Which one of the following sets of inequations describes the shaded region (boundaries included) on the graph shown?

Question

BlackTom AI · Accepted Answer

The two slanted boundary lines meet the axes at (0,8) & (5,0) and (0,6) & (7,0), so write their equations from intercept form and pick the side that contains the shaded region (the origin).

1. Relevant concepts
- A line through x-intercept a and y-intercept b has equation (x/a) + (y/b) = 1, or equivalently bx + ay = ab.  
- A linear inequality picks one side of the line; test any point (here the origin (0,0) lies in the shaded region) to determine ≤ or ≥.  
- Boundaries included mean use ≤ or ≥, not strict < or >.

2. Step-by-step reasoning and intermediate calculations
- For the steeper (red) line: intercepts (5,0) and (0,8). Using bx + ay = ab with a = 5, b = 8 gives 8x + 5y = 40.  
  Check which side is shaded: substitute (0,0): 8(0)+5(0)=0 and 0 ≤ 40, so the shaded side is 8x + 5y ≤ 40.
- For the shallower (blue) line: intercepts (7,0) and (0,6). Using the same form with a = 7, b = 6 gives 6x + 7y = 42.  
  Test (0,0): 6(0)+7(0)=0 and 0 ≤ 42, so the shaded side is 6x + 7y ≤ 42.
- The shaded region is clearly in the first quadrant with axes included, so x ≥ 0 and y ≥ 0.

3. Why this answer is correct
Combining these gives x ≥ 0, y ≥ 0, 8x + 5y ≤ 40, 6x + 7y ≤ 42, which exactly describes the triangular shaded region bounded by the two lines and the axes, with boundaries included.

4. Why other options are incorrect (brief)
- (a) reverses the inequalities (≥) for the slanted lines; that would select the region away from the origin.  
- (b) uses equalities (=), which are only the lines themselves, not the filled region.  
- (c) swaps coefficients (5x+8y and 7x+6y), giving lines with different intercepts (y‑intercepts 5 and 7), not matching the graph.

Conclusion: The correct set is x ≥ 0, y ≥ 0, 8x + 5y ≤ 40, 6x + 7y ≤ 42, which matches option d.

Which one of the following sets of inequations describes the shaded region (boundaries included) on the graph shown?单项选择题

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