https://file.aiquickdraw.com/mathbot/file/1726698545774cmplh4ar.jpg

Let's analyze the graph and the problems based on the image.

The figure shows two equations:
1. The red curve represents the quadratic function \( y = 3x - x^2 \).
2. The blue horizontal line represents \( y = 2 \).

We are asked to do two things:

### (a) Find the solutions to the equation \( 3x - x^2 = 2 \).
This is where the red curve intersects the blue line (y = 2). To find the solutions, we look for the x-values at the intersection points.

From the graph, it looks like the red curve intersects the blue line at two points:
- The first intersection is between \( x = 0 \) and \( x = 1 \).
- The second intersection is between \( x = 2 \) and \( x = 3 \).

Let me solve the equation \( 3x - x^2 = 2 \) algebraically to get the exact solutions.

Rearranging the equation:
\[
3x - x^2 = 2 \quad \Rightarrow \quad -x^2 + 3x - 2 = 0
\]
Multiply by -1 to make it easier:
\[
x^2 - 3x + 2 = 0
\]
Factoring the quadratic:
\[
(x - 1)(x - 2) = 0
\]
Thus, the solutions are \( x = 1 \) and \( x = 2 \).

### (b) Find the solutions of the inequality \( 3x - x^2 > 2 \).

This inequality means we need to find the x-values where the red curve is above the blue line.

Looking at the graph, the red curve is above the blue line between the two intersection points at \( x = 1 \) and \( x = 2 \). Therefore, the solution to the inequality is:
\[
1 < x < 2
\]

### Summary:
- (a) The solutions to \( 3x - x^2 = 2 \) are \( x = 1 \) and \( x = 2 \).
- (b) The solution to \( 3x - x^2 > 2 \) is \( (1, 2) \) in interval notation.

Would you like further details or explanations? Here are some related questions:

1. How do you solve quadratic equations by factoring?
2. What other methods can you use to solve quadratic equations?
3. How can you solve quadratic inequalities graphically?
4. Can you explain the difference between solving an equation and solving an inequality?
5. How do you determine where a curve is above or below a line?

**Tip:** When solving quadratic inequalities, graphing can give a quick visual representation of the solution set!

Find the solutions to the equation and inequality using the graph of y = 3x - x^2 and y = 2.

Learn how to solve the quadratic equation 3x - x^2 = 2 and the inequality 3x - x^2 > 2 using graphing methods. This problem explores algebraic techniques such as factoring and graphical solutions. Suitable for students in Grades 8-10.

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