Solving a System of Linear Inequalities: 3x + 4y ≥ 150, x + y ≥ 40

Math Problem Statement

Solve the system of inequalities shown in the image.

The image you uploaded contains the following system of inequalities:

3x + 4y &\geq 150 \\ x + y &\geq 40 \\ x &\geq 0 \\ y &\geq 0 \end{aligned}$$ This system of inequalities represents a feasible region in two dimensions, where $$x$$ and $$y$$ must satisfy these constraints. Typically, this would be used in a linear programming problem or optimization to find a solution such as maximizing or minimizing a function, subject to these constraints. Would you like further details on solving these inequalities, graphing the solution, or applying them to an optimization problem? Here are 5 related questions: 1. How can we graph these inequalities and find the feasible region? 2. What is the significance of the boundary lines in the system of inequalities? 3. Can we apply this system to solve a linear programming problem? 4. How do you interpret the solution set of this system? 5. What changes if the inequality signs are reversed or strict (e.g., $$>$$ instead of $$\geq$$)? **Tip**: When graphing systems of inequalities, always start by graphing the boundary lines (replace inequalities with equalities) and then determine which side of the boundary satisfies the inequality.

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Linear Inequalities
Systems of Inequalities
Optimization

Ax + By ≥ C
x ≥ 0
y ≥ 0

Feasible Region in Linear Programming
Boundary Line and Half-Plane

Grades 9-12

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