Answer :
Sure, let's analyze and graph the solution for the system of inequalities:
[tex]\[ \begin{array}{l} x + y < 4 \\ 2x - 3y \geq 12 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the Boundary Lines:
- For the inequality [tex]\(x + y < 4 \)[/tex]:
- Convert the inequality to an equation: [tex]\(x + y = 4\)[/tex].
- To graph [tex]\(x + y = 4\)[/tex], find the intercepts:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex].
- When [tex]\(y = 0\)[/tex], [tex]\(x = 4\)[/tex].
- Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((4, 0)\)[/tex] and draw a dashed line through them because the inequality is strict (less than, not less than or equal to).
- For the inequality [tex]\(2x - 3y \geq 12 \)[/tex]:
- Convert the inequality to an equation: [tex]\(2x - 3y = 12\)[/tex].
- To graph [tex]\(2x - 3y = 12\)[/tex], find the intercepts:
- When [tex]\(x = 0\)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \][/tex]
- When [tex]\(y = 0\)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((6, 0)\)[/tex] and draw a solid line through them because the inequality is inclusive (greater than or equal to).
2. Determine the Shaded Regions:
- For [tex]\(x + y < 4 \)[/tex]:
- Choose a test point not on the line, like [tex]\((0, 0)\)[/tex]:
[tex]\[ 0 + 0 < 4 \implies 0 < 4 \quad \text{(True)} \][/tex]
- Shade the region below the line, as it satisfies the inequality.
- For [tex]\(2x - 3y \geq 12 \)[/tex]:
- Choose a test point not on the line, like [tex]\((0, 0)\)[/tex]:
[tex]\[ 2(0) - 3(0) \geq 12 \implies 0 \geq 12 \quad \text{(False)} \][/tex]
- Shade the region above the line, as it satisfies the inequality.
3. Identify the Solution Region:
- The solution is the intersection of the two shaded regions. This requires both conditions to be satisfied simultaneously.
Given these steps, let’s visualize the appropriate graph:
- A dashed line for [tex]\(x + y = 4\)[/tex] with shading below the line.
- A solid line for [tex]\(2x - 3y = 12\)[/tex] with shading above the line.
- The solution is the area where these two shaded regions overlap.
I don’t have visual content to show you, but you can draw these on the coordinate plane and find the overlapping region by shading accordingly. The correct graph will accurately reflect these details.
[tex]\[ \begin{array}{l} x + y < 4 \\ 2x - 3y \geq 12 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the Boundary Lines:
- For the inequality [tex]\(x + y < 4 \)[/tex]:
- Convert the inequality to an equation: [tex]\(x + y = 4\)[/tex].
- To graph [tex]\(x + y = 4\)[/tex], find the intercepts:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex].
- When [tex]\(y = 0\)[/tex], [tex]\(x = 4\)[/tex].
- Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((4, 0)\)[/tex] and draw a dashed line through them because the inequality is strict (less than, not less than or equal to).
- For the inequality [tex]\(2x - 3y \geq 12 \)[/tex]:
- Convert the inequality to an equation: [tex]\(2x - 3y = 12\)[/tex].
- To graph [tex]\(2x - 3y = 12\)[/tex], find the intercepts:
- When [tex]\(x = 0\)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \][/tex]
- When [tex]\(y = 0\)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((6, 0)\)[/tex] and draw a solid line through them because the inequality is inclusive (greater than or equal to).
2. Determine the Shaded Regions:
- For [tex]\(x + y < 4 \)[/tex]:
- Choose a test point not on the line, like [tex]\((0, 0)\)[/tex]:
[tex]\[ 0 + 0 < 4 \implies 0 < 4 \quad \text{(True)} \][/tex]
- Shade the region below the line, as it satisfies the inequality.
- For [tex]\(2x - 3y \geq 12 \)[/tex]:
- Choose a test point not on the line, like [tex]\((0, 0)\)[/tex]:
[tex]\[ 2(0) - 3(0) \geq 12 \implies 0 \geq 12 \quad \text{(False)} \][/tex]
- Shade the region above the line, as it satisfies the inequality.
3. Identify the Solution Region:
- The solution is the intersection of the two shaded regions. This requires both conditions to be satisfied simultaneously.
Given these steps, let’s visualize the appropriate graph:
- A dashed line for [tex]\(x + y = 4\)[/tex] with shading below the line.
- A solid line for [tex]\(2x - 3y = 12\)[/tex] with shading above the line.
- The solution is the area where these two shaded regions overlap.
I don’t have visual content to show you, but you can draw these on the coordinate plane and find the overlapping region by shading accordingly. The correct graph will accurately reflect these details.
