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The number of solutions of the equation [2x] - [x + 1] = 2x must be equal to (where [.] denotes the greatest integer function)
    Correct answer is '2'. Can you explain this answer?
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    The number of solutions of the equation [2x] - [x + 1] = 2xmust be equ...
    Given equation: [2x] - [x 1] = 2x

    We need to find the number of solutions of the given equation, where [.] denotes the greatest integer function.

    Let's solve the equation step by step:

    Step 1: Simplify the equation by expanding the greatest integer function.

    [2x] - [x] - [1] = 2x

    Step 2: Simplify the greatest integer functions.

    [2x] - x - 1 = 2x

    Step 3: Move all the terms containing x to one side of the equation.

    [2x - x - 1] - 2x = 0

    [2x - x - 2x] - 1 = 0

    -x - 1 = 0

    Step 4: Solve for x.

    x = -1

    Step 5: Check the validity of the solution.

    Substituting x = -1 back into the original equation:

    [2(-1)] - [-1] - [1] = 2(-1)

    -2 - (-1) - 1 = -2

    -2 + 1 - 1 = -2

    -2 = -2

    The equation is satisfied when x = -1.

    Number of Solutions:

    From the above solution, we can see that there is only one solution for the given equation, which is x = -1. However, we need to consider the greatest integer function.

    When x is an integer, the greatest integer function [x] does not affect the value of x. So, for integer values of x, the equation remains the same.

    When x is not an integer, the greatest integer function [x] rounds down x to the nearest integer. In this case, the equation simplifies to:

    -1 - 1 = -2

    -2 = -2

    So, for non-integer values of x, the equation is also satisfied.

    Therefore, the equation [2x] - [x 1] = 2x has infinitely many solutions, but only one integer solution.

    Conclusion:

    The number of solutions of the equation [2x] - [x 1] = 2x is equal to 2.
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