https://r2.erweima.ai/mathbot/image/1726514360867qwalxx52.pdf

The project you uploaded contains several mathematical problems, focused on equations, function modeling, and geometry. Here’s an outline of the key parts:

### Problem 1: Solving Equations
1. **Graphical and numerical solutions** for equations like:
   - \( f(x) = g(x) \), which is reduced to solving \( F(x) = f(x) - g(x) = 0 \).
   - Solutions are found where the graph intersects the x-axis, using both graphing and algebraic methods.
2. Specific problems include:
   - \( x - 1 + 2\sqrt{x - 2} + x - 1 - 2\sqrt{x - 2} = 2 \)
   - \( |2x - 9x - 5| = 3x - 5 \)
   - A system of equations:
     \[
     x + xy + y = 4 \quad \text{and} \quad x + xy + y = 2
     \]

### Problem 2: Modeling Functions
This problem involves analyzing and modeling the concentration of \( \text{CO}_2 \) in a lab experiment over 3 days. The concentration is modeled with:
1. **Graphical representation** of the data.
2. **Polynomial models** to fit the data.
3. Linear functions to model the concentration during **dark cycles** and periods of **decay**.
4. **Piecewise functions** describing the full experiment.

### Problem 3: Geometry
Tasks here include:
1. Finding shaded areas supported by computer algebra systems (CAS).
   
For detailed help, I can work through the specific problems, such as solving the equations or building the model for the data.

Let me know if you want to focus on a particular part! 

**Relative questions:**
1. How do you solve the system of equations in Problem 1 using graphing?
2. What are the algebraic steps to solve the equation \( |2x - 9x - 5| = 3x - 5 \)?
3. How do you model a piecewise function from real-world data like in Problem 2?
4. What software (CAS) do you plan to use for graphing and solving these equations?
5. How do you compute the shaded area in geometry problems using a CAS?

**Tip:** When solving systems of equations graphically, always check for the number of intersections—each intersection represents a solution.

Explore how to solve algebraic equations, systems of equations, and model piecewise functions using computer algebra systems (SAC) like GeoGebra or Mathematica. This project covers CO2 concentration modeling, graphical solutions, and geometry. Suitable for undergraduate students.

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