Determine which of the following points are included in the solution set to the following system of linear inequalities. $$ \begin{aligned} x+3 y & -8 \ -2 x+3 y & \leq 1 \end{aligned} $$ Answer $$ (8,4) $$ $$ (0,0) $$ $$ (-8,-2) $$ $$ \left(-7,-\frac{1}{3}\right) $$ $$ (7,5) $$

Question

Determine which of the following points are included in the solution set to the following system of linear inequalities. $$ \begin{aligned} x+3 y & >-8 \ -2 x+3 y & \leq 1 \end{aligned} $$ Answer $$ (8,4) $$ $$ (0,0) $$ $$ (-8,-2) $$ $$ \left(-7,-\frac{1}{3}\right) $$ $$ (7,5) $$

UpStudy AI · Accepted Answer

To determine which points satisfy the system of linear inequalities:

$$
\begin{aligned}
x + 3y &> -8 \
-2x + 3y &\leq 1
\end{aligned}
$$

we will evaluate each point individually.

### 1. Point $$ (8, 4) $$

**First Inequality:**
$$
x + 3y = 8 + 3(4) = 8 + 12 = 20 > -8 \quad \text{(True)}
$$

**Second Inequality:**
$$
-2x + 3y = -2(8) + 3(4) = -16 + 12 = -4 \leq 1 \quad \text{(True)}
$$

**Conclusion:** $$ (8, 4) $$ **is in the solution set**.

---

### 2. Point $$ (0, 0) $$

**First Inequality:**
$$
x + 3y = 0 + 3(0) = 0 > -8 \quad \text{(True)}
$$

**Second Inequality:**
$$
-2x + 3y = -2(0) + 3(0) = 0 \leq 1 \quad \text{(True)}
$$

**Conclusion:** $$ (0, 0) $$ **is in the solution set**.

---

### 3. Point $$ (-8, -2) $$

**First Inequality:**
$$
x + 3y = -8 + 3(-2) = -8 - 6 = -14 > -8 \quad \text{(False)}
$$

**Conclusion:** $$ (-8, -2) $$ **is \textbf{not} in the solution set**.

---

### 4. Point $$ \left(-7, -\frac{1}{3}\right) $$

**First Inequality:**
$$
x + 3y = -7 + 3\left(-\frac{1}{3}\right) = -7 - 1 = -8 > -8 \quad \text{(False)}
$$

**Second Inequality:**
$$
-2x + 3y = -2(-7) + 3\left(-\frac{1}{3}\right) = 14 - 1 = 13 \leq 1 \quad \text{(False)}
$$

**Conclusion:** $$ \left(-7, -\frac{1}{3}\right) $$ **is \textbf{not} in the solution set**.

---

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

**First Inequality:**
$$
x + 3y = 7 + 3(5) = 7 + 15 = 22 > -8 \quad \text{(True)}
$$

**Second Inequality:**
$$
-2x + 3y = -2(7) + 3(5) = -14 + 15 = 1 \leq 1 \quad \text{(True)}
$$

**Conclusion:** $$ (7, 5) $$ **is in the solution set**.

---

**Final Answer:**

All points except $$ (-8, -2) $$ and $$ \left(-7, -\frac{1}{3}\right) $$ satisfy the system. Thus, the solution set includes:
- $$ (8, 4) $$
- $$ (0, 0) $$
- $$ (7, 5) $$

**Answer:**
All points except (–8, –2) and (–7, –1⁄3) satisfy both inequalities. The solution set includes
 (8, 4),
 (0, 0), and
 (7, 5).
The points $$ (8, 4) $$, $$ (0, 0) $$, and $$ (7, 5) $$ are included in the solution set.

Determine which of the following points are included in the solution set to the following system of linear inequalities.

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Determine which of the following points are included in the solution set to the following system of linear inequalities. Answer