)
-a > -b
|
|
Bhargavi Sen answered |
If a person ate only fruits and vegetables for a week, they may experience several potential benefits. These can include:
1. Increased nutrient intake: Fruits and vegetables are rich in essential vitamins, minerals, and antioxidants. Eating a variety of fruits and vegetables can provide a wide range of nutrients necessary for optimal health.
2. Improved digestion: Fruits and vegetables are high in fiber, which can promote healthy digestion and prevent constipation. The high water content in many fruits and vegetables can also help maintain hydration and promote regular bowel movements.
3. Enhanced immune function: Fruits and vegetables are packed with immune-boosting nutrients like vitamin C, beta-carotene, and antioxidants. These nutrients can support a strong immune system, helping to fight off infections and diseases.
4. Weight management: Fruits and vegetables are generally low in calories and high in fiber, which can help with weight management. The high fiber content can promote feelings of fullness and reduce overall calorie intake.
5. Improved skin health: Fruits and vegetables contain antioxidants that can help protect the skin from damage caused by free radicals. Additionally, the high water content in fruits and vegetables can contribute to hydrated and healthy-looking skin.
6. Reduced risk of chronic diseases: A diet rich in fruits and vegetables has been associated with a lower risk of chronic diseases such as heart disease, certain types of cancer, and diabetes. The various nutrients and antioxidants found in fruits and vegetables may help protect against these conditions.
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.
1. Increased nutrient intake: Fruits and vegetables are rich in essential vitamins, minerals, and antioxidants. Eating a variety of fruits and vegetables can provide a wide range of nutrients necessary for optimal health.
2. Improved digestion: Fruits and vegetables are high in fiber, which can promote healthy digestion and prevent constipation. The high water content in many fruits and vegetables can also help maintain hydration and promote regular bowel movements.
3. Enhanced immune function: Fruits and vegetables are packed with immune-boosting nutrients like vitamin C, beta-carotene, and antioxidants. These nutrients can support a strong immune system, helping to fight off infections and diseases.
4. Weight management: Fruits and vegetables are generally low in calories and high in fiber, which can help with weight management. The high fiber content can promote feelings of fullness and reduce overall calorie intake.
5. Improved skin health: Fruits and vegetables contain antioxidants that can help protect the skin from damage caused by free radicals. Additionally, the high water content in fruits and vegetables can contribute to hydrated and healthy-looking skin.
6. Reduced risk of chronic diseases: A diet rich in fruits and vegetables has been associated with a lower risk of chronic diseases such as heart disease, certain types of cancer, and diabetes. The various nutrients and antioxidants found in fruits and vegetables may help protect against these conditions.
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.
By solving the inequality 3(a - 6) < 4 + a, the answer will be- a)a < 9
- b)a < 12
- c)a < 13
- d)a < 11
Correct answer is option 'D'. Can you explain this answer?
By solving the inequality 3(a - 6) < 4 + a, the answer will be
a)
a < 9
b)
a < 12
c)
a < 13
d)
a < 11
|
Harsh Rajput answered |
To solve inequality,take < as="." />
3(a+6)=4+a
3a+18=4+a
2a=22
a=11
replace = with
so a<11>
3(a+6)=4+a
3a+18=4+a
2a=22
a=11
replace = with
so a<11>
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?- a)5 ≤ × ≤ 91
- b)3 ≤ × ≤ 5
- c)3 ≤ × ≤ 91
- d)8 ≤ × ≤ 22
Correct answer is option 'D'. Can you explain this answer?
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?
a)
5 ≤ × ≤ 91
b)
3 ≤ × ≤ 5
c)
3 ≤ × ≤ 91
d)
8 ≤ × ≤ 22
|
|
Ashwin Verma answered |
Let x be the length of the shortest board.
The second length is x+3.
The third length is 2x.
The third length is at least 5 cm longer than the second: 2x ≥ (x+3)+5
2x ≥ x+8
x ≥ 8
The possible lengths for the shortest board are x ≥ 8. Answer: \boxed{8}.
The second length is x+3.
The third length is 2x.
The third length is at least 5 cm longer than the second: 2x ≥ (x+3)+5
2x ≥ x+8
x ≥ 8
The possible lengths for the shortest board are x ≥ 8. Answer: \boxed{8}.
Identify the solution set for –( x – 3 ) + 5 – 2x- a)(−∞,−1)
- b)(−∞,−2)
- c)(−∞,−5)
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
Identify the solution set for –( x – 3 ) + 5 – 2x
a)
(−∞,−1)
b)
(−∞,−2)
c)
(−∞,−5)
d)
none of these
|
|
Niharika Bose answered |
To solve the inequality 3(a - 6), we need to isolate the variable a.
First, distribute the 3 to both terms inside the parentheses:
3(a - 6) = 3a - 18
Now, we have the inequality 3a - 18.
We need to determine the range of values for a that satisfy this inequality.
To do this, we can add 18 to both sides of the inequality:
3a - 18 + 18 > 0 + 18
3a > 18
Next, divide both sides of the inequality by 3:
(3a)/3 > 18/3
a > 6
Therefore, the solution to the inequality 3(a - 6) is a > 6.
First, distribute the 3 to both terms inside the parentheses:
3(a - 6) = 3a - 18
Now, we have the inequality 3a - 18.
We need to determine the range of values for a that satisfy this inequality.
To do this, we can add 18 to both sides of the inequality:
3a - 18 + 18 > 0 + 18
3a > 18
Next, divide both sides of the inequality by 3:
(3a)/3 > 18/3
a > 6
Therefore, the solution to the inequality 3(a - 6) is a > 6.
If 3x + 22x ≥ 5x, then the solution set for x is:- a)(-∞, 2]
- b)[2, ∞)
- c)[0, 2]
- d){2}
Correct answer is option 'A'. Can you explain this answer?
If 3x + 22x ≥ 5x, then the solution set for x is:
a)
(-∞, 2]
b)
[2, ∞)
c)
[0, 2]
d)
{2}
|
KP Classes answered |
We have,
3x + 22x ≥ 5x
⇒ (3/5)x + (4/5)x ≥ 1
⇒ (3/5)x + (4/5)x ≥ (3/5)2 + (4/5)2
⇒ x ≤ 2 ⇒ x ∈ (-∞, 2]
[If ax + bx ≥ 1 and a2 + b2 = 1], then x ∈ (-∞, 2)
3x + 22x ≥ 5x
⇒ (3/5)x + (4/5)x ≥ 1
⇒ (3/5)x + (4/5)x ≥ (3/5)2 + (4/5)2
⇒ x ≤ 2 ⇒ x ∈ (-∞, 2]
[If ax + bx ≥ 1 and a2 + b2 = 1], then x ∈ (-∞, 2)
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?
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)
|
Geetika Shah answered |
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).
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).
Given that x is an integer, find the values of x which satisfy both 2x + 3 > 7 and x + 4 < 10- a)4 , 5
- b)3
- c)4
- d)3 , 4 , 5
Correct answer is option 'D'. Can you explain this answer?
Given that x is an integer, find the values of x which satisfy both 2x + 3 > 7 and x + 4 < 10
a)
4 , 5
b)
3
c)
4
d)
3 , 4 , 5
|
Learners Habitat answered |
We are given two inequalities:
- 2x + 3 > 7
- 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
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
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?
By solving the inequality 6x - 7 > 5, the answer will be
a)
x > 6
b)
x < 5
c)
x < 7
d)
x > 2
|
|
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.
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?
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
|
|
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 $c=4$.
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 $c=4$.
x2 ―3|x| + 2 < 0, then x belongs to- a)(1, 2)
- b) ( ― 2, ― 1)
- c) ( ―2, ― 1) ∪ (1,2)
- d)( ― 3, 5)
Correct answer is option 'C'. Can you explain this answer?
x2 ―3|x| + 2 < 0, then x belongs to
a)
(1, 2)
b)
( ― 2, ― 1)
c)
( ―2, ― 1) ∪ (1,2)
d)
( ― 3, 5)
|
Samridhi Kaur answered |
Understanding the Inequality
To solve the inequality x^2 + 3|x| + 2 < 0,="" we="" first="" need="" to="" analyze="" the="" expression="" step="" by="" step.="" />Breaking Down the Expression
- Identify the critical points: The expression x^2 + 3|x| + 2 can be rewritten for different cases of x.
- Case 1: When x ≥ 0, |x| = x. The expression becomes:
- x^2 + 3x + 2
- Case 2: When x < 0,="" |x|="-x." the="" expression="" becomes:="" -="" x^2="" -="" 3x="" +="" 2="" />Finding Roots
- For x ≥ 0: Set x^2 + 3x + 2 = 0.
- The roots are found using the quadratic formula, giving us roots at x = -1 and x = -2. However, since we're in the domain x ≥ 0, this case doesn't contribute any valid roots.
- For x < />:="" set="" x^2="" -="" 3x="" +="" 2="0." -="" the="" roots="" are="" x="1" and="" x="2." again,="" since="" we're="" considering="" x="">< 0,="" we="" focus="" on="" any="" intervals="" that="" satisfy="" the="" inequality.="" />Testing Intervals
- Intervals to test: The critical points divide the number line into intervals: (-∞, -2), (-2, -1), (-1, 1), (1, 2), and (2, ∞).
- Test the intervals:
- For x in (-∞, -2), the value is positive.
- For x in (-2, -1), the value is negative (which satisfies the inequality).
- For x in (-1, 1), the value is positive.
Thus, the only interval where the inequality holds true is (-2, -1).
Final Conclusion
Combining the findings from the two cases, the solution to the inequality x^2 + 3|x| + 2 < 0="" />
- x belongs to: (-2, -1) ∪ (1, 2), which corresponds to option 'c'. 0="" is:="" -="" x="" belongs="" to:="" (-2,="" -1)="" ∪="" (1,="" 2),="" which="" corresponds="" to="" option="">
- x belongs to: (-2, -1) ∪ (1, 2), which corresponds to option 'c'.>
To solve the inequality x^2 + 3|x| + 2 < 0,="" we="" first="" need="" to="" analyze="" the="" expression="" step="" by="" step.="" />Breaking Down the Expression
- Identify the critical points: The expression x^2 + 3|x| + 2 can be rewritten for different cases of x.
- Case 1: When x ≥ 0, |x| = x. The expression becomes:
- x^2 + 3x + 2
- Case 2: When x < 0,="" |x|="-x." the="" expression="" becomes:="" -="" x^2="" -="" 3x="" +="" 2="" />Finding Roots
- For x ≥ 0: Set x^2 + 3x + 2 = 0.
- The roots are found using the quadratic formula, giving us roots at x = -1 and x = -2. However, since we're in the domain x ≥ 0, this case doesn't contribute any valid roots.
- For x < />:="" set="" x^2="" -="" 3x="" +="" 2="0." -="" the="" roots="" are="" x="1" and="" x="2." again,="" since="" we're="" considering="" x="">< 0,="" we="" focus="" on="" any="" intervals="" that="" satisfy="" the="" inequality.="" />Testing Intervals
- Intervals to test: The critical points divide the number line into intervals: (-∞, -2), (-2, -1), (-1, 1), (1, 2), and (2, ∞).
- Test the intervals:
- For x in (-∞, -2), the value is positive.
- For x in (-2, -1), the value is negative (which satisfies the inequality).
- For x in (-1, 1), the value is positive.
Thus, the only interval where the inequality holds true is (-2, -1).
Final Conclusion
Combining the findings from the two cases, the solution to the inequality x^2 + 3|x| + 2 < 0="" />
- x belongs to: (-2, -1) ∪ (1, 2), which corresponds to option 'c'. 0="" is:="" -="" x="" belongs="" to:="" (-2,="" -1)="" ∪="" (1,="" 2),="" which="" corresponds="" to="" option="">
- x belongs to: (-2, -1) ∪ (1, 2), which corresponds to option 'c'.>
The number of real values of parameter k for which (log16x)2−log16x+log16k=0 will have exactly one solution is- a)0
- b)2
- c)1
- d)4
Correct answer is option 'C'. Can you explain this answer?
The number of real values of parameter k for which (log16x)2−log16x+log16k=0 will have exactly one solution is
a)
0
b)
2
c)
1
d)
4
|
Aditya Sengupta answered |
Understanding the Equation
The equation given is:
(log16x)² - log16x + log16k = 0.
Let y = log16x. Then, we can rewrite the equation as:
y² - y + log16k = 0.
This is a quadratic equation in terms of y.
Condition for Exactly One Solution
For a quadratic equation of the form ay² + by + c = 0 to have exactly one solution, the discriminant must be zero. The discriminant (D) is given by:
D = b² - 4ac.
In our equation:
- a = 1
- b = -1
- c = log16k
Thus, the discriminant becomes:
D = (-1)² - 4(1)(log16k) = 1 - 4log16k.
Setting the discriminant to zero for exactly one solution:
1 - 4log16k = 0.
This simplifies to:
log16k = 1/4.
Solving for k
To solve for k, we convert the logarithmic equation to its exponential form:
k = 16^(1/4).
Calculating this gives:
k = 2.
Finding Real Values of k
The value k = 2 is the only solution that satisfies the condition for exactly one solution of y.
Thus, there is only one real value of k that results in the quadratic equation having exactly one solution.
Final Answer
Therefore, the number of real values of parameter k for which the original equation has exactly one solution is 1. Thus, the correct option is C.
The equation given is:
(log16x)² - log16x + log16k = 0.
Let y = log16x. Then, we can rewrite the equation as:
y² - y + log16k = 0.
This is a quadratic equation in terms of y.
Condition for Exactly One Solution
For a quadratic equation of the form ay² + by + c = 0 to have exactly one solution, the discriminant must be zero. The discriminant (D) is given by:
D = b² - 4ac.
In our equation:
- a = 1
- b = -1
- c = log16k
Thus, the discriminant becomes:
D = (-1)² - 4(1)(log16k) = 1 - 4log16k.
Setting the discriminant to zero for exactly one solution:
1 - 4log16k = 0.
This simplifies to:
log16k = 1/4.
Solving for k
To solve for k, we convert the logarithmic equation to its exponential form:
k = 16^(1/4).
Calculating this gives:
k = 2.
Finding Real Values of k
The value k = 2 is the only solution that satisfies the condition for exactly one solution of y.
Thus, there is only one real value of k that results in the quadratic equation having exactly one solution.
Final Answer
Therefore, the number of real values of parameter k for which the original equation has exactly one solution is 1. Thus, the correct option is C.
If x < 0 then | x | is equal to- a)x
- b)0
- c)1
- d)−x
Correct answer is option 'D'. Can you explain this answer?
If x < 0 then | x | is equal to
a)
x
b)
0
c)
1
d)
−x
|
|
Kalyan Yadav answered |
Is greater than 5, then x + 2 is greater than 7.
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?
What is the solution set for
a)
(2 , ∞)
b)
(0 , 2)
c)
(-∞ , - 2)
d)
none of these
|
|
Raghav Bansal answered |
|x-2|/(x-2) > 0
=> x - 2 > 0
x > 2
x denotes (2,∞)
=> x - 2 > 0
x > 2
x denotes (2,∞)
What is the solution set for 
- a)(5 , 6)
- b)(- 6 , 6)
- c)(- 6 , 0)
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
What is the solution set for
a)
(5 , 6)
b)
(- 6 , 6)
c)
(- 6 , 0)
d)
none of these
|
Tarun Kaushik answered |
Given Inequality:
Step 1: Break down the inequality
We need to solve both parts of the compound inequality:
Step 2: Solve each part separately
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.
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.
x - 1)(x² - 5x + 7) < (x - 1), then x belongs to- a)(1, 2) ∪ (3, ∞)
- b)(2, 3)
- c)(-∞, 1) ∪ (2, 3)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
x - 1)(x² - 5x + 7) < (x - 1), then x belongs to
a)
(1, 2) ∪ (3, ∞)
b)
(2, 3)
c)
(-∞, 1) ∪ (2, 3)
d)
None of these
|
|
KP Classes answered |
(x - 1)(x² - 5x + 7) < (x - 1)
⇒ (x - 1)(x² - 5x + 6) < 0
⇒ (x - 1)(x - 2)(x - 3) < 0
∴ x ∈ (-∞, 1) ∪ (2, 3)
⇒ (x - 1)(x² - 5x + 6) < 0
⇒ (x - 1)(x - 2)(x - 3) < 0
∴ x ∈ (-∞, 1) ∪ (2, 3)
The set of admissible values of x such that (2x + 3) / (2x - 9) < 0 is:- a)(-∞, -3/2) ∪ (9/2, ∞)
- b)(-∞, 0) ∪ (9/2, ∞)
- c)(-3/2, 0)
- d)(-3/2, 9/2)
Correct answer is option 'D'. Can you explain this answer?
The set of admissible values of x such that (2x + 3) / (2x - 9) < 0 is:
a)
(-∞, -3/2) ∪ (9/2, ∞)
b)
(-∞, 0) ∪ (9/2, ∞)
c)
(-3/2, 0)
d)
(-3/2, 9/2)
|
Nipuns Institute answered |
Given, (2x + 3) / (2x - 9) < 0
⇒ 2x + 3 < 0 and 2x - 9 > 0
Or 2x + 3 > 0 and 2x - 9 < 0 and x ≠ 9/2
⇒ x < -3/2 and x > 9/2
or x > -3/2 and x < 9/2
⇒ x ∈ (-3/2, 9/2)
⇒ 2x + 3 < 0 and 2x - 9 > 0
Or 2x + 3 > 0 and 2x - 9 < 0 and x ≠ 9/2
⇒ x < -3/2 and x > 9/2
or x > -3/2 and x < 9/2
⇒ x ∈ (-3/2, 9/2)
x = 4, 5 and 6 are the solutions for:
- a)x > 4 and x < 7
- b)x ≥ 4 and x ≤ 7
- c)x ≥ 4 and x < 7
- d)x > 4 and x > 7
Correct answer is option 'C'. Can you explain this answer?
x = 4, 5 and 6 are the solutions for:
a)
x > 4 and x < 7
b)
x ≥ 4 and x ≤ 7
c)
x ≥ 4 and x < 7
d)
x > 4 and x > 7
|
Foothill Academy answered |
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
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?
By solving the inequality 6x - 7 > 5, the answer will be
a)
x > 6
b)
x < 5
c)
x < 7
d)
x > 2
|
|
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.="" />
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.="" />
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 'A'. Can you explain this answer?
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.
|
|
Akshara Deshpande answered |
Yes, I can help you with that. What do you need assistance with?
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