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Let S be the set of all real roots of the equation 3x(3x - 1) + 2 = |3x - 1| + |3x - 2|. Then S:
- a)Contains at least four elements
- b)Is a singleton
- c)Contains exactly two elements
- d)Is an empty set
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Let S be the set of all real roots of the equation 3x(3x- 1) + 2 = |3x...
Given:
The equation is 3x(3x-1)^2 = |3x-1| |3x-2|
To find:
The set S, which represents all real roots of the equation.
Solution:
Step 1: Simplify the equation
Let's simplify the given equation to make it easier to solve.
3x(3x-1)^2 = |3x-1| |3x-2|
We notice that the terms |3x-1| and |3x-2| can both be positive or negative depending on the value of x. So, we can rewrite the equation as follows:
3x(3x-1)^2 = (3x-1)(3x-2)
Step 2: Solve for x
Now, we can solve the equation by equating the factors on both sides:
3x = 3x-2 (Case 1)
(3x-1)^2 = 1 (Case 2)
Case 1: 3x = 3x-2
We subtract 3x from both sides:
0 = -2
This equation has no solution since 0 cannot equal -2. Therefore, Case 1 does not provide any real roots.
Case 2: (3x-1)^2 = 1
Taking the square root of both sides, we get:
3x-1 = ±1
Simplifying further, we have two cases:
Case 2a: 3x-1 = 1
Adding 1 to both sides:
3x = 2
Dividing by 3:
x = 2/3
Case 2b: 3x-1 = -1
Adding 1 to both sides:
3x = 0
Dividing by 3:
x = 0
Step 3: Determine the set S
The set S represents all real roots of the equation. From the solutions obtained in Case 2, we have:
S = {2/3, 0}
Conclusion:
The set S contains exactly two elements, making option (c) the correct answer.
The equation is 3x(3x-1)^2 = |3x-1| |3x-2|
To find:
The set S, which represents all real roots of the equation.
Solution:
Step 1: Simplify the equation
Let's simplify the given equation to make it easier to solve.
3x(3x-1)^2 = |3x-1| |3x-2|
We notice that the terms |3x-1| and |3x-2| can both be positive or negative depending on the value of x. So, we can rewrite the equation as follows:
3x(3x-1)^2 = (3x-1)(3x-2)
Step 2: Solve for x
Now, we can solve the equation by equating the factors on both sides:
3x = 3x-2 (Case 1)
(3x-1)^2 = 1 (Case 2)
Case 1: 3x = 3x-2
We subtract 3x from both sides:
0 = -2
This equation has no solution since 0 cannot equal -2. Therefore, Case 1 does not provide any real roots.
Case 2: (3x-1)^2 = 1
Taking the square root of both sides, we get:
3x-1 = ±1
Simplifying further, we have two cases:
Case 2a: 3x-1 = 1
Adding 1 to both sides:
3x = 2
Dividing by 3:
x = 2/3
Case 2b: 3x-1 = -1
Adding 1 to both sides:
3x = 0
Dividing by 3:
x = 0
Step 3: Determine the set S
The set S represents all real roots of the equation. From the solutions obtained in Case 2, we have:
S = {2/3, 0}
Conclusion:
The set S contains exactly two elements, making option (c) the correct answer.
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Let S be the set of all real roots of the equation 3x(3x- 1) + 2 = |3x- 1| + |3x- 2|. Then S:a)Contains at least four elementsb)Is a singletonc)Contains exactly two elementsd)Is an empty setCorrect answer is option 'B'. Can you explain this answer?
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