https://file.aiquickdraw.com/mathbot/file/1727106852949mibpbt1o.jpg

Let's solve the inequality provided in the image:

The inequality is:

\[
x^2 + 2x - 8 > -5
\]

### Step 1: Simplify the inequality
First, let's move the constant \(-5\) to the other side:

\[
x^2 + 2x - 8 + 5 > 0
\]

This simplifies to:

\[
x^2 + 2x - 3 > 0
\]

### Step 2: Solve the corresponding equation
Now, solve the quadratic equation \(x^2 + 2x - 3 = 0\). We can factor this:

\[
(x + 3)(x - 1) = 0
\]

So, the solutions are \(x = -3\) and \(x = 1\).

### Step 3: Analyze the inequality
We now have the critical points \(x = -3\) and \(x = 1\). These points divide the number line into three intervals: 
- \((-\infty, -3)\)
- \((-3, 1)\)
- \((1, \infty)\)

We need to determine in which intervals the inequality \(x^2 + 2x - 3 > 0\) is true. To do this, we test points from each interval:

- For \(x = -4\) (in \((-\infty, -3)\)): 
  \[
  (-4)^2 + 2(-4) - 3 = 16 - 8 - 3 = 5 > 0
  \]
  So, the inequality holds for \(x \in (-\infty, -3)\).

- For \(x = 0\) (in \((-3, 1)\)): 
  \[
  (0)^2 + 2(0) - 3 = -3 < 0
  \]
  The inequality does not hold for \(x \in (-3, 1)\).

- For \(x = 2\) (in \((1, \infty)\)): 
  \[
  (2)^2 + 2(2) - 3 = 4 + 4 - 3 = 5 > 0
  \]
  So, the inequality holds for \(x \in (1, \infty)\).

### Step 4: Graph the solution
The solution to the inequality is \(x \in (-\infty, -3) \cup (1, \infty)\). This means the solution includes all numbers less than \(-3\) and all numbers greater than \(1\), but not between \(-3\) and \(1\). The points \(x = -3\) and \(x = 1\) are not included in the solution, as we have a strict inequality.

### Step 5: Compare to the options
Looking at the number lines in the image:
- Option A shows the correct intervals: \(x < -3\) and \(x > 1\) with open circles at \(-3\) and \(1\), representing that these points are not included in the solution.

Thus, the correct answer is **A**.

---

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

Here are 5 related questions you might explore:
1. How do you solve quadratic inequalities with strict inequalities versus non-strict inequalities?
2. What is the difference between solving a quadratic equation and a quadratic inequality?
3. How do you factor quadratic expressions effectively?
4. How can graphing a quadratic function help in solving inequalities?
5. What is the significance of open and closed circles on a number line?

**Tip:** Always check your critical points and test intervals carefully to ensure the solution to inequalities is accurate!

Solve the inequality x^2 + 2x - 8 > -5. Graph the solution on a number line.

Learn how to solve the quadratic inequality x^2 + 2x - 8 > -5 with a detailed step-by-step solution. This problem covers key algebraic concepts such as factoring and analyzing number lines to graph solutions.

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