To solve the inequality $(x - 7)(x + 9) > 0$, we will follow these steps:

Step 1: Find the critical points

The inequality involves a product of two binomials. To find the critical points, set each factor equal to zero:

1. $x - 7 = 0 \Rightarrow x = 7$
2. $x + 9 = 0 \Rightarrow x = -9$

These values divide the number line into three intervals: $(-\infty, -9)$, $(-9, 7)$, and $(7, \infty)$.

Step 2: Test each interval

We need to test the sign of the product $(x - 7)(x + 9)$ in each of these intervals.

1. Interval $(-\infty, -9)$: Pick a test point like $x = -10$:

   • $(x - 7)(x + 9) = (-10 - 7)(-10 + 9) = (-17)(-1) = 17$
   • Positive.

2. Interval $(-9, 7)$: Pick a test point like $x = 0$:

   • $(x - 7)(x + 9) = (0 - 7)(0 + 9) = (-7)(9) = -63$
   • Negative.

3. Interval $(7, \infty)$: Pick a test point like $x = 8$:

   • $(x - 7)(x + 9) = (8 - 7)(8 + 9) = (1)(17) = 17$
   • Positive.

Step 3: Interpret the results

We want the product to be greater than 0 (positive). Based on the sign analysis:

• The product is positive in the intervals $(-\infty, -9)$ and $(7, \infty)$.

Step 4: Exclude the critical points

At $x = -9$ and $x = 7$, the product $(x - 7)(x + 9) = 0$, so they are not included in the solution set.

Final Solution

The solution set is $(-\infty, -9) \cup (7, \infty)$.

Interval Notation:

The solution in interval notation is $(-\infty, -9) \cup (7, \infty)$.

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To graph the solution on a real number line:

• Open circles at $-9$ and $7$, because these points are not included.
• The graph will consist of two regions: one to the left of $-9$ and one to the right of $7$.

Would you like further clarification on the process or the graphing of this solution?

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Here are some related questions for deeper understanding:

1. How would the solution change if the inequality was $(x - 7)(x + 9) \leq 0$?
2. What would happen if the inequality involved a greater than or equal to symbol instead of just greater than?
3. How can we solve a similar inequality involving three or more factors?
4. What is the general method to solve polynomial inequalities with two factors?
5. How can you check your solution using a graphing calculator or by plotting the function?

Tip: Always test points from each interval to check the signs of the expression. This is a simple but effective way to ensure accuracy in solving polynomial inequalities.