https://file.aiquickdraw.com/mathbot/file/1727058520857u1mrctoh.jpg

The system of inequalities provided in the image is:

1. \( y < -x + 3 \)
2. \( y \leq x^2 - 2x - 5 \)

The solution to this system is the region where both inequalities overlap.

### Explanation of the graph:
- The **parabola** \( y = x^2 - 2x - 5 \) (drawn as a solid curve) represents the boundary for the second inequality. The shaded region below the parabola corresponds to the values where \( y \leq x^2 - 2x - 5 \).
- The **line** \( y = -x + 3 \) (drawn as a dashed line) represents the boundary for the first inequality. The shaded region below this line corresponds to \( y < -x + 3 \).

### Interpreting the solution:
The overlapping shaded region, where the two inequalities are both satisfied, is the solution to the system. It lies:
- Below the dashed line \( y = -x + 3 \), but not on the line (since the inequality is strict, \( y < -x + 3 \)).
- Below the parabola \( y = x^2 - 2x - 5 \), including the points on the parabola (since the inequality is \( \leq \)).

This region is bound by:
- The parabola on top.
- The line on the bottom.
- It spans roughly between \( x \approx -1.5 \) and \( x \approx 4 \), where the line and parabola intersect.

### Key Points of Intersection:
To find the exact intersection points, we solve the system \( -x + 3 = x^2 - 2x - 5 \):
\[
x^2 - x - 8 = 0
\]
Solving this quadratic equation gives \( x = 4 \) and \( x = -2 \). Therefore, the regions between \( x = -2 \) and \( x = 4 \) are critical to the solution.

Would you like further details on how to solve this or any other clarifications?

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Here are five related questions:
1. What are the exact coordinates of the points where the line and parabola intersect?
2. How do we graph a system of inequalities step by step?
3. What happens when the inequality is non-strict (i.e., \( \leq \) or \( \geq \))?
4. How do we determine if a point is a solution to the system of inequalities?
5. How can you check the solution region on a graphing calculator?

**Tip:** When solving inequalities graphically, always ensure you pay attention to whether the boundary is included (solid line) or not (dashed line).

What is the solution to the system of equations: y < -x + 3 and y ≤ x² - 2x - 5?

Learn how to solve a system of inequalities involving a linear equation and a quadratic function: y < -x + 3 and y ≤ x² - 2x - 5. This detailed guide includes graphing and finding the overlapping solution region for the inequalities. Suitable for Grades 9-11.

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