Final answer:
The greatest value of x for which y can take all real values is any number less than 1.5, since the function y = (x^2 + 2x + γ)/(2x - 3) is undefined at x = 1.5.
Step-by-step explanation:
The student is asking to find the largest value of x for which the function y can take on any real value, given that y = \frac{ {x}^{2} + 2x + \gamma }{2x - 3} .
The key concept to understand here is that for y to be able to take all real values as x varies, the denominator 2x - 3 cannot be zero, as division by zero is undefined.
Thus, the greatest value of x for which y can take all real values should be just less than the point where the denominator becomes zero, which occurs when x = 1.5.
At any value greater than 1.5 for x, the function y will not be defined. Therefore, we are looking for the largest value of x less than 1.5.