Getting Started 
Misconception/Error The student does not understand how to determine the solution region of a system of inequalities. 
Examples of Student Work at this Level The student:
 Selects the graph that contains the point (0, 0) in the shaded region.
 Selects option A because the boundary lines are graphed correctly.
 Checks a solution region by testing points outside of the region (including points on a dashed boundary line).
 Tests two different points in a shaded region in only one of the inequalities.
 Determines that (0, 0) is not a solution of the first inequality but does not understand the implications for finding the solution region.

Questions Eliciting Thinking What does it mean for an ordered pair to be a solution of a system of inequalities?
If option A is correctly graphed, where would you find the solutions of this system of inequalities?
Can you explain how you determined which graph shows the solution region of this system? 
Instructional Implications Review graphing inequalities. Explain that solutions of a linear equation are points on the line, but all possible solutions of an inequality lie in a halfplane separated by a boundary line. Provide additional examples of strict (< or >) and nonstrict (Â orÂ ) inequalities for the student to graph. Emphasize the relationship between the shaded part of the graph and the solutions of the inequality. Review the method of testing a point to determine the region of the plane in which solutions are found. Caution the student not to rely on the direction of the inequality symbol to determine the solution region (e.g., concluding that the less than symbol indicates to shade below the line) as this is only true in special cases. Be sure the student understands why testing a point on a boundary line does not convey anything about the location of the solution region beyond the boundary line.
Emphasize that the solution region of a system of inequalities is the intersection of the solution regions of the individual inequalities. Explain that any point in this region satisfies each inequality in the system, so that by definition, it is a solution of the system. Guide the student to test a point in each shaded region on the worksheet using each inequality. Model explaining why options A, C, and D are incorrect while option B is correct.
Provide a graphed system of inequalities and ask the student to identify examples and nonexamples of solutions. Give the student examples of points that are in shaded regions of some, but not all, of the inequalities, on dashed boundary lines, on solid boundary lines that are part of the solution region, and on solid boundary lines that are not part of the solution region. Ask the student to decide if each point is a solution. 
Making Progress 
Misconception/Error The student makes an error implementing an effective strategy for identifying the solution region. 
Examples of Student Work at this Level The student selects points in the shaded regions to test in each inequality but:
 Makes a computational error when testing a point that leads to a wrong selection.
 Does not recognize a false statement as false. For example, the student tests the point (0, 0) in the first inequality and concludes that 0 > 1 is a true statement.

Questions Eliciting Thinking You made an error when you tested this point. Can you find and correct it?
Is 0 > 1 a true statement? 
Instructional Implications Provide feedback to the student concerning any errors made and allow the student to revise his or her answers.
Provide opportunities to graph solutions of inequalities. After each graph is completed, ask the student to identify a point on the graph that is a solution of the system and a point that is not. Then ask the student to justify his or her choices by determining whether or not the points satisfy the system of inequalities. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student chooses points in the shaded regions to test in each inequality and determines that points in the shaded region of option B satisfy both inequalities. The student concludes that option B represents the solution of the system of inequalities.

Questions Eliciting Thinking How did you determine that option B shows the correct solution region?
Suppose a point is on a dashed boundary line of the solution region. Is the point a solution of the system of inequalities?
Suppose a point is on a solid boundary line of the solution region. Is the point a solution of the system of inequalities? 
Instructional Implications Provide the student with a graphed system of inequalities. Give the student examples of points that are in shaded regions of some, but not all, of the inequalities, on dashed boundary lines, on solid boundary lines that are part of the solution region, and on solid boundary lines that are not part of the solution region. Ask the student to decide if each point is a solution.
Provide the student with systems of linear inequalities to graph. Consider implementing MFAS task Graph a System of Inequalities (AREI.4.12). 