Donna will donate up to $460 to charity. The money will be divided between two charities: the City Youth Fund and the Educational Growth Foundation. Donna would like to donate at least $160 to the Educational Growth Foundation. She would also like the amount donated to the Educational Growth Foundation to be at least three 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 y
denote 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

To find the region corresponding to all values of x and y that satisfy the given requirements, we need to plot the constraints on a graph and shade the relevant region.

Let's begin by assigning variables:
x = amount of money donated to the City Youth Fund (in dollars)
y = amount of money donated to the Educational Growth Foundation (in dollars)

Given constraints:
1. Donna will donate up to $460 to charity:
This can be represented as: x + y ≤ 460

2. Donna would like to donate at least $160 to the Educational Growth Foundation:
This can be represented as: y ≥ 160

3. The amount donated to the Educational Growth Foundation should be at least three times the amount donated to the City Youth Fund:
This can be represented as: y ≥ 3x

Now, let's graph these inequalities on a coordinate grid:

1. Graph x + y ≤ 460:

Start by plotting the line x + y = 460:
- Choose any two points on this line, for example, (0, 460) and (460, 0).
- Connect these points with a straight line.
- Shade the region below this line (since we want x + y to be less than or equal to 460).

2. Graph y ≥ 160:

Start by plotting the line y = 160:
- Choose any two points on this line, for example, (0, 160) and (460, 160).
- Connect these points with a straight line.
- Shade the region above this line (since we want y to be greater than or equal to 160).

3. Graph y ≥ 3x:

Start by plotting the line y = 3x:
- Choose any two points on this line, for example, (0, 0) and (153.33, 460).
- Connect these points with a straight line.
- Shade the region above this line (since we want y to be greater than or equal to 3x).

The shaded region where all three constraints are satisfied is the region that corresponds to all values of x and y that satisfy the requirements mentioned in the problem.

To shade the region corresponding to all values of x and y that satisfy the given requirements, we need to set up a system of inequalities based on the given conditions.

Let's break down the given conditions into equations:

Condition 1: Donna will donate up to $460:
x + y ≤ 460

Condition 2: Donna would like to donate at least $160 to the Educational Growth Foundation:
y ≥ 160

Condition 3: Donna wants the amount donated to the Educational Growth Foundation to be at least three times the amount donated to the City Youth Fund:
y ≥ 3x

Now, let's graph these inequalities on a coordinate system:

1. Graph x + y ≤ 460:

To graph this equation, rearrange it to solve for y:
y ≤ 460 - x

Plot the line y = 460 - x by plotting two points (0, 460) and (460, 0), and draw a dashed line through them.

2. Graph y ≥ 160:

To graph this equation, plot a horizontal line at y = 160.

3. Graph y ≥ 3x:

To graph this equation, rearrange it to solve for y:
y ≥ 3x

Plot the line y = 3x by plotting two points (0, 0) and (160, 480), and draw a solid line through them.

Now, shade the region that satisfies all the conditions. This region will be the overlapping area where all the shaded regions intersect.

The shaded region will be below the line y = 460 - x, above the line y = 160, and above the line y = 3x.

Note: It is not possible to accurately shade the region without the exact coordinate scale and dimensions. This explanation provides a general idea of how to shade the region based on the given conditions.

The question states "up to $460" in charity, which means it can be less than or equal to 460, but not any higher. Then,

$460 to charity (max)

$160+ to Educational (160 being the minimum)
Education is 3x (or higher) than City.

If we denote x to City, and y to Educational, then,
x + y = 460 . And we know that y >= 3x
x + 3x = 4x = 460
x = 115. This means when she donates $115 to City, she will donate 3x (or $345) to Educational. These are the max values; she will not donate higher than $460.

A minimum requirement was placed on Educational earlier. When y = 160, x will be y/3, or $53.34. (This isn't very meaningful, but: This means she donated 213.34, which is still less than $460)

Therefore, x (which is City) can be the range of $53-ish dollars (depending on your region/graph) but no higher than $115.
y (which is Educational) will be as low as $160, but no higher than $345.

-Hope this helps.