Graph the solution to this system of inequalities in the coordinate plane.
[tex]
\begin{aligned}
3 y & \ \textgreater \ 2 x+12 \\
2 x+y & \leq-5
\end{aligned}
[/tex]
Answer (1)
Solve each inequality for y : \frac{2}{3}x + 4"> y > 3 2 x + 4 and y ≤ − 2 x − 5 .
Graph the line y = 3 2 x + 4 as a dashed line and shade above.
Graph the line y = − 2 x − 5 as a solid line and shade below.
The overlapping shaded region is the solution: Overlapping shaded region
Explanation
Understanding the Problem We are given a system of two inequalities:
2 x+12 \ 2 x+y &\leq -5 \end{aligned}"> 3 y > 2 x + 12 2 x + y ≤ − 5
We need to graph the solution to this system of inequalities in the coordinate plane.
Isolating y First, let's solve each inequality for y to express them in slope-intercept form ( y = m x + b ).
For the first inequality, 2x + 12"> 3 y > 2 x + 12 , we divide both sides by 3:
\frac{2}{3}x + 4"> y > 3 2 x + 4
For the second inequality, 2 x + y ≤ − 5 , we subtract 2 x from both sides:
y ≤ − 2 x − 5
Analyzing the Inequalities Now, let's analyze the inequalities to determine how to graph them.
For the inequality \frac{2}{3}x + 4"> y > 3 2 x + 4 :
The boundary line is y = 3 2 x + 4 .
Since the inequality is strict ( "> > ), we use a dashed line to indicate that the points on the line are not included in the solution.
We shade the region above the line because y is greater than the expression.
For the inequality y ≤ − 2 x − 5 :
The boundary line is y = − 2 x − 5 .
Since the inequality is inclusive ( ≤ ), we use a solid line to indicate that the points on the line are included in the solution.
We shade the region below the line because y is less than or equal to the expression.
Finding the Solution Region The solution to the system of inequalities is the region where the shaded regions of both inequalities overlap. This region represents all the points ( x , y ) that satisfy both inequalities simultaneously.
Graphing the Solution To graph this, you would:
Draw a dashed line for y = 3 2 x + 4 and shade the region above it.
Draw a solid line for y = − 2 x − 5 and shade the region below it.
The overlapping shaded region is the solution to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as optimizing resource allocation. For example, a company might use inequalities to determine the optimal production levels of two products, given constraints on labor hours and raw materials. The solution region of the system of inequalities represents the feasible production plans that satisfy all the constraints, helping the company maximize its profit while staying within its resource limitations. This approach ensures efficient and effective decision-making in resource management.
