Final answer:
Expressions with the same value of x are equivalent if they yield the same result for all permissible values of x. We can establish equivalence through algebraic manipulation, ensuring the same mathematical operations are applied across the expressions, while considering dimensionality and reasonableness.
Step-by-step explanation:
If x has the same value in two different expressions, and we are considering whether these expressions are equivalent, we must examine the operations and terms involved. Equivalence of expressions means they yield the same value for all admissible values of their variables. For instance, should the two expressions in Exercise 25 represent two fractions with the same numerator and denominator, then each would simplify to 1, demonstrating equivalence.
However, if the expressions involve different operations or combinations of variables and constants, we would need to transform one expression to the other to verify equivalence.
This often involves algebraic manipulations such as factoring, expanding, combining like terms, or simplifying fractions. An equality is maintained as long as the same operation is applied to both sides of an equation. For example, if both sides of an equation represent a fraction with the same numerical value in the numerator and denominator, each side simplifies to a value of 1, thus maintaining equality.
It is also crucial to ensure that all terms in the expressions have the same dimensions, as adding or subtracting quantities with different dimensions is not mathematically valid. When evaluating any mathematical expressions or equations, it is essential to make sense and persevere, as well as check if the solution is reasonable, which aligns with the overarching principles in problem-solving outlined in The Common Core State Standards for Mathematics.