Which is the graph of the system [tex]x+3 y>-3[/tex] and [tex]y<\frac{1}{2} x+1[/tex]?
Answer (2)
To graph the system of inequalities, first rewrite them in slope-intercept form. The solution region is the area above the line y = − 3 1 x − 1 and below the line y = 2 1 x + 1 . The final graph will show dashed lines and the overlapping area represents the solution to the inequalities.
;
Rewrite the inequalities in slope-intercept form: -\frac{1}{3}x - 1"> y > − 3 1 x − 1 and y < 2 1 x + 1 .
Identify the boundary lines: y = − 3 1 x − 1 and y = 2 1 x + 1 (both dashed).
Determine the shaded regions: above the first line and below the second line.
The solution is the intersection of these shaded regions.
Explanation
Understanding the Problem We are given a system of two inequalities:
-3"> x + 3 y > − 3
y < 2 1 x + 1
We need to find the graph that represents the solution set of this system of inequalities.
Rewriting the Inequalities First, let's rewrite the inequalities in slope-intercept form to make them easier to graph.
For the first inequality, -3"> x + 3 y > − 3 , we can isolate y :
-x - 3"> 3 y > − x − 3
-\frac{1}{3}x - 1"> y > − 3 1 x − 1
This inequality represents the region above the dashed line y = − 3 1 x − 1 because it's a 'greater than' inequality.
For the second inequality, y < 2 1 x + 1 , it is already in slope-intercept form.
This inequality represents the region below the dashed line y = 2 1 x + 1 because it's a 'less than' inequality.
Analyzing the Inequalities Now, let's analyze the characteristics of each inequality to visualize their graphs:
-\frac{1}{3}x - 1"> y > − 3 1 x − 1 :
The boundary line is y = − 3 1 x − 1 .
The slope is − 3 1 , and the y-intercept is -1.
Since it's a 'greater than' inequality, we shade the region above the line. The line itself is dashed because the inequality is strict (i.e., not 'greater than or equal to').
y < 2 1 x + 1 :
The boundary line is y = 2 1 x + 1 .
The slope is 2 1 , and the y-intercept is 1.
Since it's a 'less than' inequality, we shade the region below the line. The line itself is dashed because the inequality is strict (i.e., not 'less than or equal to').
The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap.
Describing the Graph The graph of the system of inequalities will show two dashed lines:
y = − 3 1 x − 1
y = 2 1 x + 1
The solution region is where the area above the first line and below the second line overlaps.
Final Answer The solution to the system of inequalities is the intersection of the regions defined by each inequality. The graph should show two dashed lines, y = − 3 1 x − 1 and y = 2 1 x + 1 , with the area above the first line and below the second line shaded.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on resources like labor and materials. The feasible region, representing all possible solutions that satisfy the constraints, is determined by a system of inequalities. Graphing these inequalities helps visualize the feasible region and find the optimal solution.
