If a linear inequality involves two variables, we can represent it as a region on a coordinate plane rather than an interval on a number line.

Drag the blue point to the different regions of the graph (shaded, unshaded, boundary line) and note what happens.

Use the dropdown to change the inequality symbol and note what happens.

- What do you notice about the label when the point is in the shaded vs. the unshaded region? Why do you think that happens?
- What do you notice about the label when the point is on the boundary line? Why do you think that happens?
- How does the inequality symbol affect the graph and the label on the point? Why do you think that happens?

The solution to an inequality in two variables includes many solutions, similarly to inequalities in one variable. Instead of shading a number line to show a solution set, we now have a region that represents the solution to a linear inequality. Instead of an unfilled circle to show the boundary of a solution set that is excluded, we use a dashed line. Instead of a filled circle to show the boundary of a solution set that is included, we use a solid line.

Depending on the inequality sign, the boundary line will be solid or dashed, and region shaded will be above or below the boundary line.

y>x

y \geq x

y<x

y \leq x

We can also use linear inequalities to represent real-world situations. Recall that a **viable solution** is a valid solution that makes sense within the context of the question or problem while a **non-viable solution** is an algebraically valid solution that does not make sense within the context of the question or problem.

We were previously introduced to constraints on inequalities that present a limitation or restriction of the possible x-values when solving inequalities in one variable. We should also consider constraints in context when solving two variable linear inequalities.

Consider the inequality

y \leq -2x + 5

a

Graph the boundary line y=-2x+5.

Worked Solution

b

Determine whether the point (4,\,2) is a solution to the inequality y \leq -2x+5.

Worked Solution

c

Determine whether the point (0,\,0) is a solution to the inequality y \leq -2x+5.

Worked Solution

d

Shade the side of the graph that includes the point from part (b) or part (c) that is a solution to the inequality.

Worked Solution

Consider the linear inequality 5x + 3y > 15

a

Graph the inequality.

Worked Solution

b

Is the point (0,\,5) in the solution set of the inequality?

Worked Solution

Write the inequality that describes the region shaded on the given coordinate plane.

Worked Solution

A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs 8 pounds and each box of grapefruits weighs 12 pounds.

a

Write an inequality to represent the number of boxes of oranges and grapefruit that can be in the truck.

Worked Solution

b

Create a graph of the region containing the points corresponding to all the different numbers of orange and grapefruit boxes that can be loaded into the truck.

Worked Solution

c

Explain why the point (325,\,-100) is a non-viable solution to the inequality.

Worked Solution

Idea summary

We can use test points or the inequality in slope-intercept form to determine the region of the solution set on the coordinate plane:

- y > mx+b indicates that the solution set is above the boundary line
- y < mx+b indicates that the solution set is below the boundary line

The constraints of a context are important to consider when writing linear inequalities and interpreting viable solutions.