In this section, students solve linear inequalities in one variable. They begin by writing constraints for situations as inequalities, and then solve them by writing equivalent inequalities and reasoning about quantities. They finish the section by examining inequalities for constraints that are implied by the context, such as requiring variables to be positive.

Much of the work of this section revisits concepts from middle school, but the reasoning in this unit builds on the work that students did with equations in an earlier unit. Depending on student proficiency in solving inequalities with one variable, this section can either be moved through quickly or extra time can be spent on several optional activities available for additional practice solving inequalities.

This section expands the exploration of inequalities to include systems. Students begin by interpreting solutions to systems of inequalities as values that satisfy all the constraints simultaneously. Then, they graph the inequalities to visualize a solution region for the systems as the overlapping portions of the solution regions for the individual inequalities. In the optional last lesson of the unit, students mathematically model a situation using systems of linear inequalities and explore the solutions in that context.

Two inequalities graphed on a coordinate plane, origin O, scale from 0 to 6 on both axes. Horizontal axis, hours on trampoline. Vertical axis, hours in pool. The dashed line starts on vertical axis at 4, goes through 1 comma 3 and ends on horizontal axis at 4. The region below the dashed line is shaded. The solid line starts on the vertical axis at 5, goes through 1 comma 3 and ends on the horizontal axis at 2 point 5. The region below the solid line is shaded.

In this section, students work with inequalities that have two variables. They begin by considering what it means to be a solution of an inequality of this type, and then recognize that, on a graph, the solutions occupy a region bounded by a linear equation related to the original inequality. This understanding is used to graph solution regions for 2-variable inequalities—first by graphing an equation related to the inequality, and then by testing points to find the solution region.

Initially, students work through this process by hand, but later they are introduced to the use of graphing technology to speed up the process.