FIND THE SOLUTION SET OF x²+9x+14>0 using five steps

To find the solution set of the inequality \( x^2 + 9x + 14 > 0 \) in five steps, follow this process:

### Step 1: Solve the quadratic equation \( x^2 + 9x + 14 = 0 \)
The first step is to solve the related quadratic equation \( x^2 + 9x + 14 = 0 \) by factoring.
Factor \( x^2 + 9x + 14 \):
\[
x^2 + 9x + 14 = (x + 7)(x + 2) = 0
\]
The solutions of this equation are:
\[
x + 7 = 0 \quad \text{or} \quad x + 2 = 0
\]
Thus, \( x = -7 \) and \( x = -2 \).

### Step 2: Identify critical points
The solutions \( x = -7 \) and \( x = -2 \) are the critical points that divide the real number line into intervals.

### Step 3: Test intervals
The real line is divided into three intervals by the critical points: \( (-\infty, -7) \), \( (-7, -2) \), and \( (-2, \infty) \).

- **For \( x \in (-\infty, -7) \)**: Choose \( x = -8 \).
  Substitute into \( x^2 + 9x + 14 \):
  \[
  (-8)^2 + 9(-8) + 14 = 64 - 72 + 14 = 6 > 0
  \]
  So, \( x^2 + 9x + 14 > 0 \) for \( x \in (-\infty, -7) \).

- **For \( x \in (-7, -2) \)**: Choose \( x = -5 \).
  Substitute into \( x^2 + 9x + 14 \):
  \[
  (-5)^2 + 9(-5) + 14 = 25 - 45 + 14 = -6 < 0
  \]
  So, \( x^2 + 9x + 14 < 0 \) for \( x \in (-7, -2) \).

- **For \( x \in (-2, \infty) \)**: Choose \( x = 0 \).
  Substitute into \( x^2 + 9x + 14 \):
  \[
  0^2 + 9(0) + 14 = 14 > 0
  \]
  So, \( x^2 + 9x + 14 > 0 \) for \( x \in (-2, \infty) \).

### Step 4: Analyze the inequality
The quadratic expression \( x^2 + 9x + 14 > 0 \) is true for the intervals \( (-\infty, -7) \) and \( (-2, \infty) \).

### Step 5: Write the solution set
The solution set for \( x^2 + 9x + 14 > 0 \) is:
\[
(-\infty, -7) \cup (-2, \infty)
\]

This is the set of values of \( x \) for which the quadratic expression is positive.

Would you like more details or clarification? Here are 5 related questions to expand on:

1. How would the solution change if the inequality were \( x^2 + 9x + 14 \geq 0 \)?
2. Can we solve the inequality using the quadratic formula instead of factoring?
3. How do the signs of the factors help determine the intervals where the inequality is true?
4. What happens if the discriminant of the quadratic equation is negative?
5. How do we graph the quadratic inequality to visualize the solution?

**Tip:** Always test points in each interval to confirm whether the inequality holds, especially for complex or non-factorable expressions.

Discover how to find the solution set for the quadratic inequality x² + 9x + 14 > 0 with a comprehensive five-step approach. This problem involves key algebraic concepts, including factorization and the application of the quadratic inequality theorem, suitable for students in Grades 9-11.

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