To solve this system of linear inequalities graphically, we'll start with the first inequality:
[tex]\[ y \leq -2x - 3 \][/tex]
### Step-by-Step Solution for Graphing the First Inequality
1. Identify the Boundary Line
The boundary line for the inequality [tex]\( y \leq -2x - 3 \)[/tex] is [tex]\( y = -2x - 3 \)[/tex]. Since the inequality is "less than or equal to," we will use a solid line to represent the boundary.
2. Find Two Points on the Boundary Line
To graph the boundary line, we need at least two points.
a. When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -2(0) - 3 = -3
\][/tex]
So, one point is [tex]\( (0, -3) \)[/tex].
b. When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = -2(2) - 3 = -4 - 3 = -7
\][/tex]
So, another point is [tex]\( (2, -7) \)[/tex].
These two points are sufficient to draw the line.
3. Graph the Boundary Line
Draw a solid line passing through the points [tex]\( (0, -3) \)[/tex] and [tex]\( (2, -7) \)[/tex].
4. Shade the Appropriate Region
The inequality is [tex]\( y \leq -2x - 3 \)[/tex]. This means we shade the region below the line because it includes all points where [tex]\( y \)[/tex] is less than or equal to [tex]\( -2x - 3 \)[/tex].
### Summary for Step 1:
- Draw a solid line through the points [tex]\( (0, -3) \)[/tex] and [tex]\( (2, -7) \)[/tex].
- Shade the region below this line to represent the inequality [tex]\( y \leq -2x - 3 \)[/tex].
### Visual Representation
To visualize, here's a rough sketch of the line and shading:
```
y
10|
9|
8|
7|
6|
5| (2,-7)
4|
3|
2|
1|
0|---------------------------------- x
-1|
-2|
-3| (0,-3)
-4|
-5|
-6|
-7|
-8|
-9|
-10|
```
The area below this solid line will be shaded to represent [tex]\( y \leq -2x - 3 \)[/tex].
Next, we will move on to graph the solution set for the second inequality.
