Choose the correct solution.
Note: Only the overlapping solution region is shaded. If a graph has no shading, it means that there is no overlap.
-2x-3y≥ 12
x - 5y ≤ 20
x≥0
Answer (2)
Analyze the inequalities: -2x-3y≥ 12, x - 5y ≤ 20, and x≥0.
Rewrite the first inequality: 2x + 3y ≤ -12.
Identify the region that satisfies all three inequalities simultaneously.
The solution region is the intersection of the three regions defined by the inequalities.
Explanation
Understanding the Inequalities We are given three inequalities:
-2x-3y≥ 12
x - 5y ≤ 20
x≥0
We need to find the region that satisfies all three inequalities simultaneously.
Analyzing Each Inequality Let's analyze each inequality separately:
-2x - 3y ≥ 12 can be rewritten as 2x + 3y ≤ -12. To graph this, we first consider the equation 2x + 3y = -12. When x = 0, y = -4. When y = 0, x = -6. Since the inequality is ≤, we shade the region below the line.
x - 5y ≤ 20. To graph this, we first consider the equation x - 5y = 20. When x = 0, y = -4. When y = 0, x = 20. Since the inequality is ≤, we shade the region above the line.
x ≥ 0 means we only consider the region to the right of the y-axis.
Finding the Overlapping Region Now, we need to find the overlapping region of all three inequalities. The first inequality, 2x + 3y ≤ -12, is in the third and fourth quadrants. The second inequality, x - 5y ≤ 20, includes the fourth quadrant and extends to the first quadrant. The third inequality, x ≥ 0, restricts the region to the right of the y-axis.
Determining the Solution Region Considering all three inequalities, we look for the region that is below the line 2x + 3y = -12, above the line x - 5y = 20, and to the right of the y-axis (x ≥ 0). Since 2x + 3y ≤ -12 is always negative, and x ≥ 0, the overlapping region will be in the lower half of the coordinate plane.
Identifying the Correct Graph The solution region is the intersection of the three regions defined by the inequalities. We need to identify the graph that correctly represents this intersection.
Conclusion Without the ability to see the graphs, it's challenging to definitively choose the correct solution. However, we know the following:
The region must be to the right of the y-axis (x ≥ 0).
The region must be below the line 2x + 3y = -12.
The region must be above the line x - 5y = 20.
Examples
Understanding systems of inequalities is crucial in various real-world applications, such as resource allocation, optimization problems, and economics. For instance, a company might use inequalities to determine the optimal production levels of different products, considering constraints like available resources, production costs, and market demand. By graphing these inequalities, the company can visually identify the feasible region, representing all possible production combinations that satisfy the constraints. This helps in making informed decisions to maximize profit or minimize costs.
The solution region includes areas to the right of the y-axis where one inequality is met below the line y = − 3 2 x − 4 and above the line y = 5 1 x − 4 . When these inequalities are graphically represented, the overlapping region in the fourth quadrant should be correctly shaded. Since no visual options are provided, carefully examining the final graphical representation is key.
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