Question

Given a^3 < 0, prove: b - a / a - b < 0.

A) The proof is incorrect.
B) The proof is valid.
C) The proof is incomplete.
D) The proof is not applicable.

Answer 1

The proof that (b - a) / (a - b) < 0 is valid because a^3 < 0 implies a is negative, and factoring the expression results in -1, which is indeed less than 0.

Step-by-step explanation:

The student has presented a mathematical statement, Given a^3 < 0, prove b - a / a - b < 0. To evaluate this statement and prove the inequality, let's begin with the given information that a^3 < 0. This tells us that 'a' is a negative number because the cube of a positive number or zero cannot be less than zero.

The expression (b - a) / (a - b) can be simplified by factoring out a negative sign from the denominator, which gives us:

• (b - a) / (a - b) = (b - a) / (-(b - a))
• (b - a) / (a - b) = -1

Since -1 is less than 0, the given proof is valid. Hence, the correct answer is option B.