The shaded region shown including the boundaries shown is defined by:

Question

BlackTom AI · Accepted Answer

We start by identifying what the prompt is asking: the shaded region shown, including the boundaries, is defined by which set of inequalities or conditions.

In this case, the provided answer text lists a single option: d. x ≥ 3, y ≥ 4, x ≤ 7, y ≤ 9, 3x + 4y ≤ 45. If we take this as the candidate description, we can parse its components individually:
- x ≥ 3 means the region lies to the right of the vertical line x = 3 (including the line itself).
- y ≥ 4 means the region lies above the horizontal line y = 4 (including the line itself).
- x ≤ 7 means the region lies to the left of the vertical line x = 7 (including the line itself).
- y ≤ 9 means the region lies below the horizontal line y = 9 (including the line itself).
Combined, these four inequalities already specify a closed rectangle with corners at (3,4), (7,4), (3,9), and (7,9).
- The additional inequality 3x + 4y ≤ 45 imposes a slanted boundary that cuts across the rectangle, keeping only the portion of the rectangle that lies on or below the line 3x + 4y = 45.

To understand the interaction, consider the line 3x + 4y = 45. Its intercepts are x = 15 when y = 0, and y = 11.25 when x = 0. Within the rectangle 3 ≤ x ≤ 7 and 4 ≤ y ≤ 9, the line 3x + 4y = 45 will intersect somewhere, and the inequality ≤ 45 will include the region on or below this line, which typically forms a polygonal portion of the rectangle rather than the entire rectangle.

Thus, if we trust that the shaded region is defined by all of these constraints simultaneously, the region is the intersection of the rectangle [3,7] × [4,9] with the half-plane 3x + 4y ≤ 45. This interpretation is consistent with the combination of axis-aligned bounds and the diagonal bound provided by the linear inequality.

Now, since the question’s answer options are not provided in the data (the answer_options field is empty), we cannot compare this option to others to determine which one is correct in a test-taking sense. However, within the context of the given single option, the described region matches a plausible shaded area formed by intersecting the rectangle with the half-plane 3x + 4y ≤ 45, and therefore this option describes a specific bounded region defined by those inequalities.

If you needed to verify, you could plot the rectangle with corners (3,4), (7,4), (3,9), (7,9) and then shade the portion on or below the line 3x + 4y = 45. The resulting shaded region would illustrate exactly the area defined by these combined constraints.

The shaded region shown including the boundaries shown is defined by:

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