The graphs of the linear relations x − 2y = −4 and 3x − y = 3 are shown.A point that satisfies both the inequalities x − 2y $ \geq $ −4 and 3x − y $ \geq $ 3 is:

Question

BlackTom AI · Accepted Answer

1) Relevant knowledge points and concepts:
- A linear inequality with ≥ has as its solution the half-plane on or above (or on or to one side of) its boundary line. To test whether a specific point is a solution, substitute its coordinates into each inequality; if both inequalities hold (true), the point is in the intersection region.

2) Step-by-step reasoning (test each choice):
We must check both inequalities: x − 2y ≥ −4 and 3x − y ≥ 3.

3) Calculations with intermediate steps:

a. Point (3, 2)
- For x − 2y: 3 − 2(2) = 3 − 4 = −1. Compare: −1 ≥ −4 is true.
- For 3x − y: 3(3) − 2 = 9 − 2 = 7. Compare: 7 ≥ 3 is true.
Both inequalities are satisfied, so (3, 2) is in the solution region.

b. Point (1, 2)
- x − 2y: 1 − 2(2) = 1 − 4 = −3 → −3 ≥ −4 is true.
- 3x − y: 3(1) − 2 = 3 − 2 = 1 → 1 ≥ 3 is false.
(1, 2) fails the second inequality.

c. Point (2, 4)
- x − 2y: 2 − 2(4) = 2 − 8 = −6 → −6 ≥ −4 is false.
No need to check further; (2, 4) is not in the solution region.

d. Point (1, 2.5)
- x − 2y: 1 − 2(2.5) = 1 − 5 = −4 → −4 ≥ −4 is true (on the boundary).
- 3x − y: 3(1) − 2.5 = 3 − 2.5 = 0.5 → 0.5 ≥ 3 is false.
(1, 2.5) fails the second inequality.

4) Why the correct answer is correct:
(3, 2) satisfies both inequalities numerically (−1 ≥ −4 and 7 ≥ 3). That places it in the intersection of the two half-planes defined by the inequalities.

5) Why the other options are incorrect:
(1, 2), (2, 4), and (1, 2.5) each fail at least one of the inequalities as shown above, so none of them lie in the intersection region.

6) Conclusion:
The only point from the choices that satisfies both inequalities is (3, 2). This matches the provided correct answer: (3, 2).

The graphs of the linear relations x − 2y = −4 and 3x − y = 3 are shown.A point that satisfies both the inequalities x − 2y \( \geq \) −4 and 3x − y \( \geq \) 3 is:单项选择题

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