$$ \left.\begin{array}{l} ext { Correct } \ ext { Determine which of the following points are included in the solution set to the following system of linear inequalities. } \ \qquad \begin{array}{l}4 x+2 y-3 \ -2 x+y \leq 2\end{array} \ ext { Answer } \ \square(5,7) \ \square(-7,-5) \ \square\left(-1, \frac{1}{2} ight) \ \square(8,0)\end{array} ight] $$

Question

$$ \left.\begin{array}{l}	ext { Correct } \ 	ext { Determine which of the following points are included in the solution set to the following system of linear inequalities. } \ \qquad \begin{array}{l}4 x+2 y>-3 \ -2 x+y \leq 2\end{array} \ 	ext { Answer } \ \square(5,7) \ \square(-7,-5) \ \square\left(-1, \frac{1}{2}ight) \ \square(8,0)\end{array}ight] $$

UpStudy AI · Accepted Answer

To determine which points satisfy the given system of linear inequalities:

$$
\begin{cases}
4x + 2y > -3 \
-2x + y \leq 2
\end{cases}
$$

Let's evaluate each point individually.

---

### 1. **Point $$(5, 7)$$**

**First Inequality:**
$$
4(5) + 2(7) = 20 + 14 = 34 > -3 \quad \text{✓}
$$

**Second Inequality:**
$$
-2(5) + 7 = -10 + 7 = -3 \leq 2 \quad \text{✓}
$$

**Conclusion:** $$\boxed{(5,\ 7)}$$ **is in the solution set.**

---

### 2. **Point $$(-7, -5)$$**

**First Inequality:**
$$
4(-7) + 2(-5) = -28 - 10 = -38 > -3 \quad \text{✗}
$$

**Conclusion:** $$\boxed{(-7,\ -5)}$$ **is \textbf{not} in the solution set.**

---

### 3. **Point $$\left(-1, \frac{1}{2}\right)$$**

**First Inequality:**
$$
4(-1) + 2\left(\frac{1}{2}\right) = -4 + 1 = -3 > -3 \quad \text{✗}
$$

**Conclusion:** $$\boxed{\left(-1,\ \frac{1}{2}\right)}$$ **is \textbf{not} in the solution set.**

---

### 4. **Point $$(8, 0)$$**

**First Inequality:**
$$
4(8) + 2(0) = 32 > -3 \quad \text{✓}
$$

**Second Inequality:**
$$
-2(8) + 0 = -16 \leq 2 \quad \text{✓}
$$

**Conclusion:** $$\boxed{(8,\ 0)}$$ **is in the solution set.**

---

### **Summary**

$$
\begin{array}{ll}
\boxed{\square} & (5,\ 7) \
\boxed{\square} & ( -7,\ -5) \
\boxed{\square} & \left(-1,\ \frac{1}{2}\right) \
\boxed{\square} & (8,\ 0)
\end{array}
$$

After evaluation:

- ☑︎ (5, 7)
- ☐ (–7, –5)
- ☐ (–1, ½)
- ☑︎ (8, 0)

Only the points (5, 7) and (8, 0) are in the solution set.
The points (5, 7) and (8, 0) are in the solution set.

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