Let's break down the problem from the image.

You are asked to shade the region corresponding to all values of $x$ (area for flowers) and $y$ (area for vegetables) that satisfy two conditions:

1. The gardener can use up to 220 square feet, so the total area used for flowers and vegetables must satisfy: $x + y \leq 220$
2. The area used for vegetables must be at least four times the area used for flowers: $y \geq 4x$

We now have two inequalities:

1. $x + y \leq 220$
2. $y \geq 4x$

These inequalities describe a region on the graph that you need to shade. Let me explain how to graph this step by step:

Step 1: Graph the first inequality $x + y \leq 220$

• The boundary line is $x + y = 220$, which is a straight line with:
  • The y-intercept at $(0, 220)$ (when $x = 0$).
  • The x-intercept at $(220, 0)$ (when $y = 0$).
• Shade below this line, as the inequality is $x + y \leq 220$.

Step 2: Graph the second inequality $y \geq 4x$

• The boundary line is $y = 4x$, which is a straight line through the origin with slope 4. This line passes through points like:
  • $(0, 0)$
  • $(50, 200)$
• Shade above this line, as the inequality is $y \geq 4x$.

Step 3: Find the intersection

The region that satisfies both inequalities is where the shaded areas overlap.

Would you like a more detailed explanation of how to solve this graphically, or do you need help with any particular part of this?

Here are some additional related questions to consider:

1. How would the solution change if the total area was increased to 300 square feet?
2. What if the requirement for vegetable area was only twice the flower area instead of four times?
3. Can you express the solution as a system of equations and inequalities?
4. How would you interpret the region where the gardener uses exactly 220 square feet?
5. How can you verify the solution graphically using test points?

Tip: When solving inequalities graphically, always identify boundary lines and test regions to ensure correct shading.