The shaded region of the graph is described by:

Question

BlackTom AI · Accepted Answer

The line shown passes through (0, 1) on the y-axis and (3, 0) on the x-axis, so use those to find the equation.

1) Relevant concepts
- Equation of a line from two points, slope-intercept form, and converting to standard form.
- To describe a shaded region, determine which side of the boundary line is shaded and whether the boundary is included (≥ or ≤).

2) Step-by-step reasoning and calculations
- Slope m = (0 − 1) / (3 − 0) = −1/3.
- Point-slope using (0,1): y − 1 = (−1/3)(x − 0) → y = 1 − x/3.
- Multiply both sides by 3 to get standard form: 3y = 3 − x.
- Rearranged: 3y + x = 3. This is the boundary line (matches the red line).

3) Determine the correct inequality direction
- The shaded region lies above (higher y-values than) the red line. Test a point in the shaded region, e.g. (0, 2): substitute into 3y + x = 3 → 3(2) + 0 = 6, and 6 ≥ 3 is true.
- Since points in the shaded region make 3y + x greater than or equal to 3, the correct inequality is 3y + x ≥ 3. The boundary is included (solid red line), so use ≥ rather than >.

4) Why the correct answer is correct
- The derived line 3y + x = 3 exactly matches option (a). Testing an interior shaded point satisfies 3y + x ≥ 3, confirming orientation and inclusion of the boundary.

5) Why the other options are incorrect (brief)
- (b) 3y − x ≥ 0 rearranges to 3y = x, i.e. y = x/3, a line through the origin with slope 1/3 — does not match the given boundary.
- (c) y + 3x ≥ 0 is y = −3x for the boundary — slope −3 through origin, not matching intercepts (0,1) and (3,0).
- (d) y ≥ x + 1 has slope 1 and intercept (0,1) but would pass through (3,4), not (3,0). Thus it does not match the red line.

Conclusion: The shaded region is described by 3y + x ≥ 3, which corresponds to option a.

The shaded region of the graph is described by:单项选择题

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