Final answer:
Option (d) (4-3x)/(2) is the rational expression that is defined for all real numbers, as it does not contain the variable 'x' in the denominator, which ensures that the expression will never be undefined.
Step-by-step explanation:
The question is asking to identify the rational expression that is defined for all real numbers. A rational expression is undefined when its denominator is zero since division by zero is undefined in mathematics. We must look at each option and determine if there is some real number x that will make the denominator equal to zero.
- Option (a) (6x-12)/(7x-14) will be undefined when 7x - 14 = 0, which happens when x = 2.
- Option (b) (3x-6x²)/(3x) will be undefined when 3x = 0, which occurs when x = 0.
- Option (c) (x²+1)/(2x) will be undefined when 2x = 0, which is when x = 0.
- Option (d) (4-3x)/(2) is the only expression that does not have x in the denominator, and therefore it will never be zero, so this expression is defined for all real numbers.
Therefore, the rational expression that is defined for all real numbers is option (d) (4-3x)/(2).