Students recognize and identify solutions to two-variable inequalities. They represent the solution set graphically. They create two variable inequalities to represent a situation.
Students understand that a half-plane bounded by the line ax + by = c is a visual representation of the solution set to a linear inequality such as ax + by < c. They interpret the inequality symbol correctly to determine which portion of the coordinate plane is shaded to represent the solution.

Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically.

Students use systems of equations or inequalities to solve contextual problems and interpret solutions within a particular context.

In the context of examining combinations of cats and dogs that satisfy all of the “Pet Sitter” constraints students will solidify the following mathematics:
• The solution set to a system of inequalities represents the set of points that satisfy all of the inequalities simultaneously. That is, a point in the solution set of a system of inequalities makes all of the inequalities true. Contexts, such as “Pet Sitters” may imply additional constraints, such as y ≥ 0 or x ≥ 0, that are not explicitly stated.
• A shaded region on the coordinate grid is used to represent the solution set for a system of inequalities in two-variables. In this case, the points on the boundary lines that trace out the region are also included in the solution set, indicated by drawing a solid line for the boundaries. A shaded polygonal region is used to represent the solution to a system of constraints.

Students will graph one linear inequality and then graph a system of linear inequalities as they apply to an area of land. Students will practice graphing systems of inequalities and determining if a points lies in the solution. In the extension, students are challenged to write a system of inequalities that describe the area of Mississippi.

CK-12 Foundation's Algebra FlexBook is an introduction to algebraic concepts for the high school student. Topics include: Equations & Functions, Real Numbers, Equations of Lines, Solving Systems of Equations & Quadratic Equations.

This lesson unit is intended to help teachers assess how well students are able to use linear inequalities to create a set of solutions. In particular, the lesson will help teachers identify and assist students who have difficulties in: representing a constraint by shading the correct side of the inequality line; and understanding how combining inequalities affects a solution space.

This is an interactive investigation using the Desmos.com calculator. It helps students understand the meaning of the inequality signs as it relates to the graphs of linear inequalities in two variables. This is used as a precursor to the actual graphing of linear inequalities in two variables.

The purpose of this task is to practice solving systems of linear inequalities by identifying the overlapping (or intersecting) region of the half-planes that form the solution sets of each of the two-variable inequalities in the system.

This lesson introduces the notion of a system of inequalities and demonstrates how to graph such a system.

As students continue to work with the pet sitter context they will examine and extend many ideas, strategies and representations related to solving systems of equations and inequalities. For example, each constraint (space, start-up costs) can be represented by an inequality since Carlos and Clarita don’t have to use up all of the available space or money. The solution set of an inequality (in quadrant I) consists of all of the points that satisfy the constraint. The boundary between points that satisfy the constraint and those that do not is a line. The slope of the line represents the “exchange rate” between cat pens and dog runs (e.g., for every 2 fewer dog runs we buy, we can purchase 5 more cat pens), and the x- and y-intercepts of the line represent purchasing either all cat pens and no dog runs, or all dog runs and no cat pens. A shaded half-plane in the coordinate grid is being used to represent the solution set for an inequality in two-variables. In this case, the points on the boundary line are also included in the solution set, indicated by drawing a solid line for the boundary.