Solve Systems of Linear Inequalities by Graphing - Algebra-1

1. Fundamental Concepts

System of Linear Inequalities: A set consisting of multiple linear inequalities, such as the one mentioned in the document: . Its solution is an ordered pair that satisfies all the inequalities simultaneously.
Graphical Significance: Each linear inequality corresponds to a half - plane region in the coordinate plane. The solution to a system of linear inequalities is the intersection of these half - plane regions, which is the overlapping part of all the regions corresponding to each inequality.

Boundary Line: The straight line $Ax + By = C$ corresponding to each linear inequality is called the boundary line.
- If the inequality symbol is “” or “”, the boundary line is a solid line, indicating that the points on the boundary line are solutions to the inequality.
- If the inequality symbol is ” or “”, the boundary line is a dashed line, meaning that the points on the boundary line are not solutions to the inequality.
Region Determination: For each inequality, select a test point (usually the origin if the boundary line does not pass through the origin) and substitute it into the inequality. If the test point satisfies the inequality, the side of the boundary line where the test point lies is the solution region of the inequality; otherwise, the other side is the solution region.
Representation of Solutions: The solution to a system of linear inequalities on the graph is the overlapping part of all the solution regions of the inequalities, and all points in this region are solutions to the system of inequalities.

Easy Level
- Solve the system of inequalities
- Steps:
  1. Draw the line as a dashed line (because of the ” symbol). Take the test point and substitute it into , which is not true. So the solution region is above the line.
  2. Draw the line as a solid line (because of the “” symbol). Take the test point and substitute it into , we get , which is true. So the solution region is below the line and including the line itself.
  3. The overlapping part of the two regions is the solution to the system of inequalities.
Medium Level
- Solve the system of inequalities
- Steps:
  1. Draw the line as a solid line (because of the “” symbol). Take the test point and substitute it into , we get , which is not true. So the solution region is above the line.
  2. Draw the line as a dashed line (because of the “” symbol). Take the test point and substitute it into , we get , which is true. So the solution region is below the line.
  3. The overlapping part of the two regions is the solution to the system of inequalities.
Hard Level
- Solve the system of inequalities
- Steps:
  1. Draw the boundary lines corresponding to each inequality respectively: (dashed line), (solid line), (dashed line), and (dashed line).
  2. Determine the solution region for each inequality.
  3. Find the overlapping part of the four regions, which is the solution to this system of inequalities.

Step 1: Draw Boundary Lines: According to the symbol of each inequality, determine whether the boundary line is a solid line or a dashed line, and draw each boundary line accurately.
Step 2: Determine Solution Regions: For each inequality, select an appropriate test point (such as the origin), judge whether the point satisfies the inequality, so as to determine which side of the boundary line the solution region of the inequality is, and mark it (for example, with a shadow).
Step 3: Find the Intersection: The overlapping part of all the solution regions of the inequalities is the solution to the system of linear inequalities, which should be clearly presented on the graph.
Verification: A point in the solution region can be selected and substituted into all the inequalities to verify whether it satisfies all of them, so as to ensure the correctness of the result.