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Number of solutions of the equation [2x]−3{2x}=1 is?
(where [⋅] and {⋅} denote greatest integer an fractional part function respectively).?
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Number of solutions of the equation [2x]−3{2x}=1 is?(where [⋅] and {⋅}...
Number of solutions of the equation [2x]−3{2x}=1


Introduction:

In this question, we are asked to find the number of solutions of the equation [2x]−3{2x}=1, where [⋅] and {⋅} denote the greatest integer and fractional part function, respectively. Let us solve this equation step by step.

Solution:


Step 1:

Let us assume that [2x] = k and {2x} = p, where k is an integer and 0 ≤ p < />

Step 2:

From the given equation, we have k - 3p = 1.

Step 3:

Since k is an integer, we can rewrite the above equation as p = (k-1)/3.

Step 4:

Now, we know that 0 ≤ p < 1.="" therefore,="" (k-1)/3="" />< 1="" or="" k="" />< />

Step 5:

Also, as k is an integer, the possible values of k are 1, 2, and 3.

Step 6:

For k = 1, we have p = 0. Therefore, x = 0.

Step 7:

For k = 2, we have p = 1/3. Therefore, x = 1/3, 5/6.

Step 8:

For k = 3, we have p = 2/3. Therefore, x = 2/3, 7/6.

Step 9:

Thus, the solutions of the given equation are x = 0, 1/3, 5/6, 2/3, and 7/6.

Step 10:

Therefore, the number of solutions of the equation [2x]−3{2x}=1 is 5.

Conclusion:

Thus, we have found the number of solutions of the given equation using the greatest integer and fractional part function.
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Number of solutions of the equation [2x]−3{2x}=1 is?(where [⋅] and {⋅} denote greatest integer an fractional part function respectively).?
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