Final Answer:
The given expression ((a-b)⋅(b-a) > 0) is always true when (a < b).
Step-by-step explanation:
In mathematics, when a < b, subtracting b from a results in a negative value, denoted as (a-b), and subtracting a from b results in a positive value, denoted as (b-a). Multiplying a negative value by a positive value always yields a negative product. Therefore, ((a-b)⋅(b-a) < 0) is true when a < b.
Let's break down the expression further:
1. (a-b): Since a < b, (a-b) is negative.
2. (b-a): As a < b, (b-a) is positive.
3. Multiplication: When you multiply a negative value by a positive value, the result is always negative.
The inequality ((a-b)⋅(b-a) < 0) implies that the product of (a-b) and (b-a) is negative. This holds true because the negative product reflects the multiplication of a negative and a positive number.
In summary, when (a < b), the given expression evaluates to ((a-b)⋅(b-a) < 0), which is always true. The mathematical reasoning behind this conclusion lies in the properties of negative and positive numbers, ensuring the validity of the inequality.