A **system of inequalities** is a set of inequalities that have the same variables.

Drag the point and notice what happens to the label in the different regions of the graph. To generate a new graph, click the arrows in the top right corner.

- Drag the point to a spot where the label says "Solution". What do you notice about this region?
- Drag the point all the way along the dashed boundary line. What happens to the label and why do you think that is?
- Drag the point all the way along the solid boundary line. What happens to the label and why do you think that is?
- How could you verify your assumptions algebraically?

The solution set of a system of inequalities is the region where the solution sets of both linear inequalities overlap. The ordered pairs in this shaded region make both inequalities in the system true.

A solution can also be represented graphically as the region of the plane that satisfies all inequalities in the system. This is shown in the overlapping shaded regions. A point on the solid boundary line that borders the overlapping shaded region is included in the solution set, while any point on a dashed boundary line is not included in the solution set. We can verify these statements by substituting any ordered pair into the system algebraically.

\begin{cases} y < x \\y\leq- 1 \end{cases}

\begin{cases} y\geq x \\y\geq -x \end{cases}

Conventional systems of two linear inequalities will have four distinct regions- where both inequalities are true, where only the first inequality is true, where only the second inequality is true, and where neither inequality is true. Unconventional systems occur when the boundary lines are parallel, resulting in fewer distinct regions.

\begin{cases} y < x-2 \\y\leq x+3 \end{cases}

\begin{cases} y\geq 2x+6 \\y\leq 2x-6 \end{cases}

The solution to a system of inequalities in a given context is **viable** if the solution makes sense in the context, and is **non-viable** if it does not make sense.

Consider the graph of a system of linear inequalities:

a

Write the system of inequalities.

Worked Solution

b

Determine which of the following points are solutions to the system of inequalities: (-1,\,-1),\,(4,\,-4),\,(2,\,-1),\,(-1,\,-3)

Worked Solution

Consider the following system of inequalities:

\begin{cases} y\leq 3 \\y > 4 x + 5\end{cases}

a

Sketch a graph of the solution set to the system of inequalities.

Worked Solution

b

Is the point (-1,\,3) a solution to the system?

Worked Solution

Applicants for a particular university are asked to sit a quantitative reasoning test and verbal reasoning test. Successful applicants must obtain a minimum score of 14 on a quantitative reasoning test and a minimum combined score of 29 for both tests.

a

Write a system of inequalities for this scenario, where x represents the quantitative reasoning test score and y represents the verbal reasoning test score.

Worked Solution

b

Sketch a graph of the system of inequalities.

Worked Solution

c

Suppose the maximum of the verbal reasoning test was a score of 50. Is the solution (15,\,56) a viable solution in the context?

Worked Solution

d

Update the system of inequalities that models the new information about the tests.

Worked Solution

Idea summary

The solution to a system of inequalities lies in the region where the solutions of more than one linear inequality overlaps. Since solutions to systems of inequalities can have many solutions, we use a graph to show the solution set.