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Find the solution set of each of the following quadratic inequalities then graph.

x² + 9x + 14 > 0

x^2 – 5x-14 ≥ 0

These are the steps
Steps in Solving a Quadratic Inequality:

Step 1: Express the quadratic inequality as a quadratic equation in the form of ax2 + bx + c = 0 and then solve for x.

Step 2: Locate the numbers found in step 1 on a number line. They serve as the boundary points. The number line will be divided into three regions.

Step 3: Choose a one number from each region as a test point. Substitute the test point to the original inequality.

Step 4: If the inequality holds true for the test point, then that region belongs to the solution set, otherwise, it is not part of the solution set of the inequality.

Step 5: Write the solution set as interval notation.

Let's solve each quadratic inequality step by step, following the outlined procedure:

### 1. Inequality: \( x^2 + 9x + 14 > 0 \)

#### Step 1: Solve the corresponding quadratic equation
We start by solving the equation:

\[
x^2 + 9x + 14 = 0
\]

We can factor this quadratic expression:

\[
(x + 7)(x + 2) = 0
\]

Thus, the solutions are:

\[
x = -7 \quad \text{or} \quad x = -2
\]

These are the boundary points that divide the number line into three regions.

#### Step 2: Locate boundary points and divide the number line
The boundary points are \( x = -7 \) and \( x = -2 \). These points divide the number line into three regions:
1. \( (-\infty, -7) \)
2. \( (-7, -2) \)
3. \( (-2, \infty) \)

#### Step 3: Test points from each region
Now, we'll pick test points from each region and substitute them into the original inequality \( x^2 + 9x + 14 > 0 \):

- For region \( (-\infty, -7) \), pick \( x = -8 \):
  
  \[
  (-8)^2 + 9(-8) + 14 = 64 - 72 + 14 = 6 \quad (6 > 0)
  \]
  This region satisfies the inequality.

- For region \( (-7, -2) \), pick \( x = -5 \):
  
  \[
  (-5)^2 + 9(-5) + 14 = 25 - 45 + 14 = -6 \quad (-6 < 0)
  \]
  This region does not satisfy the inequality.

- For region \( (-2, \infty) \), pick \( x = 0 \):
  
  \[
  0^2 + 9(0) + 14 = 14 \quad (14 > 0)
  \]
  This region satisfies the inequality.

#### Step 4: Determine the solution set
The solution set includes the regions where the inequality holds true: \( (-\infty, -7) \) and \( (-2, \infty) \).

#### Step 5: Write the solution set in interval notation
Since the inequality is strict (i.e., \( > 0 \)), the boundary points \( x = -7 \) and \( x = -2 \) are not included in the solution set. Therefore, the solution set is:

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

### 2. Inequality: \( x

Find the solution set of each of the following quadratic inequalities then graph.
x² + 9x + 14 > 0
x^2 – 5x-14 ≥ 0

Learn how to solve quadratic inequalities like x² + 9x + 14 > 0 and x² – 5x - 14 ≥ 0 with a detailed step-by-step solution. This problem explores key algebraic concepts such as quadratic equations, interval notation, and number line analysis, suitable for students in Grades 9-11.

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