To solve the inequality:

\[
(x - 1)(x + 1) > 0
\]

This is a **quadratic inequality**. Follow these steps:

### Step 1: Find the critical points
The critical points are the values of \(x\) that make each factor equal to 0. Set each factor to 0 and solve:

- \(x - 1 = 0 \quad \Rightarrow \quad x = 1\)
- \(x + 1 = 0 \quad \Rightarrow \quad x = -1\)

So, the critical points are \(x = 1\) and \(x = -1\).

### Step 2: Test the intervals
These critical points divide the number line into three intervals:
- \( (-\infty, -1) \)
- \( (-1, 1) \)
- \( (1, \infty) \)

Now, test a value from each interval to see where the inequality holds.

1. **Interval \( (-\infty, -1) \):** Choose \(x = -2\).
   \[
   (-2 - 1)(-2 + 1) = (-3)(-1) = 3 \quad \Rightarrow \quad 3 > 0 \quad (\text{True})
   \]

2. **Interval \( (-1, 1) \):** Choose \(x = 0\).
   \[
   (0 - 1)(0 + 1) = (-1)(1) = -1 \quad \Rightarrow \quad -1 > 0 \quad (\text{False})
   \]

3. **Interval \( (1, \infty) \):** Choose \(x = 2\).
   \[
   (2 - 1)(2 + 1) = (1)(3) = 3 \quad \Rightarrow \quad 3 > 0 \quad (\text{True})
   \]

### Step 3: Exclude the critical points
Since the inequality is strictly greater than 0 (i.e., \(>\), not \(\geq\)), the critical points \(x = -1\) and \(x = 1\) are not included in the solution.

### Final Answer:
The solution is:
\[
x \in (-\infty, -1) \cup (1, \infty)
\]

This means \(x\) can be any value less than \(-1\) or greater than \(1\), but not equal to \(-1\) or \(1\).

Would you like further clarification on any steps?

Here are 5 related questions for further practice:

1. Solve \( (x - 2)(x + 2)

Learn how to solve the quadratic inequality (x-1)(x+1) > 0 with a step-by-step solution. This problem covers key algebraic concepts such as factoring and testing intervals to find where the inequality holds. Suitable for students in Grades 9-10.

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