https://file.aiquickdraw.com/mathbot/file/1726883771147odm0qyx3.jpg

The inequality given in the image is:

\[
-x^2 + x + 6 > 0
\]

Let's solve this step-by-step:

### Step 1: Rearrange the inequality
The inequality is already in standard form. To make it easier to solve, we can multiply both sides of the inequality by \(-1\), which will reverse the inequality sign:

\[
x^2 - x - 6 < 0
\]

Now, we will solve the inequality \(x^2 - x - 6 < 0\).

### Step 2: Solve the related quadratic equation
First, solve the quadratic equation \(x^2 - x - 6 = 0\). This can be factored:

\[
x^2 - x - 6 = (x - 3)(x + 2) = 0
\]

The solutions to this equation are:

\[
x = 3 \quad \text{and} \quad x = -2
\]

### Step 3: Analyze the sign of the quadratic expression
Now, we analyze the sign of \( (x - 3)(x + 2) \) for different intervals of \(x\), based on the roots \(-2\) and \(3\). The intervals are:

- \( x < -2 \)
- \( -2 < x < 3 \)
- \( x > 3 \)

We test the sign of \( (x - 3)(x + 2) \) in each interval:

- For \( x < -2 \) (e.g., \(x = -3\)): \( (x - 3) < 0 \) and \( (x + 2) < 0 \), so \( (x - 3)(x + 2) > 0 \)
- For \( -2 < x < 3 \) (e.g., \(x = 0\)): \( (x - 3) < 0 \) and \( (x + 2) > 0 \), so \( (x - 3)(x + 2) < 0 \)
- For \( x > 3 \) (e.g., \(x = 4\)): \( (x - 3) > 0 \) and \( (x + 2) > 0 \), so \( (x - 3)(x + 2) > 0 \)

### Step 4: Solution to the inequality
We want the quadratic expression to be negative, so we select the interval where \( (x - 3)(x + 2) < 0 \), which is:

\[
-2 < x < 3
\]

### Final Answer:
The solution to the inequality \( -x^2 + x + 6 > 0 \) is:

\[
x \in (-2, 3)
\]

Would you like more details or explanations on any step?

Here are 5 related questions you might want to explore:
1. How do we solve quadratic inequalities in general?
2. What is the geometric interpretation of this inequality on a graph?
3. Can we apply the quadratic formula instead of factoring?
4. What happens if the quadratic inequality had \( \geq \) instead of \( > \)?
5. How do we check the solution using a test point method?

**Tip:** Always remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign reverses!

Learn how to solve the quadratic inequality -x^2 + x + 6 > 0. This step-by-step solution involves key algebraic concepts such as factoring quadratic equations and analyzing the sign of expressions over intervals. Ideal for students in Grades 8-10.

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