solve the following rational inequality: 1/(x^2+7x+12)>0

To solve the inequality \(\frac{1}{x^2 + 7x + 12} > 0\), we need to determine when the rational expression is positive.

### Step 1: Factor the Denominator

First, factor the quadratic expression in the denominator:

\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

So the inequality becomes:

\[
\frac{1}{(x + 3)(x + 4)} > 0
\]

### Step 2: Identify Critical Points

The critical points occur where the denominator is zero:

\[
x + 3 = 0 \quad \text{or} \quad x + 4 = 0
\]

So, the critical points are:

\[
x = -3 \quad \text{and} \quad x = -4
\]

These points divide the real line into three intervals:

1. \( (-\infty, -4) \)
2. \( (-4, -3) \)
3. \( (-3, \infty) \)

### Step 3: Test the Sign in Each Interval

We need to test the sign of \(\frac{1}{(x + 3)(x + 4)}\) in each interval:

1. For \( x \in (-\infty, -4) \):
   - Choose \( x = -5 \): \(\frac{1}{(-5 + 3)(-5 + 4)} = \frac{1}{(-2)(-1)} = \frac{1}{2} > 0\)
   
2. For \( x \in (-4, -3) \):
   - Choose \( x = -3.5 \): \(\frac{1}{(-3.5 + 3)(-3.5 + 4)} = \frac{1}{(-0.5)(0.5)} = \frac{1}{-0.25} < 0\)
   
3. For \( x \in (-3, \infty) \):
   - Choose \( x = 0 \): \(\frac{1}{(0 + 3)(0 + 4)} = \frac{1}{12} > 0\)

### Step 4: Determine Where the Expression is Positive

From the tests:

- The expression is positive in the intervals \((- \infty, -4)\) and \((-3, \infty)\).
- The expression is undefined at \(x = -4\) and \(x = -3\) since the denominator becomes zero.

### Final Solution

\[
x \in (-\infty, -4) \cup (-3, \infty)
\]

Would you like more details on any step or have any questions?

Here are some related questions:

1. How do you solve rational inequalities with different signs in the numerator and denominator?
2. What is the significance of critical points in rational inequalities?
3. How do you determine when a rational function is undefined?
4. What is the process of testing intervals in inequalities?
5. How can factoring help in solving inequalities?

**Tip:** When solving inequalities, always be careful to check the sign in each interval between the critical points.

Learn how to solve the rational inequality 1/(x^2 + 7x + 12) > 0 with a comprehensive step-by-step solution. This problem involves factoring quadratic expressions, identifying critical points, and testing intervals. Perfect for students in Grades 9-11.

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