Animated Solution for Mathematics - Basic Mathematics: If $p,q,r$ are any real numbers, then

Today, we are exploring the fundamental properties of real numbers using three points: $p$, $q$, and $r$. Our goal is to identify the universal truth among the provided options by testing their validity across different scenarios.

Option A suggests that $\max (p,q)<\max (p,q,r)$. Let us test this with a counter-example by setting $p=5$, $q=3$, and $r=2$.

In this case, $\max (5,3)=5$ and $\max (5,3,2)=5$. Since $5$ is not strictly less than $5$, this option is proven false.

Option C claims that $\max (p,q)<\min (p,q,r)$. This is conceptually impossible because the minimum of a set is always less than or equal to any element within that set. Therefore, it can never be greater than the maximum of a subset.

The core of this problem lies in the identity:

This formula acts as a geometric map. The term $\frac{p+q}{2}$ represents the midpoint, while $|p-q|$ represents the distance between the two points. To find the minimum, we start at the midpoint and subtract half the distance.

We can verify this identity by considering two cases:

If $p<q$, then $|p-q|=q-p$. Substituting this into the formula, we get:

If $p>q$, then $|p-q|=p-q$. Substituting this into the formula, we get:

The identity holds true in all cases, making it a powerful tool in your JEE arsenal. As a useful corollary, remember that the maximum can be expressed similarly:

By visualizing these relationships on the number line, you can master these concepts and solve complex inequalities with ease.