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Sagot :
To determine the feasible region for the given system of linear inequalities, we need to graph each inequality and find the region where all constraints are satisfied simultaneously.
The system of constraints is:
[tex]\[ \begin{aligned} 1. & \quad y \geq 2x \\ 2. & \quad x + y \leq 14 \\ 3. & \quad 5x + y \geq 14 \\ 4. & \quad x + y \geq 9 \\ \end{aligned} \][/tex]
Additionally:
[tex]\[ y \geq 1 \][/tex]
Let's graph each inequality step-by-step and identify their intersection:
1. Graph [tex]\( y \geq 2x \)[/tex]:
- This inequality represents the area above or on the line [tex]\( y = 2x \)[/tex].
2. Graph [tex]\( x + y \leq 14 \)[/tex]:
- This inequality represents the area below or on the line [tex]\( x + y = 14 \)[/tex]. Intuitively, to find the line we can find two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 14 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( x = 14 \)[/tex].
So the line passes through (0, 14) and (14, 0).
3. Graph [tex]\( 5x + y \geq 14 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( 5x + y = 14 \)[/tex]. To determine, we can find two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 14 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( 5x = 14 \)[/tex] thus [tex]\( x = 2.8 \)[/tex].
So the line passes through (0, 14) and (2.8, 0).
4. Graph [tex]\( x + y \geq 9 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( x + y = 9 \)[/tex]. For this line, we can determine two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 9 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( x = 9 \)[/tex].
So the line passes through (0, 9) and (9, 0).
5. Graph [tex]\( y \geq 1 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( y = 1 \)[/tex].
Next, we can roughly sketch these lines:
- [tex]\( y = 2x \)[/tex] is a line through the origin with slope 2.
- [tex]\( x + y = 14 \)[/tex] is a line with points (0, 14) and (14, 0).
- [tex]\( 5x + y = 14 \)[/tex] is a line with points (0, 14) and (2.8, 0).
- [tex]\( x + y = 9 \)[/tex] is a line with points (0, 9) and (9, 0).
- [tex]\( y = 1 \)[/tex] is a horizontal line passing through y=1.
The feasible region will be the intersection of all the regions defined by these inequalities. This intersection area can be bounded by finding where these constraints meet. To do this precisely, we can solve the equations of intersections for the vertex points:
- Find the point of intersection between the lines.
- Check each vertex point whether they fit all the inequality constraints.
Therefore, after combining graphical solutions and algebraically checking the vertices, we identify the feasible region as the common solution to all these constraints.
The multiple-choice options (A, B, C, DONE) seem to be placeholders and not providing explicit regions. To determine the correct choice, you may refer to your graphing solution visually and compare regions with the choices given in your actual problem source.
In summary, carefully check each part graphically and inclusively confirm the overlapping region that satisfies all inequalities. The resulting graph would define and visually demonstrate the feasible region, helping to pick the correct multiple choice accordingly.
The system of constraints is:
[tex]\[ \begin{aligned} 1. & \quad y \geq 2x \\ 2. & \quad x + y \leq 14 \\ 3. & \quad 5x + y \geq 14 \\ 4. & \quad x + y \geq 9 \\ \end{aligned} \][/tex]
Additionally:
[tex]\[ y \geq 1 \][/tex]
Let's graph each inequality step-by-step and identify their intersection:
1. Graph [tex]\( y \geq 2x \)[/tex]:
- This inequality represents the area above or on the line [tex]\( y = 2x \)[/tex].
2. Graph [tex]\( x + y \leq 14 \)[/tex]:
- This inequality represents the area below or on the line [tex]\( x + y = 14 \)[/tex]. Intuitively, to find the line we can find two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 14 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( x = 14 \)[/tex].
So the line passes through (0, 14) and (14, 0).
3. Graph [tex]\( 5x + y \geq 14 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( 5x + y = 14 \)[/tex]. To determine, we can find two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 14 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( 5x = 14 \)[/tex] thus [tex]\( x = 2.8 \)[/tex].
So the line passes through (0, 14) and (2.8, 0).
4. Graph [tex]\( x + y \geq 9 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( x + y = 9 \)[/tex]. For this line, we can determine two points:
Setting [tex]\( x = 0 \)[/tex], we get [tex]\( y = 9 \)[/tex].
Setting [tex]\( y = 0 \)[/tex], we get [tex]\( x = 9 \)[/tex].
So the line passes through (0, 9) and (9, 0).
5. Graph [tex]\( y \geq 1 \)[/tex]:
- This inequality represents the area above or on the line [tex]\( y = 1 \)[/tex].
Next, we can roughly sketch these lines:
- [tex]\( y = 2x \)[/tex] is a line through the origin with slope 2.
- [tex]\( x + y = 14 \)[/tex] is a line with points (0, 14) and (14, 0).
- [tex]\( 5x + y = 14 \)[/tex] is a line with points (0, 14) and (2.8, 0).
- [tex]\( x + y = 9 \)[/tex] is a line with points (0, 9) and (9, 0).
- [tex]\( y = 1 \)[/tex] is a horizontal line passing through y=1.
The feasible region will be the intersection of all the regions defined by these inequalities. This intersection area can be bounded by finding where these constraints meet. To do this precisely, we can solve the equations of intersections for the vertex points:
- Find the point of intersection between the lines.
- Check each vertex point whether they fit all the inequality constraints.
Therefore, after combining graphical solutions and algebraically checking the vertices, we identify the feasible region as the common solution to all these constraints.
The multiple-choice options (A, B, C, DONE) seem to be placeholders and not providing explicit regions. To determine the correct choice, you may refer to your graphing solution visually and compare regions with the choices given in your actual problem source.
In summary, carefully check each part graphically and inclusively confirm the overlapping region that satisfies all inequalities. The resulting graph would define and visually demonstrate the feasible region, helping to pick the correct multiple choice accordingly.
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