https://file.aiquickdraw.com/mathbot/file/1727103239411ygzdg748.jpg

The problem in the image asks which point \((x, y)\) satisfies the system of inequalities:

\[
y \leq x + 7
\]
\[
y \geq -2x - 1
\]

We will check each option by substituting the given points into both inequalities to see if both are satisfied.

### Option A: \((-14, 0)\)
1. **First inequality:** \( 0 \leq -14 + 7 \Rightarrow 0 \leq -7 \) (False)
2. Since the first inequality is false, this point is not a solution.

### Option B: \((0, -14)\)
1. **First inequality:** \(-14 \leq 0 + 7 \Rightarrow -14 \leq 7 \) (True)
2. **Second inequality:** \(-14 \geq -2(0) - 1 \Rightarrow -14 \geq -1 \) (False)
3. The second inequality is false, so this point is not a solution.

### Option C: \((0, 14)\)
1. **First inequality:** \(14 \leq 0 + 7 \Rightarrow 14 \leq 7 \) (False)
2. Since the first inequality is false, this point is not a solution.

### Option D: \((14, 0)\)
1. **First inequality:** \(0 \leq 14 + 7 \Rightarrow 0 \leq 21 \) (True)
2. **Second inequality:** \(0 \geq -2(14) - 1 \Rightarrow 0 \geq -28 - 1 \Rightarrow 0 \geq -29 \) (True)

Both inequalities are satisfied for \((14, 0)\), so the correct answer is **Option D**.

Would you like any further details or have any questions?

Here are some related questions you might explore:
1. What is the graphical interpretation of solving a system of inequalities?
2. How do you find the solution region for a system of inequalities?
3. How do the inequalities divide the plane into regions?
4. How can we confirm a solution by graphing?
5. What types of systems of inequalities are there, and how are they solved?

**Tip:** When solving inequalities, always verify each point by substituting values into all inequalities in the system.

Which point (x, y) is a solution to the given system of inequalities in the xy-plane?

Find which point (x, y) satisfies the system of inequalities y ≤ x + 7 and y ≥ -2x - 1. This problem explores algebraic graphing and systems of inequalities, making it suitable for students in Grades 8-10.

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