Problem 34

Given the inverse proportion function $y=\frac{m-1}{x}$, we are tasked with determining the range of $m$ given that one branch of its graph is located in the third quadrant.

Step 1: Understand the characteristics of the third quadrant. In the third quadrant, both $x$ and $y$ are negative. This implies that for the function $y=\frac{m-1}{x}$ to have its graph in the third quadrant, the numerator $m-1$ must be positive because a negative divided by a negative yields a positive result, which contradicts the condition of being in the third quadrant.

Step 2: Set up the inequality based on the above understanding. Since $m-1$ must be positive for the graph to be in the third quadrant, we have:
$$m-1 > 0$$

Step 3: Solve the inequality for $m$. To find the range of $m$, we solve the inequality:
\begin{align*}
m-1 &> 0 \\
m &> 1
\end{align*}

Therefore, for the graph of the given inverse proportion function to be located in the third quadrant, the range of $m$ must be greater than 1.

Final Answer:
$$\boxed{m > 1}$$