Objective:

To graph linear inequalities in two variables.

Notes:

Steps to graph linear inequalities on a coordinate plane.
  1. Use a graphing method from sections 4.1, 4.2, 4.3, or 4.5 to graph the line.
    • Use a dashed line for < and >.
    • Use a solid line for and .
  2. Decide which side to shade.
    • If y is by itself in the equation (as in slope-intercept form).
      • If y < a or y ≤ a then you will shade below the line.
      • If y > a or y ≥ a then you will shade above the line.
    • If the equation is in standard form then choose a test point.
      • Do not choose a point that the line goes through.
      • Choose a point that is easy to work with such as (0, 0).
      • If the test point makes the inequality true then you will shade on the side with the point.
      • If the test point makes the inequality false then you will shade on the other side.
  3. Shade.

Examples:

Sketch each inequality on a coordinate plane.

  1. Sketch the graph of y < 4.

    Ex 1 See section 4.1 for help graphing the line. Use a dashed line since there is no =.

    This inequality is in slope-intercept form. Since y is less than we should shade below the dashed line.

    If you would like to choose a test point, the origin (point (0, 0)) can be used. Substituting 0 in for y in the equation yields a true statement. This verifies that we should shade below the dashed line.


  2. Sketch the graph of 6x +5y ≥ 30.

    Ex 2 See section 4.3 for help graphing the line. Use a solid line since there is an =.

    This inequality is in standard form. We should choose a test point. The origin (point (0, 0)) can be used. Substituting 0 in for x and y in the equation yields a false statement (see below). This tells us that we should not shade on the side including the origin. We should shade above the solid line.

    Ex 2


  3. Sketch the graph of y > 3x.

    Ex 3 See section 4.5 for help graphing the line. Use a dashed line since there is no =.

    This inequality is in slope-intercept form. Since y is greater than we should shade above the dashed line.

    If you would like to choose a test point, the origin (point (0, 0)) cannot be used this time since the line goes through it. Choose another easy point such as (1, 1). Substituting 1 in for y and x in the equation yields a false statement (see below). This tells us that we should not shade on the side with the point (1, 1). We should shade above the dashed line.

    Ex 3

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This page was last updated on 08/09/13.