Problem 88 - A Graphical Approach to College Algebra

Question

Solve each problem. For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _______ and when \(x\) decreases, \(y\) _______.

Step-by-Step Solution

Answer

When \(x\) increases, \(y\) decreases; when \(x\) decreases, \(y\) increases.

Step 1: Understanding Inverse Variation

When two variables, say \(y\) and \(x\), vary inversely, it means they are related in such a way that the product \(xy = k\). Here \(k\) is a constant and positive. This relationship implies that as one variable increases, the other decreases to maintain the constant product.

Step 2: Analyzing the Effect of Increasing \(x\)

If \(x\) increases in an inverse variation relationship, \(y\) must decrease in order to keep the product \(xy\) constant. This decrease in \(y\) compensates for the increase in \(x\), ensuring that their product remains \(k\).

Step 3: Analyzing the Effect of Decreasing \(x\)

Conversely, if \(x\) decreases, \(y\) must increase. This increase occurs because a smaller \(x\) means that \(y\) needs to be larger to maintain the constant product \(xy = k\).

Key Concepts

Constant Product

In the context of inverse variation, one of the key concepts is the idea of a constant product. When we say that two variables, such as \( y \) and \( x \), vary inversely, it means that the product of these two variables is always equal to a constant, denoted as \( k \). This relationship is represented mathematically as \( xy = k \).

This constant value, \( k \), does not change as long as the values of \( x \) and \( y \) adhere to the inverse variation relationship. If \( x \) changes, \( y \) must adjust in a way that their product remains equal to \( k \).

Thus, the constant product is central to understanding inverse variation because it ensures that any increase in one variable is balanced by a proportional decrease in the other, and vice versa.

Variable Relationship

Understanding the variable relationship in inverse variation helps us see how the two variables are interconnected. Here, the relationship is characterized by mutual compensation – meaning if one variable rises, the other falls.

For example, if \( y \) varies inversely with \( x \), the equation \( xy = k \) guides this relationship. For every increase in \( x \), \( y \) must decrease. This is because the inverse relationship ties the two variables together to maintain the constant product \( k \).

Conversely, if \( x \) decreases, \( y \) must increase. It's this sharing of the changing values that underpins the nature of their inverse relationship. This interplay is crucial for problems that deal with situations where increasing one factor necessitates decreasing another, such as the speed and time taken for a fixed-distance travel scenario.

Inverse Proportionality

Inverse proportionality is the hallmark of an inverse variation scenario and it describes a specific kind of proportional relationship. When two variables are inversely proportional, their product is constant and changes in response to each other.

In simpler terms, it means that if you multiply one variable by a number, you must multiply the other variable by its reciprocal to keep their product constant. For instance, if \( x \) doubles, \( y \) must halve for the product \( xy = k \) to remain unchanged.

This concept of inverse proportionality is not only fundamental in mathematics but also appears in real-world contexts like physics and economics, where it helps in modeling situations where one measurement needs to adjust to changes in another, such as resistance and current in electrical circuits.