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Let's solve the inequality \((x - 7)(x - 1) < 0\) step by step to understand how it should be graphed on the number line.

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### Step 1: Find the Critical Points
The product \((x - 7)(x - 1)\) changes sign at the values that make either factor equal to zero:
- \(x - 7 = 0 \Rightarrow x = 7\)
- \(x - 1 = 0 \Rightarrow x = 1\)

These values, \(x = 1\) and \(x = 7\), divide the number line into **three intervals**:  
- \((-\infty, 1)\)  
- \((1, 7)\)  
- \((7, \infty)\)  

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### Step 2: Test the Sign in Each Interval
We need to determine where the product \((x - 7)(x - 1)\) is negative.

- **For \(x \in (-\infty, 1)\)** (e.g., \(x = 0\)):
  \((0 - 7)(0 - 1) = (-7)(-1) = 7\) (positive)

- **For \(x \in (1, 7)\)** (e.g., \(x = 2\)):
  \((2 - 7)(2 - 1) = (-5)(1) = -5\) (negative)

- **For \(x \in (7, \infty)\)** (e.g., \(x = 8\)):
  \((8 - 7)(8 - 1) = (1)(7) = 7\) (positive)

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### Step 3: Determine the Solution
We are looking for where the product is **negative**, which happens in the interval:
- **\((1, 7)\)**

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### Step 4: Graph the Solution
- **At \(x = 1\) and \(x = 7\)**, the expression is equal to 0, so these points are **not included** in the solution. Use **open circles** at \(x = 1\) and \(x = 7\).

- **Draw a line segment** between \(1\) and \(7\) (not including the endpoints).

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### Final Answer
On the number line:
1. Place **open circles** at \(x = 1\) and \(x = 7\).
2. Draw a **line segment** between \(1\) and \(7\).

Let me know if you need further clarifications!

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**Related Questions:**
1. How do we solve inequalities with more than two factors?
2. What changes if the inequality were \((x - 7)(x - 1) \leq 0\)?
3. How can this method be extended to quadratic inequalities?
4. What is the importance of testing values between critical points?
5. How do open and closed circles affect solutions on a number line?

**Tip:** Always test points from different regions to verify the behavior of expressions!

Learn how to solve the inequality (x - 7)(x - 1) < 0 by finding critical points and testing intervals. This step-by-step guide explains factoring and sign analysis, ideal for students in Grades 9-11.

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