Which graph shows the solution to the system of linear inequalities?
[tex]\begin{array}{l}
2 x-3 y \leq 12 \\
y\ \textless \ -3
\end{array}[/tex]
Answer (1)
Analyze the first inequality 2 x − 3 y ≤ 12 and rewrite it as y ≥ 3 2 x − 4 .
Analyze the second inequality y < − 3 .
Find the intersection of the two regions, which is the region above the line 2 x − 3 y = 12 and below the line y = − 3 .
Identify the graph that shows this region.
Explanation
Understanding the Problem We are given the following system of linear inequalities:
2 x − 3 y ≤ 12 y < − 3
We need to find the graph that represents the solution set of this system.
Analyzing the Inequalities First, let's analyze the first inequality: 2 x − 3 y ≤ 12 . We can rewrite this as y ≥ 3 2 x − 4 . This means the region above the line y = 3 2 x − 4 is the solution region.
Now, let's analyze the second inequality: y < − 3 . This means the region below the line y = − 3 is the solution region.
Finding the Intersection To find the solution to the system of inequalities, we need to find the intersection of the two regions. This is the region that satisfies both inequalities.
The line 2 x − 3 y = 12 has x-intercept at x = 6 (when y = 0 ) and y-intercept at y = − 4 (when x = 0 ). The region satisfying 2 x − 3 y ≤ 12 is above this line.
The line y = − 3 is a horizontal line. The region satisfying y < − 3 is below this line.
Identifying the Correct Graph The solution is the region that is both above the line 2 x − 3 y = 12 and below the line y = − 3 . We are looking for a graph that shows this region.
Conclusion The correct graph will show the region below the horizontal line y = − 3 and above the line 2 x − 3 y = 12 .
Examples
Systems of linear inequalities are used in various real-world applications, such as optimizing resource allocation, determining feasible regions in linear programming, and modeling constraints in economics and engineering. For example, a company might use a system of inequalities to determine the optimal production levels of two products, given constraints on resources like labor and materials. By graphing the inequalities, the company can visualize the feasible region and identify the production levels that maximize profit while satisfying all constraints.
