how do i find the interval notation (x+2)(x+7)>0

To find the interval notation for \((x + 2)(x + 7) > 0\), follow these steps:

### 1. **Find the critical points**  
The inequality \((x + 2)(x + 7) > 0\) is satisfied when the product is positive. First, identify the critical points by setting each factor equal to 0:

- \(x + 2 = 0 \Rightarrow x = -2\)
- \(x + 7 = 0 \Rightarrow x = -7\)

These points, \(x = -2\) and \(x = -7\), divide the number line into intervals.

### 2. **Determine the sign in each interval**
Now, test the sign of \((x + 2)(x + 7)\) in each of the three intervals formed by the critical points: 
- \( (-\infty, -7) \)
- \( (-7, -2) \)
- \( (-2, \infty) \)

Pick test points from each interval and substitute them into the expression \((x + 2)(x + 7)\):

- For \(x = -8\) (in \( (-\infty, -7) \)):  
  \((-8 + 2)(-8 + 7) = (-6)(-1) = 6\), which is **positive**.

- For \(x = -5\) (in \( (-7, -2) \)):  
  \((-5 + 2)(-5 + 7) = (-3)(2) = -6\), which is **negative**.

- For \(x = 0\) (in \( (-2, \infty) \)):  
  \((0 + 2)(0 + 7) = (2)(7) = 14\), which is **positive**.

### 3. **Interpret the solution**
We are looking for where \((x + 2)(x + 7) > 0\), so we want the intervals where the product is positive. From the tests:

- The product is positive on \( (-\infty, -7) \) and \( (-2, \infty) \).

### 4. **Write the interval notation**
Since the inequality is strictly greater than 0, we exclude the critical points \(x = -7\) and \(x = -2\). Therefore, the solution in interval notation is:

\[
(-\infty, -7) \cup (-2, \infty)
\]

This is the interval where \((x + 2)(x + 7) > 0\).

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Would you like more details or have any questions about this process?

### Related Questions:
1. How do you solve inequalities involving rational expressions?
2. What is the difference between strict inequalities and non-strict inequalities?
3. How can you solve inequalities graphically?
4. How does the solution of quadratic inequalities differ from linear inequalities?
5. Can you explain how to solve inequalities with absolute values?

### Tip:
Always check the sign of the expression in each interval carefully to ensure the inequality's solution is correct.

Discover how to solve the inequality (x + 2)(x + 7) > 0 using interval notation. This step-by-step guide covers critical points, sign analysis, and writing the solution in proper interval form. Perfect for high school students studying algebra.

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