Explanation:

Step 1: Define the inequalities from the problem.

The problem states that the gardener will use up to 250 square feet for both flowers, $x$, and vegetables, $y$: $x+y \le 250$. Additionally, the area used for vegetables, $y$, should be at least twice the area used for flowers, $x$: $y \ge 2x$. Lastly, since we are considering area, we know that both $x$ and $y$ must be greater than or equal to zero: $x\ge0$ and $y\ge0$.

Step 2: Convert the inequalities into equations to plot the boundary lines.

Replace the inequality symbols with equal signs to find the equations of the lines: $x+y=250$, $y=2x$, $x=0$, $y=0$.

Step 3: Plot the boundary lines on the coordinate plane.

• For $x + y = 250$, when $x=0$, then $y=250$, and when $y=0$, then $x=250$. Connect points $(0,250)$ and $(250,0)$ to graph the line, noting that the intercepts are outside of the provided graph.
• For $y=2x$, when $x=0$, then $y=0$, and when $x=100$, then $y=200$. Connect points $(0,0)$ and $(100,200)$ to graph the line.
• $x=0$ is the same as the y-axis.
• $y=0$ is the same as the x-axis.

Step 4: Determine the shading for each inequality.

• For $x+y\le250$, test point $(0,0)$ in the inequality. Since $0+0\le250$ is true, shade the region below the line $x+y=250$ that contains the point $(0,0)$.
• For $y\ge2x$, test point $(100,0)$. Since $0\ge2(100)$ is false, shade the region above the line $y=2x$ that does not contain $(100,0)$.
• For $x\ge0$ and $y\ge0$, shade the region in the first quadrant where both $x$ and $y$ are positive.

Step 5: Identify the overlapping shaded region.

The solution to the system of inequalities is the region where all shaded areas overlap. This will be a triangular region bounded by the lines $y=2x$, $x+y=250$, and the y-axis.

Answer:

The shaded region on the graph below represents all possible combinations of area for flowers ($x$) and vegetables ($y$) that satisfy the gardener's requirements.