Answer :
To determine the correct graph representing the solution to the system of inequalities
[tex]\[ \begin{array}{l} x+y<4 \\ 2 x-3 y \geq 12 \end{array} \][/tex]
let's proceed by analyzing each inequality one at a time.
### Analyzing Inequality [tex]\( x + y < 4 \)[/tex]
1. Find the boundary line: The boundary line can be found by converting the inequality into an equation:
[tex]\[ x + y = 4 \][/tex]
2. Finding points on the line:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 + y = 4 \implies y = 4 \quad \Rightarrow \text{Point} (0, 4) \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x + 0 = 4 \implies x = 4 \quad \Rightarrow \text{Point} (4, 0) \][/tex]
3. Plot the line: Plot the line passing through points [tex]\( (0, 4) \)[/tex] and [tex]\( (4, 0) \)[/tex].
4. Shade the region: Since the inequality is [tex]\( x + y < 4 \)[/tex], shade the region below the line [tex]\( x + y = 4 \)[/tex].
### Analyzing Inequality [tex]\( 2x - 3y \geq 12 \)[/tex]
1. Find the boundary line: The boundary line can be found by converting the inequality into an equation:
[tex]\[ 2x - 3y = 12 \][/tex]
2. Finding points on the line:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \quad \Rightarrow \text{Point} (0, -4) \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \quad \Rightarrow \text{Point} (6, 0) \][/tex]
3. Plot the line: Plot the line passing through points [tex]\( (0, -4) \)[/tex] and [tex]\( (6, 0) \)[/tex].
4. Shade the region: Since the inequality is [tex]\( 2x - 3y \geq 12 \)[/tex], shade the region above the line [tex]\( 2x - 3y = 12 \)[/tex].
### Combine the Solutions
To represent the solution to the system of inequalities, identify the region that satisfies both conditions:
- The region below the line [tex]\( x + y < 4 \)[/tex]
- The region above the line [tex]\( 2x - 3y \geq 12 \)[/tex]
Combining these, the feasible region is where the shaded areas of both inequalities overlap.
Using the determined points and shading instructions, the points used on the graph will be:
[tex]\[ \text{Line 1: through } (0, 4) \text{ and } (4, 0) \][/tex]
[tex]\[ \text{Line 2: through } (0, -4) \text{ and } (6, 0) \][/tex]
The graph should show the lines connecting these points and shading for the specified regions. Find the graph that aligns with these plot points and shading instructions to find the correct representation.
[tex]\[ \begin{array}{l} x+y<4 \\ 2 x-3 y \geq 12 \end{array} \][/tex]
let's proceed by analyzing each inequality one at a time.
### Analyzing Inequality [tex]\( x + y < 4 \)[/tex]
1. Find the boundary line: The boundary line can be found by converting the inequality into an equation:
[tex]\[ x + y = 4 \][/tex]
2. Finding points on the line:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 + y = 4 \implies y = 4 \quad \Rightarrow \text{Point} (0, 4) \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x + 0 = 4 \implies x = 4 \quad \Rightarrow \text{Point} (4, 0) \][/tex]
3. Plot the line: Plot the line passing through points [tex]\( (0, 4) \)[/tex] and [tex]\( (4, 0) \)[/tex].
4. Shade the region: Since the inequality is [tex]\( x + y < 4 \)[/tex], shade the region below the line [tex]\( x + y = 4 \)[/tex].
### Analyzing Inequality [tex]\( 2x - 3y \geq 12 \)[/tex]
1. Find the boundary line: The boundary line can be found by converting the inequality into an equation:
[tex]\[ 2x - 3y = 12 \][/tex]
2. Finding points on the line:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \quad \Rightarrow \text{Point} (0, -4) \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \quad \Rightarrow \text{Point} (6, 0) \][/tex]
3. Plot the line: Plot the line passing through points [tex]\( (0, -4) \)[/tex] and [tex]\( (6, 0) \)[/tex].
4. Shade the region: Since the inequality is [tex]\( 2x - 3y \geq 12 \)[/tex], shade the region above the line [tex]\( 2x - 3y = 12 \)[/tex].
### Combine the Solutions
To represent the solution to the system of inequalities, identify the region that satisfies both conditions:
- The region below the line [tex]\( x + y < 4 \)[/tex]
- The region above the line [tex]\( 2x - 3y \geq 12 \)[/tex]
Combining these, the feasible region is where the shaded areas of both inequalities overlap.
Using the determined points and shading instructions, the points used on the graph will be:
[tex]\[ \text{Line 1: through } (0, 4) \text{ and } (4, 0) \][/tex]
[tex]\[ \text{Line 2: through } (0, -4) \text{ and } (6, 0) \][/tex]
The graph should show the lines connecting these points and shading for the specified regions. Find the graph that aligns with these plot points and shading instructions to find the correct representation.
