If `(x - y)^(3) > (x - y)^(2)`, then which one of the following must be true?

Options

A. `x^3 lt y^2`
B. `x^5 lt y^4`
C. `x^3 gt y^2`
D. `x^3 gt y^3`

Answer

To solve the inequality \((x - y)^3 > (x - y)^2\), we can follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ (x - y)^3 > (x - y)^2 \]

### Step 2: Divide Both Sides by \((x - y)^2\)
Assuming \((x - y) \neq 0\), we can divide both sides by \((x - y)^2\). Since \((x - y)^2\) is always positive (as it is a square), we do not need to reverse the inequality sign:
\[ \frac{(x - y)^3}{(x - y)^2} > \frac{(x - y)^2}{(x - y)^2} \]
This simplifies to:
\[ x - y > 1 \]

### Step 3: Rearranging the Inequality
Now we can rearrange the inequality:
\[ x - y > 1 \implies x > y + 1 \]

### Step 4: Analyzing the Result
From the inequality \(x > y + 1\), we can conclude that:
\[ x > y \]
This means that \(x\) must be greater than \(y\).

### Conclusion
Thus, the condition that must be true given the original inequality is:
\[ x > y \]