Given the system of inequalities:

1. $x \geq 0$
2. $y \geq 0$
3. $y < -4x + 16$
4. $y < -\frac{1}{3}x + 5$

These inequalities represent the constraints on $x$ and $y$. Let's analyze them:

1. $x \geq 0$:

• This indicates that $x$ must be non-negative. The solution lies in the right half-plane.

2. $y \geq 0$:

• This indicates that $y$ must be non-negative. The solution lies in the upper half-plane.

3. $y < -4x + 16$:

• This is a linear inequality representing the region below the line $y = -4x + 16$. The line has a slope of $-4$ and a y-intercept at 16.

4. $y < -\frac{1}{3}x + 5$:

• This is another linear inequality representing the region below the line $y = -\frac{1}{3}x + 5$. The line has a slope of $-\frac{1}{3}$ and a y-intercept at 5.

Intersection of Inequalities:

The feasible region that satisfies all these inequalities is where the regions overlap. Specifically:

• $x \geq 0$ and $y \geq 0$ confine the solution to the first quadrant.
• The intersection of the areas below the lines $y = -4x + 16$ and $y = -\frac{1}{3}x + 5$ in this quadrant defines the solution region.

Would you like to see a graphical representation or need further details?

Here are five related questions you might find helpful:

1. How do you graph linear inequalities on the coordinate plane?
2. What is the significance of the slope and y-intercept in linear equations?
3. How do you determine the points of intersection for two lines?
4. What are the key differences between solving linear inequalities and linear equations?
5. How do changes in inequalities affect the solution region?

Tip: Always start by sketching the boundary lines when solving inequalities; it helps to visualize the solution region clearly.