Expressions are considered non-integral when they do not consistently yield integer values, as seen in the example $\frac{4y}{x}$, which fails to be integral in x unless x is specifically 1 or a common factor. Non-rational expressions, like $3x\sqrt{y}z^3$, do not always produce rational values; in this case, y must be a perfect square of a rational number for the expression to be rational. The distinction between integral and rational is crucial in understanding the behavior of algebraic expressions. Non-integral and non-rational expressions can lead to undefined or non-standard results in mathematical contexts. Understanding these concepts is essential for accurate mathematical analysis and problem-solving.
