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How Do You Solve a System of Inequalities by Graphing?

Solve the system of inequalities by graphing.

Summary

  1. 'x' and 'y' are the variables in this system of two inequalities
  2. To find the graphical solution, graph each inequality and find the region of intersection
  3. Solving for y in each inequality gives you slope-intercept form, which is convenient for graphing
  4. The inequality sign determines whether you draw a dashed or solid line
  5. Pick a convenient test point, like the origin (0,0) to see which half plane to shade for each inequality
  6. The blue shaded section shows the solution to the given system of inequalities

Notes

    1. To make your job easier, pretend these inequalities are equations, and rewrite them in slope-intercept form
    2. Slope-intercept form is y=mx+b. Remember, m is slope and b is the y-intercept of the line
    3. y>2x-2 is already in 'slope-intercept' form!
    1. Your goal is to convert -2x-y≤6 into y=mx+b, or slope-intercept form
    2. Use the addition property of inequality to add 2x to both sides and isolate -y
    1. Divide both sides by -1 to get a positive y.
    2. Before copying the sign, recall the division property of inequality, which tells us to flip the sign.
    1. Use the y-intercept (-2 and -6) to find out where each line crosses the y-axis, and the slope (up 2 over 1 and up 2 over -1) to plot additional points.
    1. Remember, the lines that you graph when plotting inequalities are just the border of the inequality!
    2. If you have a ≥ or ≤ inequality, then include the border line in you inequality and draw it solid
    3. If you have a > or < inequality, then don't include the border line in you inequality and draw it dashed
    1. Remember, the lines that you graph when plotting inequalities are just the border of the inequality!
    2. If you have a ≥ or ≤ inequality, then include the border line in you inequality and draw it solid
    3. If you have a > or < inequality, then don't include the border line in you inequality and draw it dashed
    1. If your test point obeys the inequality, then shade the half plane that includes the test point.
    2. If your test point doesn't obey the inequality, then shade the half plane outside of the test point!
    1. The region shaded by both inequalities tells you the solution to this system of inequalities
    2. Notice that your shaded intersection region has a solid line on the left and a dashed line on the right!