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 lies within the solution set of this system of inequalities, we’ll evaluate each of the given points for both inequalities in the system:

### System of inequalities:
1. \( y < -2x + 1 \)
2. \( x + 2y \geq -4 \)

### Testing each option:

**Option A: \((-9, 1)\)**

1. Substitute \((-9, 1)\) into the first inequality:
   \[
   y < -2x + 1 \Rightarrow 1 < -2(-9) + 1 \Rightarrow 1 < 18 + 1 \Rightarrow 1 < 19 \quad (\text{True})
   \]
2. Substitute \((-9, 1)\) into the second inequality:
   \[
   x + 2y \geq -4 \Rightarrow -9 + 2(1) \geq -4 \Rightarrow -9 + 2 \geq -4 \Rightarrow -7 \geq -4 \quad (\text{False})
   \]

Since the second inequality fails, **Option A is not in the solution set.**

---

**Option B: \((-5, 4)\)**

1. Substitute \((-5, 4)\) into the first inequality:
   \[
   y < -2x + 1 \Rightarrow 4 < -2(-5) + 1 \Rightarrow 4 < 10 + 1 \Rightarrow 4 < 11 \quad (\text{True})
   \]
2. Substitute \((-5, 4)\) into the second inequality:
   \[
   x + 2y \geq -4 \Rightarrow -5 + 2(4) \geq -4 \Rightarrow -5 + 8 \geq -4 \Rightarrow 3 \geq -4 \quad (\text{True})
   \]

Since both inequalities are satisfied, **Option B is in the solution set**.

---

**Option C: \((2, -3)\)**

1. Substitute \((2, -3)\) into the first inequality:
   \[
   y < -2x + 1 \Rightarrow -3 < -2(2) + 1 \Rightarrow -3 < -4 + 1 \Rightarrow -3 < -3 \quad (\text{False})
   \]

Since the first inequality fails, **Option C is not in the solution set**.

---

**Option D: \((6, -8)\)**

1. Substitute \((6, -8)\) into the first inequality:
   \[
   y < -2x + 1 \Rightarrow -8 < -2(6) + 1 \Rightarrow -8 < -12 + 1 \Rightarrow -8 < -11 \quad (\text{False})
   \]

Since the first inequality fails, **Option D is not in the solution set**.

---

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

---

Would you like more details on how each inequality was solved, or do you have any questions?

---

1. How can we solve a system of linear inequalities graphically?
2. What methods can verify a point's inclusion in multiple inequalities?
3. Why does the combination of “<” and “≥” limit solution sets for inequalities?
4. How does changing inequality direction affect solution boundaries?
5. What strategies help quickly determine valid points in a solution set?

**Tip:** Always check each inequality separately to verify all conditions for a solution point.

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)

Explore how to find the solution set for the system of inequalities y < -2x + 1 and x + 2y ≥ -4. Includes step-by-step verification of each point, illustrating how to check solutions in algebraic inequalities. Ideal for Grades 8-10 students.

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