GETTING BIGGER? GETTING SMALLER?

(Direct and Inverse Variation)

(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
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$\,y = 2x\,$.

When $\,x\,$ gets bigger,
$\,y\,$ gets bigger.

When
[beautiful math coming... please be patient]
$\,y\,$ 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
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$\,k\,$ for which
$\,y = kx\,$,
then we say that
‘$\,y\,$ varies directly as $\,x\,$’.

When $\,x\,$ gets bigger, $\,y\,$ gets smaller.

When [beautiful math coming... please be patient] $\,x\,$ gets smaller, $\,y\,$ gets bigger.

In this type of relationship, [beautiful math coming... please be patient] $\,x\,$ 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
[beautiful math coming... please be patient]
$\,k\,$ for which
$\displaystyle \,y = \frac{k}{x}\,$,
then we say that
‘$\,y\,$ varies inversely as
$\,x\,$’.

EXAMPLES:

Question:
Consider the formula
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$\,PV = nRT\,$.

As $\,T\,$ gets bigger, what happens to $\,V\,\,$?

(Assume all other variables are held constant.)

As $\,T\,$ gets bigger, what happens to $\,V\,\,$?

(Assume all other variables are held constant.)

Solution:
[beautiful math coming... please be patient]
$V\,$ 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.

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
[beautiful math coming... please be patient]
$\,P = \frac{nRT}{V}\,$.

As $\,P\,$ gets bigger, what happens to $\,V\,\,$?

(Assume all other variables are held constant.)

As $\,P\,$ gets bigger, what happens to $\,V\,\,$?

(Assume all other variables are held constant.)

Solution:
[beautiful math coming... please be patient]
$V\,$ 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.

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.

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Scientific Notation

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Scientific Notation

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.