The greatest integer function [x] returns the largest integer less than or equal to x, while the fractional part function {x} returns the decimal part of x.
Let's break down the equation and solve it step by step:
Given equation: 4[x] = x + {x}
Let's consider two cases:
Case 1: x is an integer When x is an integer, the fractional part {x} is 0. Therefore, the equation becomes: 4[x] = x + 0 4[x] = x
In this case, we have a direct relationship between [x] and x. The solution is any value of x that satisfies the equation, such as x = 0.
Case 2: x is not an integer When x is not an integer, the fractional part {x} is non-zero. Therefore, the equation becomes: 4[x] = x + {x}
Since [x] is an integer, the left side of the equation is an integer, while the right side (x + {x}) is not an integer. This implies that there are no solutions for x in this case.
To summarize:
- When x is an integer, the solution is any value of x that satisfies the equation.
- When x is not an integer, there are no solutions for x.
