Graphing Linear Inequalities in Two Variables

September 22, 2010 at 11:56 am Leave a comment

Is it possible for one topic to be a favorite topic and a hated topic at the same time, for me it would be graphing linear inequalities. There are parts that I love, and one part that drives me batty on occasion.

The Good

My favorite part of teaching these inequalities is that it gives me a chance to go over efficient graphing techniques with students.

  • If the equation is in slope-intercept form, graph using the y-intercept and the slope.
  • If the equation is of the form Ax+By=C, find the x– and y-intercepts and graph.
  • If the equation only has one variable, then it is either horizontal (y=b) or a vertical line (x=a).

Since the graphing exam is just around the corner, this is a jump-start on reviewing for the exam.

I also love these problems because of the critical thinking involved: determining the most efficient way to graph the line, determining whether the line should be solid or dashed, determining which half-plane to shade. This is the type of thinking required to be successful in an intro stat course and is a great preparation for students heading in that direction.

The Bad

Some students have a mental block when it comes to determining which side of the line to shade. I can walk them through the process of choosing a test point, and why the origin is a great choice if available. Most can substitute the coordinates of the test point into the original inequality, and even determine whether the inequality is true or false. But at that point it breaks down – “Which side should we shade?”

Don’t get me wrong. Many students understand this portion of the problem as well. But for the few who do not, it is a struggle trying to find a way to help those students to understand. I have tried

  • Having students choose 2 test points, 1 on each side of the line. After substituting, I require them to write “False” or “True” next to each test point. Then we always “Shade the Truth!”
  • Having the students write “True/False” next to their chosen test point. After substituting, they circle the correct word. If you circle “True”, shade that side of the line. If you circle “False”, move across the line and shade the opposite side.
  • Having students that do understand explain their technique to classmates that are struggling. I do this in pairs or in groups up to size 4. (This is when the light bulb often turns on for students.)


In my experience I have to keep coming up with alternative explanations until I find one that each student understands. (In a separate blog I will lay out the general tips I offer my online students.)

Do you have any techniques that work for you and your students? Please share your experience and thoughts by leaving a comment, or reaching me through the contact page at my web site –


I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website –


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Test Review And The Half Test Student Pointers for Graphing Linear Inequalities

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