Let’s solve the system of inequalities step by step to understand the feasible region where the solutions lie.
### Step 1: Understand the Inequalities
We are given two inequalities:
1. [tex]\( y \leq 2x + 1 \)[/tex]
2. [tex]\( y < -x - 1 \)[/tex]
### Step 2: Graph the Lines
First, we graph the corresponding lines:
1. Line 1: [tex]\( y = 2x + 1 \)[/tex]
2. Line 2: [tex]\( y = -x - 1 \)[/tex]
### Step 3: Identify the Regions for the Inequalities
1. For the inequality [tex]\( y \leq 2x + 1 \)[/tex]:
- We shade the region below or on the line [tex]\( y = 2x + 1 \)[/tex].
2. For the inequality [tex]\( y < -x - 1 \)[/tex]:
- We shade the region strictly below the line [tex]\( y = -x - 1 \)[/tex]. Note that this does not include the line itself because it is a strict inequality.
### Step 4: Find the Intersection Point of the Lines
To find the intersection point of the two lines, set [tex]\( y = 2x + 1 \)[/tex] equal to [tex]\( y = -x - 1 \)[/tex]:
[tex]\[
2x + 1 = -x - 1
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
2x + x + 1 = -1
\][/tex]
[tex]\[
3x + 1 = -1
\][/tex]
[tex]\[
3x = -2
\][/tex]
[tex]\[
x = -\frac{2}{3}
\][/tex]
Now, substitute [tex]\( x = -\frac{2}{3} \)[/tex] back into either equation to find [tex]\( y \)[/tex]. Let's use [tex]\( y = 2x + 1 \)[/tex]:
[tex]\[
y = 2\left(-\frac{2}{3}\right) + 1
\][/tex]
[tex]\[
y = -\frac{4}{3} + 1
\][/tex]
[tex]\[
y = -\frac{4}{3} + \frac{3}{3}
\][/tex]
[tex]\[
y = -\frac{1}{3}
\][/tex]
So, the intersection point is [tex]\( \left( -\frac{2}{3}, -\frac{1}{3} \right) \)[/tex].
### Step 5: Graphical Interpretation of the Feasible Region
1. Graph [tex]\( y = 2x + 1 \)[/tex]:
- It is a line with a slope of 2 and a y-intercept of 1.
- Shade below this line including the line itself.
2. Graph [tex]\( y = -x - 1 \)[/tex]:
- It is a line with a slope of -1 and a y-intercept of -1.
- Shade strictly below this line, but do not include the line itself.
### Step 6: Combine the Shaded Regions
The feasible region is the overlap of the shaded areas:
- It is below and including the line [tex]\( y = 2x + 1 \)[/tex].
- It is strictly below the line [tex]\( y = -x - 1 \)[/tex].
### Conclusion
The intersection provides us with the boundary point of the feasible region:
[tex]\[
\left( -\frac{2}{3}, -\frac{1}{3} \right) \approx (-0.67, -0.33)
\][/tex]
Note that the feasible region is to the left and below this intersection point.
