30) Graph each system of linear inequalities: a. $$ \begin{array}{l}y \leq-x-2 \ y \geq-5 x+2\end{array} $$

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To graph the system of linear inequalities:

$$
\begin{cases}
y \leq -x - 2 \
y \geq -5x + 2
\end{cases}
$$

follow these step-by-step instructions:

### 1. Graph the Boundary Lines

First, convert each inequality into an equation to graph the boundary lines.

#### a. $$ y = -x - 2 $$

- **Slope (m):** -1
- **y-intercept (b):** -2

**Plotting Points:**
- When $$ x = 0 $$, $$ y = -0 - 2 = -2 $$ → Point (0, -2)
- When $$ x = 2 $$, $$ y = -2 - 2 = -4 $$ → Point (2, -4)

**Draw the Line:**
Since the inequality is $$ \leq $$, use a **solid line** to include the boundary.

#### b. $$ y = -5x + 2 $$

- **Slope (m):** -5
- **y-intercept (b):** 2

**Plotting Points:**
- When $$ x = 0 $$, $$ y = -0 + 2 = 2 $$ → Point (0, 2)
- When $$ x = 1 $$, $$ y = -5(1) + 2 = -3 $$ → Point (1, -3)

**Draw the Line:**
Since the inequality is $$ \geq $$, use a **solid line** to include the boundary.

### 2. Determine the Shading Regions

Each inequality determines a region of the plane to shade.

#### a. $$ y \leq -x - 2 $$

- **Shade Below:** Choose a test point not on the line, such as (0,0).
  
  Substitute into the inequality:
  
  $$ 0 \leq -0 - 2 $$ → $$ 0 \leq -2 $$ → **False**

- **Therefore, shade the region **below** the line $$ y = -x - 2 $$.

#### b. $$ y \geq -5x + 2 $$

- **Shade Above:** Choose a test point not on the line, such as (0,0).
  
  Substitute into the inequality:
  
  $$ 0 \geq -5(0) + 2 $$ → $$ 0 \geq 2 $$ → **False**

- **Therefore, shade the region **above** the line $$ y = -5x + 2 $$.

### 3. Identify the Solution Region

The solution to the system of inequalities is the **overlap** of the two shaded regions:

- **Below** the line $$ y = -x - 2 $$
- **Above** the line $$ y = -5x + 2 $$

### 4. Find the Intersection Point (Optional)

To better understand the region, find where the two lines intersect:

Set $$ -x - 2 = -5x + 2 $$:

$$
\begin{align*}
- x - 2 &= -5x + 2 \
- x + 5x &= 2 + 2 \
4x &= 4 \
x &= 1 \
y &= -1 - 2 = -3 \
\end{align*}
$$

**Intersection Point:** (1, -3)

### 5. Sketch the Graph

Here's a rough sketch to visualize the system:

```
y
|
4 |                
3 |                
2 |-------* (0,2)-------- y = -5x + 2
1 |                /
0 |              /
-1|            / 
-2|-------*--------- y = -x - 2
-3|          \
-4|            \
-5|               
  |------------------------- x
     -2  -1   0   1   2
```

- **Solid Lines:** Represent the boundary lines $$ y = -x - 2 $$ and $$ y = -5x + 2 $$.
- **Shaded Region:** The area below $$ y = -x - 2 $$ and above $$ y = -5x + 2 $$. This region is unbounded and extends indefinitely in the directions that satisfy both inequalities.

### 6. Final Tips

- **Solid vs. Dashed Lines:** Use solid lines for $$ \leq $$ and $$ \geq $$ inequalities (inclusive). Use dashed lines for $$ < $$ and $$ > $$ (exclusive).
- **Test Points:** Always use a test point to determine which side of the line to shade, especially if the shading isn't immediately clear.
- **Intersection Points:** Calculating where boundary lines intersect can help in accurately sketching the solution region.

By following these steps, you can successfully graph the system of inequalities and identify all solutions that satisfy both conditions.
To graph the system of inequalities: $$ \begin{cases} y \leq -x - 2 \ y \geq -5x + 2 \end{cases} $$ 1. **Graph the boundary lines** $$ y = -x - 2 $$ and $$ y = -5x + 2 $$ as solid lines. 2. **Shade below** $$ y = -x - 2 $$ and **above** $$ y = -5x + 2 $$. 3. **Find the intersection point** at (1, -3). 4. The **overlapping shaded region** represents the solution to the system. Here's a rough sketch: ``` y | 4 | 3 | 2 |-------* (0,2)-------- y = -5x + 2 1 | / 0 | / -1| / -2|-------*--------- y = -x - 2 -3| \ -4| \ -5| |------------------------- x -2 -1 0 1 2 ``` This shaded area is the solution to the system of inequalities.

30) Graph each system of linear inequalities: a. \( \begin{array}{l}y \leq-x-2 \\ y \geq-5 x+2\end{array} \)

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