A system of inequalities is shown below:

y<−2x+1y<−2x+1

x+2y≥−4x+2y≥−4


Which of the following is in the solution set of this system of inequalities?

A.(−9,1)(−9,1)

B.(−5,4)(−5,4)

C.(2,−3)(2,−3)

D.(6,−8)(6,−8)


To determine which point is in the solution set of this system of inequalities, we need to test each given point against the two inequalities:

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

### Test Point A: \((-9, 1)\)
- For \( y < -2x + 1 \):
  \[
  1 < -2(-9) + 1
  \]
  \[
  1 < 18 + 1
  \]
  \[
  1 < 19 \quad \text{(True)}
  \]
- For \( x + 2y \geq -4 \):
  \[
  -9 + 2(1) \geq -4
  \]
  \[
  -9 + 2 \geq -4
  \]
  \[
  -7 \geq -4 \quad \text{(False)}
  \]
Thus, point A does not satisfy both inequalities.

### Test Point B: \((-5, 4)\)
- For \( y < -2x + 1 \):
  \[
  4 < -2(-5) + 1
  \]
  \[
  4 < 10 + 1
  \]
  \[
  4 < 11 \quad \text{(True)}
  \]
- For \( x + 2y \geq -4 \):
  \[
  -5 + 2(4) \geq -4
  \]
  \[
  -5 + 8 \geq -4
  \]
  \[
  3 \geq -4 \quad \text{(True)}
  \]
Thus, point B satisfies both inequalities.

### Test Point C: \((2, -3)\)
- For \( y < -2x + 1 \):
  \[
  -3 < -2(2) + 1
  \]
  \[
  -3 < -4 + 1
  \]
  \[
  -3 < -3 \quad \text{(False, because -3 is not less than -3)}
  \]
Thus, point C does not satisfy the first inequality.

### Test Point D: \((6, -8)\)
- For \( y < -2x + 1 \):
  \[
  -8 < -2(6) + 1
  \]
  \[
  -8 < -12 + 1
  \]
  \[
  -8 < -11 \quad \text{(False)}
  \]
Thus, point D does not satisfy the first inequality.

### Conclusion:
The only point that satisfies both inequalities is **B: \((-5, 4)\)**.

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Do you need further explanation or any additional details? Here are five related questions to expand your understanding:

1. How do you graph systems of inequalities on a coordinate plane?
2. How would the solution set look visually for this system?
3. How can you determine if a system of inequalities has no solution?
4. What happens if both inequalities have the same slope?
5. Can inequalities with different variables (like \(x\) and \(y\)) be solved similarly?

**Tip:** When testing a point in an inequality, plug the coordinates into both inequalities and check for consistency with the inequality signs.

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. (−9, 1) B. (−5, 4) C. (2, −3) D. (6, −8)

Find out how to solve a system of inequalities with the example of y < −2x + 1 and x + 2y ≥ −4. We test points like (−5, 4) and more to see which lies in the solution set. Suitable for students in Grades 8-10.

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