A system of inequalities is shown below:y<−2x+1x+2y≥−4Which of the following is in the solution set of this system of inequalities?A.(−5,4)B.(−9,1)C.(6,−8)D.(2,−3)

We are given the system of inequalities:

1. \( y < -2x + 1 \)
2. \( x + 2y \geq -4 \)

We need to test which of the points provided satisfies **both** inequalities.

### Option A: \( (-5, 4) \)

1. First inequality: \( y < -2x + 1 \)
   \[
   4 < -2(-5) + 1
   \]
   \[
   4 < 10 + 1
   \]
   \[
   4 < 11 \quad \text{(True)}
   \]

2. Second inequality: \( x + 2y \geq -4 \)
   \[
   -5 + 2(4) \geq -4
   \]
   \[
   -5 + 8 \geq -4
   \]
   \[
   3 \geq -4 \quad \text{(True)}
   \]

Since both inequalities are satisfied, \( (-5, 4) \) **is** in the solution set.

### Option B: \( (-9, 1) \)

1. First inequality: \( y < -2x + 1 \)
   \[
   1 < -2(-9) + 1
   \]
   \[
   1 < 18 + 1
   \]
   \[
   1 < 19 \quad \text{(True)}
   \]

2. Second inequality: \( x + 2y \geq -4 \)
   \[
   -9 + 2(1) \geq -4
   \]
   \[
   -9 + 2 \geq -4
   \]
   \[
   -7 \geq -4 \quad \text{(False)}
   \]

Since the second inequality is **not** satisfied, \( (-9, 1) \) is **not** in the solution set.

### Option C: \( (6, -8) \)

1. First inequality: \( y < -2x + 1 \)
   \[
   -8 < -2(6) + 1
   \]
   \[
   -8 < -12 + 1
   \]
   \[
   -8 < -11 \quad \text{(False)}
   \]

Since the first inequality is **not** satisfied, \( (6, -8) \) is **not** in the solution set.

### Option D: \( (2, -3) \)

1. First inequality: \( y < -2x + 1 \)
   \[
   -3 < -2(2) + 1
   \]
   \[
   -3 < -4 + 1
   \]
   \[
   -3 < -3 \quad \text{(False)}
   \]

Since the first inequality is **not** satisfied, \( (2, -3) \) is **not** in the solution set.

### Conclusion:

The point that satisfies both inequalities is **Option A: \( (-5, 4) \)**.

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Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:

1. How can you graph the system of inequalities?
2. What happens when a point lies on the boundary line of an inequality?
3. How do you solve a system of inequalities with more than two constraints?
4. How can you determine the feasible region for a system of inequalities?
5. Can the solution set of a system of inequalities be empty?

**Tip**: Always test inequalities by substituting points back into the system to verify their validity!

A system of inequalities is shown below:y<−2x+1 x+2y≥−4 Which of the following is in the solution set of this system of inequalities? A.(−5,4) B.(−9,1) C.(6,−8) D.(2,−3)

Explore how to solve a system of linear inequalities y < -2x + 1 and x + 2y ≥ -4. This example checks which point among (−5,4), (−9,1), (6,−8), and (2,−3) satisfies both inequalities. Step-by-step breakdown included for Grades 8-10.

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