How Do You Solve Compound Inequalities?

How Do You Solve Compound Inequalities?

To solve compound inequalities, graph each line of the inequality system, determine the solutions that apply to each and find which set of solutions overlaps on the graph. Pay attention to whether the inequalities are joined by "and" or "or" operations, because it alters the set of solutions.

  1. Graph the first inequality on a coordinate plane

    If necessary, convert the inequality to slope-intercept form y =/= mx + b, where m is the slope and b is the y-intercept. Remember to use the right type of line for the inequality. Dotted lines do not include the value lying on the line itself, but rather values infinitely close to it.

  2. Find the set of solutions, and shade that portion of the graph

    To make finding the solution easier, substitute 0 for the x and y values of the inequality. If it holds true, shade the portion of the graph holding the origin point.

  3. Repeat for all inequalities in the system

    To make it easier to read, shade each inequality in a different pattern or a different color.

  4. Find the portion of the graph that satisfies all inequalities

    This applies only for "and" operations. For "or" operations, it can be any of the solutions. Shade the overlapping portion darkest, pick a point and substitute the values into all inequalities. If they all hold true, you have found your set of solutions.