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Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function?
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Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest in...
Solution:
To solve the equation [X/2] [2x/3]=x, we need to understand the greatest integer function. The greatest integer function means that we take the largest integer that is less than or equal to the given number. For example, [3.7] = 3 and [-2.5] = -3.
Approach:
We will solve the given equation by using the following approach:
1. Simplify the given equation.
2. Break the equation into different intervals.
3. Find the solution for each interval.
4. Combine the solutions for all intervals.
1. Simplify the given equation:
The given equation is [X/2] [2x/3]=x.
Let's simplify this equation by assuming that X/2 = a, and 2x/3 = b. Then we have:
[a] [b] = 3a/2.
Now, let's consider two cases:
Case 1: When a and b are both positive or zero.
In this case, we have:
[a] [b] = ab.
Thus, the given equation becomes:
ab = 3a/2.
Therefore, we have:
b = 3/2.
This means that:
2x/3 = 3/2.
Thus, x = 9/4.
Case 2: When a and b are both negative.
In this case, we have:
[a] [b] = (a-1)(b-1).
Thus, the given equation becomes:
(a-1)(b-1) = 3a/2.
Therefore, we have:
b = (3a+2)/(2a-1).
Now, let's consider two sub-cases:
Sub-case 1: When b < />
In this sub-case, we have:
(3a+2)/(2a-1) < />
Thus, we have:
a < -2/3="" or="" a="" /> 1/2.
Sub-case 2: When b > 0.
In this sub-case, we have:
(3a+2)/(2a-1) > 0.
Thus, we have:
-2/3 < a="" />< />
2. Break the equation into different intervals:
Based on the above analysis, we can break the given equation into three intervals:
Interval 1: a < />
Interval 2: -2/3 < a="" />< />
Interval 3: a > 1/2.
3. Find the solution for each interval:
Now, let's find the solution for each interval:
Interval 1: a < />
In this interval, we have:
[b] < />
Thus, there is no solution for this interval.
Interval 2: -2/3 < a="" />< />
In this interval, we have:
[b] = 0.
Thus, we have:
(a-1)(b-1) = 3a/2.
This equation can be simplified as:
-5a + 3 = 0.
Thus, we have:
a = 3/5.
This means that:
X/2 = 3/5.
Thus, we have:
X = 6/5.
Interval 3: a > 1/2.
In this interval, we have:
[b]
To solve the equation [X/2] [2x/3]=x, we need to understand the greatest integer function. The greatest integer function means that we take the largest integer that is less than or equal to the given number. For example, [3.7] = 3 and [-2.5] = -3.
Approach:
We will solve the given equation by using the following approach:
1. Simplify the given equation.
2. Break the equation into different intervals.
3. Find the solution for each interval.
4. Combine the solutions for all intervals.
1. Simplify the given equation:
The given equation is [X/2] [2x/3]=x.
Let's simplify this equation by assuming that X/2 = a, and 2x/3 = b. Then we have:
[a] [b] = 3a/2.
Now, let's consider two cases:
Case 1: When a and b are both positive or zero.
In this case, we have:
[a] [b] = ab.
Thus, the given equation becomes:
ab = 3a/2.
Therefore, we have:
b = 3/2.
This means that:
2x/3 = 3/2.
Thus, x = 9/4.
Case 2: When a and b are both negative.
In this case, we have:
[a] [b] = (a-1)(b-1).
Thus, the given equation becomes:
(a-1)(b-1) = 3a/2.
Therefore, we have:
b = (3a+2)/(2a-1).
Now, let's consider two sub-cases:
Sub-case 1: When b < />
In this sub-case, we have:
(3a+2)/(2a-1) < />
Thus, we have:
a < -2/3="" or="" a="" /> 1/2.
Sub-case 2: When b > 0.
In this sub-case, we have:
(3a+2)/(2a-1) > 0.
Thus, we have:
-2/3 < a="" />< />
2. Break the equation into different intervals:
Based on the above analysis, we can break the given equation into three intervals:
Interval 1: a < />
Interval 2: -2/3 < a="" />< />
Interval 3: a > 1/2.
3. Find the solution for each interval:
Now, let's find the solution for each interval:
Interval 1: a < />
In this interval, we have:
[b] < />
Thus, there is no solution for this interval.
Interval 2: -2/3 < a="" />< />
In this interval, we have:
[b] = 0.
Thus, we have:
(a-1)(b-1) = 3a/2.
This equation can be simplified as:
-5a + 3 = 0.
Thus, we have:
a = 3/5.
This means that:
X/2 = 3/5.
Thus, we have:
X = 6/5.
Interval 3: a > 1/2.
In this interval, we have:
[b]
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Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest in...
Is both gif terms are attached with product sign
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Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function?
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Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function?.
Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Challenge. [X/2] [2x/3]=x Find no.of sol.where [.] Denotes greatest integer function?.
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