Certainly! Let's solve and graph the system of inequalities step-by-step.
### Step 1: Understand the inequalities
We have the following inequalities:
1. [tex]\( x + y \geq 4 \)[/tex]
2. [tex]\( y - x \leq 1 \)[/tex]
### Step 2: Rewrite each inequality in a more familiar form
#### Inequality 1: [tex]\( x + y \geq 4 \)[/tex]
Rewriting this inequality, we get:
[tex]\[ y \geq 4 - x \][/tex]
This means, for any point [tex]\((x, y)\)[/tex] in the coordinate plane, [tex]\( y \)[/tex] must be greater than or equal to [tex]\( 4 - x \)[/tex]. The boundary line for this inequality is [tex]\( y = 4 - x \)[/tex], and the region of interest lies above this line.
#### Inequality 2: [tex]\( y - x \leq 1 \)[/tex]
Rewriting this inequality, we get:
[tex]\[ y \leq x + 1 \][/tex]
This means, for any point [tex]\((x, y)\)[/tex] in the coordinate plane, [tex]\( y \)[/tex] must be less than or equal to [tex]\( x + 1 \)[/tex]. The boundary line for this inequality is [tex]\( y = x + 1 \)[/tex], and the region of interest lies below this line.
### Step 3: Plot the boundary lines
- [tex]\( y = 4 - x \)[/tex] (from Inequality 1)
- [tex]\( y = x + 1 \)[/tex] (from Inequality 2)
### Step 4: Determine the regions to shade
- For [tex]\( y \geq 4 - x \)[/tex], shade the area above the line [tex]\( y = 4 - x \)[/tex].
- For [tex]\( y \leq x + 1 \)[/tex], shade the area below the line [tex]\( y = x + 1 \)[/tex].
### Step 5: Find the intersection of shaded regions
The solution to the system of inequalities will be the region where the shaded areas overlap.
### Step 6: Plot the solution
Let's combine everything into a comprehensive graph:
1. Plot the line [tex]\( y = 4 - x \)[/tex]. This line will have intercepts at [tex]\( (4, 0) \)[/tex] and [tex]\( (0, 4) \)[/tex].
2. Plot the line [tex]\( y = x + 1 \)[/tex]. This line will have intercepts at [tex]\( (-1, 0) \)[/tex] and [tex]\( (0, 1) \)[/tex].
To ensure clarity, let's outline both lines clearly:
[tex]\[
\begin{cases}
\text{Line 1: } y = 4 - x \\
\text{Line 2: } y = x + 1
\end{cases}
\][/tex]
### Step 7: Shade the regions appropriately
- For [tex]\( y \geq 4 - x \)[/tex], shade above the line [tex]\( y = 4 - x \)[/tex].
- For [tex]\( y \leq x + 1 \)[/tex], shade below the line [tex]\( y = x + 1 \)[/tex].
The overlapping shaded region will represent the set of points [tex]\((x, y)\)[/tex] that satisfy both inequalities.
### Final Graph Interpretation:
- Region above [tex]\( y = 4 - x \)[/tex] and below [tex]\( y = x + 1 \)[/tex]: This shared area represents the solution to the system.
To sum up, the graph plots two lines intersecting at some points and the intersection of the shaded areas above [tex]\( y = 4 - x \)[/tex] and below [tex]\( y = x + 1 \)[/tex] gives us the feasible region for the developed and open space in the planned community.
