Solving a System of Inequalities From a Graph

🔑 Key Concepts

  • The solution to a system of inequalities is the region where both inequalities are true at the same time.
  • A dashed boundary line means the line is not included in the solution.
  • A solid boundary line means the line is included in the solution.
  • For \( y < \frac{1}{3}x + 3 \), shade below the dashed line.
  • For \( y \ge -\frac{1}{3}x + 1 \), shade above the solid line.

💡 Main Idea

To find the solution to a system of inequalities on a graph, look for the overlapping shaded region. In this example, the solution is all points that are below the dashed line \( y < \frac{1}{3}x + 3 \) and also on or above the solid line \( y \ge -\frac{1}{3}x + 1 \). The overlap between those two shaded regions is the solution set.

✏️ Worked Examples — Solving a System of Inequalities From the Graph

🧠 Math Vocabulary

  • System of Inequalities: Two or more inequalities considered together.
  • Boundary Line: The line that separates the solutions from the non-solutions on the graph.
  • Dashed Line: A boundary line used when the inequality is \( < \) or \( > \), meaning the line is not included.
  • Solid Line: A boundary line used when the inequality is \( \le \) or \( \ge \), meaning the line is included.
  • Overlap: The region where both shaded areas match. This is the solution to the system.

📚 What You Should Already Know

Before solving a system of inequalities from a graph, you should know how to read points on a coordinate plane, identify whether a line is solid or dashed, and understand what the symbols \( < \), \( > \), \( \le \), and \( \ge \) mean.

🚀 What Comes Next

After identifying the solution region on a graph, the next step is graphing systems of inequalities from scratch. You can also move into writing inequalities from real-world situations and interpreting what the overlapping region means in context.

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