Debra will donate up to $460 to charity. The money will be divided between two charities: the City Youth Fund and the Educational Growth Foundation. Debra would like the amount donated to the Educational Growth Foundation to be at least four times the amount donated to the City Youth Fund. Let x denote the amount of money (in dollars) donated to the City Youth Fund. Let denote y the amount of money (in dollars) donated to the Educational Growth Foundation. Shade the region corresponding to all values of x and y that satisfy these requirements.

We need to express the problem mathematically and then describe the solution.

### Step 1: Write the equations and inequalities
We have two conditions based on the problem:

1. Debra will donate up to $460 in total. Therefore, the sum of the donations to both charities is limited by:
   \[
   x + y \leq 460
   \]
   where $x$ is the amount donated to the City Youth Fund, and $y$ is the amount donated to the Educational Growth Foundation.

2. The amount donated to the Educational Growth Foundation must be at least four times the amount donated to the City Youth Fund:
   \[
   y \geq 4x
   \]

### Step 2: Set the constraints on the variables
- $x \geq 0$ (the amount donated to the City Youth Fund cannot be negative)
- $y \geq 0$ (the amount donated to the Educational Growth Foundation cannot be negative)

### Step 3: Graph the inequalities
To shade the region that satisfies these constraints, follow these steps:

1. Plot the line $x + y = 460$. This line represents the boundary where the total donation equals $460. The area below this line (including the line itself) satisfies the inequality $x + y \leq 460$.

2. Plot the line $y = 4x$. This line represents the boundary where the donation to the Educational Growth Foundation is exactly four times the donation to the City Youth Fund. The area above this line satisfies the inequality $y \geq 4x$.

3. The shaded region will be the area where all these conditions are true:
   - Below or on the line $x + y = 460$.
   - Above or on the line $y = 4x$.
   - In the first quadrant, where $x \geq 0$ and $y \geq 0$.

### Step 4: Find the intersection points
To identify the boundaries of the region:
- Set $x + y = 460$ and $y = 4x$, then solve for $x$ to find the intersection point of these lines.

\[
x + 4x = 460 \quad \Rightarrow \quad 5x = 460 \quad \Rightarrow \quad x = 92
\]
Now, substitute $x = 92$ into $y = 4x$ to find $y$:
\[
y = 4(92) = 368
\]
Thus, the lines intersect at the point $(92, 368)$.

### Step 5: Conclusion
The region satisfying all constraints will be the area below the line $x + y = 460$, above the line $y = 4x$, and limited to the first quadrant. The boundaries are:
- $x$ ranges from 0 to 92.
- $y$ ranges from $4x$ to $460 - x$.

Would you like more details on how to graph this, or any other clarifications?

### Related Questions:
1. How do you calculate the slope of a line?
2. How do you determine the feasible region in a system of inequalities?
3. What are the steps to solving a system of equations graphically?
4. How can we interpret the intersection point of two lines?
5. How does linear programming apply to optimization problems like this?

### Tip:
Always check if constraints involve strict inequalities (like $>$) or inclusive inequalities (like $\geq$) to correctly identify the boundaries of the feasible region.

Solve the donation allocation problem where Debra will donate up to $460 between the City Youth Fund and the Educational Growth Foundation, with specific constraints. Learn to express inequalities, graph the solution, and find the intersection points. This problem covers concepts of inequalities and linear equations, suitable for Grades 9-12.

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