Select the section of the graph that represents the solution to the system of inequalities: y > 3x − 52 and y < −12x + 1.
Understand the Problem
The question is asking to identify the region on a graph that represents the solution set for the given system of inequalities. This involves graphing each inequality and finding the area where they both overlap.
Answer
The solution set is the overlapping shaded region on the graph of the inequalities.
Answer for screen readers
The solution set for the system of inequalities is the area where the shaded regions for both inequalities overlap on the graph.
Steps to Solve
- Graph the first inequality
Identify the first inequality from the system. Convert it to an equation to find the boundary line. For example, if the inequality is $y \leq 2x + 3$, graph the line $y = 2x + 3$ as a dashed line for $<$ or $\leq$ and a solid line for $>$ or $\geq$.
- Shade the appropriate region
Determine which side of the line to shade. For $y \leq 2x + 3$, shade the area below the line, since it represents all the points where $y$ is less than or equal to $2x + 3$.
- Graph the second inequality
Repeat the process for the second inequality. Suppose it's $y > -x + 1$. First, graph the line $y = -x + 1$ as a dashed line.
- Shade the correct region for the second inequality
Since the inequality is $y > -x + 1$, shade the area above the line.
- Identify the overlap
Find the region where the shaded areas from both inequalities intersect. This overlapping area represents the solution set for the system of inequalities.
The solution set for the system of inequalities is the area where the shaded regions for both inequalities overlap on the graph.
More Information
Identifying the solution set for a system of inequalities visually helps in understanding how multiple constraints affect possible solutions. The overlapping region can represent various scenarios, such as feasible solutions in optimization problems.
Tips
- Forgetting to use dashed lines for inequalities that do not include equality (like $<$ or $>$).
- Misjudging which side of the line to shade based on the inequality direction.
- Overlooking the necessity of finding the overlap between the regions of the inequalities.
