Math Problem Statement

Which set of constraints below correctly bound the feasible region shown in the graph?

Which set of constraints below correctly bound the feasible region shown in the graph?

Solution

To determine the correct set of inequalities that bound the feasible region shown in the graph, let's analyze each line and shading in the graph.

Analysis of the Graph

  1. Green Line (Bottom Boundary): This line seems to be the x-axis constraint, meaning y0y \geq 0.
  2. Blue Line (Left Boundary): The line has a negative slope and appears to represent a linear inequality. It restricts the region to the left and below this line.
  3. Black Line (Diagonal Boundary): This line appears to have a positive slope and seems to bound the region on the right side.

Step-by-Step Solution

1. Checking Inequality Conditions

  • x-axis Constraint: As this feasible region is in the upper part of the graph, y0y \geq 0 is likely one of the constraints.
  • y-axis Constraint: The region is to the right of the y-axis, so we expect x0x \geq 0 to also be a constraint.
  • Diagonal Boundaries: To determine which inequalities match the blue and black lines, we would test the points or compare with the options.

2. Options Analysis

Let's match each option to see which correctly bounds the feasible region:

  • Option (a):

    • 2x3y122x - 3y \leq 12
    • y2x+4y \leq -2x + 4
    • y0y \geq 0
    • x0x \geq 0
  • Option (b):

    • 2x+3y122x + 3y \leq 12
    • y2x+4y \leq -2x + 4
    • y0y \geq 0
    • x0x \geq 0
  • Option (c):

    • 2x+3y122x + 3y \leq 12
    • y2x+4y \geq -2x + 4
    • y0y \geq 0
    • x0x \geq 0
  • Option (d):

    • 2x3y122x - 3y \leq 12
    • y2x+4y \geq -2x + 4
    • y0y \geq 0
    • x0x \geq 0

Conclusion

Based on the boundaries provided and the shading, Option (b) seems to be the correct answer as it matches the constraints on the feasible region's boundaries.

Would you like further details on how each boundary was matched to the inequalities?


Related Questions

  1. How do you determine the feasible region in a system of linear inequalities?
  2. What methods can be used to solve systems of linear inequalities graphically?
  3. How can you verify if a point is within the feasible region of a system of inequalities?
  4. Why is it important to check boundary conditions when working with inequalities?
  5. How do constraints affect the shape and location of the feasible region in a graph?

Tip:

Always start by identifying the basic boundaries, like the axes, when interpreting the feasible region on a graph.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Inequalities
Graphing Inequalities

Formulas

Standard form of a linear equation: Ax + By ≤ C
Inequalities on x-axis and y-axis (x ≥ 0, y ≥ 0)

Theorems

System of Linear Inequalities
Boundary Conditions in Graphs

Suitable Grade Level

Grade 9-11