Answer :
Let's graph the solution of the given system step by step.
The system of inequalities is:
[tex]\[ \begin{array}{lr} x - 3y \geq -3 \\ 2x + y \leq 6 \end{array} \][/tex]
We will graph each inequality one by one and then determine the region of the graph that satisfies both inequalities.
### Step 1: Graph [tex]\( x - 3y \geq -3 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ x - 3y = -3 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ x - 3(0) = -3 \implies x = -3 \][/tex]
So, the x-intercept is [tex]\((-3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 0 - 3y = -3 \implies y = 1 \][/tex]
So, the y-intercept is [tex]\((0, 1)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≥).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 0 - 3(0) \geq -3 \implies 0 \geq -3 \][/tex]
This is true, so we shade the region that includes the point [tex]\((0, 0)\)[/tex].
### Step 2: Graph [tex]\( 2x + y \leq 6 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ 2x + y = 6 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ 2x + 0 = 6 \implies x = 3 \][/tex]
So, the x-intercept is [tex]\((3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 2(0) + y = 6 \implies y = 6 \][/tex]
So, the y-intercept is [tex]\((0, 6)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≤).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 2(0) + 0 \leq 6 \implies 0 \leq 6 \][/tex]
This is true, so we shade the region that includes the point [tex]\( (0, 0) \)[/tex].
### Step 3: Determine the solution region
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region satisfies both [tex]\( x - 3y \geq -3 \)[/tex] and [tex]\( 2x + y \leq 6 \)[/tex].
To graph the solution region:
1. Draw the boundary lines based on the intercept points and the information obtained from the inequalities' boundary lines.
2. Shade accordingly: The final region of interest that satisfies both inequalities is the overlapping shaded region from both inequalities.
Here is a sketch of the graph:
- Draw the boundary line for [tex]\( x - 3y = -3 \)[/tex] passing through [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex], and shade above the line.
- Draw the boundary line for [tex]\( 2x + y = 6 \)[/tex] passing through [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex], and shade below the line.
The intersection of these shaded regions represents the solution to the system.
The system of inequalities is:
[tex]\[ \begin{array}{lr} x - 3y \geq -3 \\ 2x + y \leq 6 \end{array} \][/tex]
We will graph each inequality one by one and then determine the region of the graph that satisfies both inequalities.
### Step 1: Graph [tex]\( x - 3y \geq -3 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ x - 3y = -3 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ x - 3(0) = -3 \implies x = -3 \][/tex]
So, the x-intercept is [tex]\((-3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 0 - 3y = -3 \implies y = 1 \][/tex]
So, the y-intercept is [tex]\((0, 1)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≥).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 0 - 3(0) \geq -3 \implies 0 \geq -3 \][/tex]
This is true, so we shade the region that includes the point [tex]\((0, 0)\)[/tex].
### Step 2: Graph [tex]\( 2x + y \leq 6 \)[/tex]
1. Convert to equality: Rewrite the inequality as an equation to find the boundary line:
[tex]\[ 2x + y = 6 \][/tex]
2. Find intercepts:
- For the x-intercept (when [tex]\( y = 0 \)[/tex]):
[tex]\[ 2x + 0 = 6 \implies x = 3 \][/tex]
So, the x-intercept is [tex]\((3, 0)\)[/tex].
- For the y-intercept (when [tex]\( x = 0 \)[/tex]):
[tex]\[ 2(0) + y = 6 \implies y = 6 \][/tex]
So, the y-intercept is [tex]\((0, 6)\)[/tex].
3. Draw the boundary line: Plot the points [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex] and draw a solid line through them, as the inequality includes the boundary line (≤).
4. Shade the region: To determine which side of the line to shade, use a test point (e.g., [tex]\( (0, 0) \)[/tex]):
[tex]\[ 2(0) + 0 \leq 6 \implies 0 \leq 6 \][/tex]
This is true, so we shade the region that includes the point [tex]\( (0, 0) \)[/tex].
### Step 3: Determine the solution region
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region satisfies both [tex]\( x - 3y \geq -3 \)[/tex] and [tex]\( 2x + y \leq 6 \)[/tex].
To graph the solution region:
1. Draw the boundary lines based on the intercept points and the information obtained from the inequalities' boundary lines.
2. Shade accordingly: The final region of interest that satisfies both inequalities is the overlapping shaded region from both inequalities.
Here is a sketch of the graph:
- Draw the boundary line for [tex]\( x - 3y = -3 \)[/tex] passing through [tex]\((-3, 0)\)[/tex] and [tex]\((0, 1)\)[/tex], and shade above the line.
- Draw the boundary line for [tex]\( 2x + y = 6 \)[/tex] passing through [tex]\((3, 0)\)[/tex] and [tex]\((0, 6)\)[/tex], and shade below the line.
The intersection of these shaded regions represents the solution to the system.
