Getting Bigger? Getting Smaller? (Direct and Inverse Variation)

Bigger means farther away from zero and smaller means closer to zero. (This is discussed in more detail in a future section.)

Suppose that $\,y = 2x\,$. When $\,x\,$ gets bigger, $\,y\,$ gets bigger. When $\,y\,$ gets bigger, $\,x\,$ gets bigger. In this type of relationship, $\,x\,$ and $\,y\,$ ‘follow each other’ in size: when one gets bigger, so does the other. When one gets smaller, so does the other.

This kind of relationship between two variables is called direct variation: if there is a nonzero number $\,k\,$ for which $\,y = kx\,$, then we say that ‘$\,y\,$ varies directly as $\,x\,$’.

Now suppose that $\,y = \frac{2}{x}\,$. When $\,x\,$ gets bigger, $\,y\,$ gets smaller. When $\,x\,$ gets smaller, $\,y\,$ gets bigger. In this type of relationship, $\,x\,$ and $\,y\,$ have sizes that go in different directions: when one gets bigger, the other gets smaller. When one gets smaller, the other gets bigger.

This kind of relationship between two variables is called inverse variation: if there is a nonzero number $\,k\,$ for which $\displaystyle \,y = \frac{k}{x}\,$, then we say that ‘$\,y\,$ varies inversely as $\,x\,$’.

Examples

Question: Consider the formula $\,PV = nRT\,$. As $\,T\,$ gets bigger, what happens to $\,V\,\,$? (Assume all other variables are held constant.)
Solution: $V\,$ gets bigger. There is a direct relationship between $\,T\,$ and $\,V\,$. As $\,T\,$ gets bigger, so does $\,V\,$.

Intuition: Both variables are ‘upstairs’ on opposite sides of the equation.
Question: Consider the formula $\,P = \frac{nRT}{V}\,$. As $\,P\,$ gets bigger, what happens to $\,V\,\,$? (Assume all other variables are held constant.)
Solution: $V\,$ gets smaller. There is an inverse relationship between $\,P\,$ and $\,V\,$. As $\,P\,$ gets bigger, $\,V\,$ gets smaller.

Intuition: One variable is ‘upstairs’ and the other ‘downstairs’ on opposite sides of the equation.