Pregunta
Consider the following system of linear inequalities and corresponding graph:
Ask by Macdonald Klein. in the United States
Jan 26,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The feasible region is the area on the graph where all three inequalities
Solución
Certainly! Let’s analyze and graph the given system of linear inequalities:
Step 1: Graph the Boundary Lines
-
First Inequality:
- Boundary Line:
(a solid line because the inequality is ) - Slope (m): 2
- Y-intercept (b):
- Boundary Line:
-
Second Inequality:
- Boundary Line:
(a dashed vertical line because the inequality is )
- Boundary Line:
-
Third Inequality:
- Boundary Line:
(a dashed horizontal line because the inequality is )
- Boundary Line:
Step 2: Plot Each Boundary Line
-
Plotting
: - Start at
on the y-axis. - Use the slope to find another point: from
, rise 2 units and run 1 unit to the right to reach . - Draw a solid line through these points.
- Start at
-
Plotting
: - Draw a dashed vertical line crossing the x-axis at
.
- Draw a dashed vertical line crossing the x-axis at
-
Plotting
: - Draw a dashed horizontal line crossing the y-axis at
.
- Draw a dashed horizontal line crossing the y-axis at
Step 3: Determine the Shaded Regions
-
For
: - Shade above the line
.
- Shade above the line
-
For
: - Shade to the right of the line
.
- Shade to the right of the line
-
For
: - Shade below the line
.
- Shade below the line
Step 4: Identify the Feasible Region
The feasible region is where all shaded areas overlap. This region satisfies all three inequalities simultaneously.
Step 5: Find Intersection Points (Vertices)
To better understand the feasible region, let’s find where the boundary lines intersect:
-
Intersection of
and : -
Intersection of
and : -
Intersection of
and :
Step 6: Sketch the Graph
While I can’t provide a visual plot here, here’s a description to help you sketch it:
-
Draw the axes with appropriate scales.
-
Plot the boundary lines:
- Draw a solid line for
. - Draw a dashed vertical line at
. - Draw a dashed horizontal line at
.
- Draw a solid line for
-
Shade the regions:
- Shade above the solid line
. - Shade to the right of the dashed line
. - Shade below the dashed line
.
- Shade above the solid line
-
Identify the overlapping area, which is bounded:
- On the left by
. - Below by
. - Above by
.
- On the left by
This overlapping (feasible) region represents all the points
Step 7: Analyze the Feasible Region
-
Boundaries:
- The vertical boundary
limits the region to . - The line
serves as the lower boundary. - The horizontal line
serves as the upper boundary.
- The vertical boundary
-
Vertices (Corner Points):
- Intersection of
and : - Intersection of
and : - Intersection of
and :
- Intersection of
These vertices can be useful for optimization problems or further analysis within the feasible region.
Summary
The solution to the system of inequalities is the set of all points
- To the right of
. - Above the line
. - Below the line
.
This region is a trapezoidal area on the coordinate plane bounded by the lines mentioned above.
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución Mind Expander
The given system of linear inequalities represents a region in the Cartesian plane that we can visualize! The inequality
When solving systems like this, it’s easy to misinterpret the inequalities. A common mistake is to forget that the boundary lines are “included” or “not included” based on whether the inequality is <= or < . For instance, in the case of
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