What graph shows the correct solution for the following system of linear inequalities?
[tex]
\begin{array}{l}
y-x \geq 5 \\
y-2 x\ \textgreater \ 3
\end{array}
[/tex]
Answer (1)
Rewrite the inequalities in slope-intercept form: y g e q x + 5 and 2x + 3"> y > 2 x + 3 .
Identify the graph of y g e q x + 5 as a solid line with the region above it shaded.
Identify the graph of 2x + 3"> y > 2 x + 3 as a dashed line with the region above it shaded.
The solution is the overlapping shaded region. The graph shows the correct solution for the system of linear inequalities. The solution is the overlapping shaded region. T h e g r a p h s h o w s t h e correc t so l u t i o n f or t h e sys t e m o f l in e a r in e q u a l i t i es .
Explanation
Understanding the Problem We are given the following system of linear inequalities:
3 \end{array}"> y − xg e q 5 y − 2 x > 3
Our goal is to find the graph that represents the solution set of this system.
Rewriting the Inequalities First, let's rewrite the inequalities in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
For the first inequality, y − xg e q 5 , we add x to both sides to get:
y g e q x + 5
For the second inequality, 3"> y − 2 x > 3 , we add 2 x to both sides to get:
2x + 3"> y > 2 x + 3
Analyzing the Inequalities Now, let's analyze the first inequality, y g e q x + 5 . This represents the region above the line y = x + 5 . Since the inequality includes the equals sign, the line should be solid.
Next, let's analyze the second inequality, 2x + 3"> y > 2 x + 3 . This represents the region above the line y = 2 x + 3 . Since the inequality does not include the equals sign, the line should be dashed.
Finding the Solution Region The solution to the system of inequalities is the region where the shaded regions of both inequalities overlap. We need to find the graph that shows the correct lines (solid or dashed) and the correct shaded region.
Conclusion Therefore, the graph should show:
A solid line for y = x + 5 and the region above it shaded.
A dashed line for y = 2 x + 3 and the region above it shaded.
The overlapping shaded region represents the solution to the system of inequalities.
Examples
Systems of linear inequalities are used in various real-world applications, such as optimizing resource allocation, determining feasible production plans, and modeling constraints in economics and engineering. For example, a company might use a system of inequalities to determine the optimal combination of labor and capital to maximize profit, subject to constraints on available resources and production capacity. By graphing the inequalities, the company can visualize the feasible region and identify the combination of labor and capital that yields the highest profit.
