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 (2)
Rewrite the first inequality: 2 x − 3 y ≤ 12 as y ≥ 3 2 x − 4 .
The solution region for y ≥ 3 2 x − 4 is above or on the line y = 3 2 x − 4 .
The solution region for y < − 3 is below the line y = − 3 .
The solution to the system is the intersection of these two regions.
Explanation
Understanding the Problem We are given a system of two 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 First Inequality First, let's analyze the inequality 2 x − 3 y ≤ 12 . We can rewrite this inequality as:
2 x − 3 y ≤ 12 ⟹ − 3 y ≤ − 2 x + 12 ⟹ y ≥ 3 2 x − 4
This inequality represents the region above the line y = 3 2 x − 4 , including the line itself.
Analyzing the Second Inequality Now, let's analyze the inequality y < − 3 . This inequality represents the region below the horizontal line y = − 3 , but not including the line itself.
Finding the Intersection The solution to the system of inequalities is the intersection of the two regions. We are looking for the region that is both above or on the line y = 3 2 x − 4 and below the line y = − 3 .
Visualizing the Solution The line y = 3 2 x − 4 has a y-intercept of -4 and a slope of 3 2 . The line y = − 3 is a horizontal line passing through y = − 3 . The region y < − 3 is below the line y = − 3 . The region 2 x − 3 y ≤ 12 is above or on the line 2 x − 3 y = 12 .
Examples
Linear inequalities are used in various real-world applications, such as optimizing resource allocation, determining feasible regions in production planning, and setting constraints in financial modeling. For example, a company might use a system of linear inequalities to determine the optimal production levels of different products, given constraints on available resources like labor, materials, and capital. By graphing these inequalities, the company can visualize the feasible region and identify the production levels that maximize profit while satisfying all constraints.
The solution to the system of inequalities is represented by the region above the line y = 3 2 x − 4 and below the line y = − 3 . This intersection is the feasible region of the combined inequalities. Visual representation on a graph shows the shaded area indicating this solution set.
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