Final answer:
The two different algebraic expressions for "1/4 the sum of X and 7" are (1/4)(X + 7) and (X + 7)/4, as these both correctly represent a quarter of the sum of X and 7.
Step-by-step explanation:
When asked to write two different algebraic expressions for the phrase "1/4 the sum of X and 7," two correct expressions would be:
- (1/4)(X + 7)
- (X + 7)/4
The first expression, (1/4)(X + 7), is obtained by multiplying the sum of X and 7 by 1/4. This is following the distributive property of multiplication over addition. The second expression, (X + 7)/4, demonstrates that the sum of X and 7 is being divided by 4, which also represents "1/4 the sum of X and 7." The other options presented, (X/4) + 7 and X/4 + 7/4, do not represent the phrase accurately as they suggest adding 7 after taking a quarter of X alone or taking a quarter of X and 7 separately, which is not the same as taking a quarter of the sum of X and 7.
Thus, the correct expressions are a. (1/4)(X + 7) and c. (X + 7)/4, both of which accurately capture the given phrase.