Topics
3. Systems of Linear Equations and Inequalities
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United States of America
Grades 9-12

3.05 Systems of linear inequalities

Lesson
Practice

Introduction

Systems of linear inequalities take and extend what we learned in lesson  4.01 Writing and graphing linear systems,  where we focused on graphing systems, and lesson  4.04 Two variable linear inequalities  where we worked with inequalities in the coordinate plane. Systems of linear inequalities give us an opportunity to more accurately model real-world scenarios.

Systems of linear inequalities

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

Exploration

Drag the point and notice what happens to the label in the different regions of the graph.

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  1. Drag the point to a spot where the label says "Solution". What do you notice about this region?
  2. Drag the point all the way along the dashed boundary line. What happens to the label and why do you think that is?
  3. Drag the point all the way along the solid boundary line. What happens to the label and why do you think that is?
  4. 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.

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

Conventional systems of two linear equalities 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.

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

Examples

Example 1

Consider the graph of a system of linear inequalities:

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a

Write the system of inequalities.

Worked Solution
Create a strategy

Start by writing the equation of each line, then determine the inequality symbol, depending on the location of the shaded region and whether or not the boundary line is solid or dashed.

Apply the idea

We will start with the solid blue line. We can count the slope from the graph to find that it is \dfrac{4}{1} or 4 and the y-intercept is 3.

That gives us the slope-intercept form equation y=4x+3. Since the shaded region lies below the blue line on the graph we need to use a less than inequality symbol and since the boundary line is solid it must be \leq to include all of the points on the line in its solution.

So the blue inequality is y\leq 4x+3.

Moving onto the dashed green line. We can count the slope from the graph to find that it is -\dfrac{3}{4} and the y-intercept is -1.

That gives us the slope-intercept form equation y=-\dfrac{3}{4}x-1. Since the shaded region lies below the green line on the graph we need to use a less than inequality symbol and since the boundary line is dashed it must be \lt (but not equal to) to exclude all of the points on the line in its solution.

So the green inequality is y\lt -\dfrac{3}{4}x-1.

Writing it as a system we get:

\begin{cases} y \leq 4x + 3 \\ y < -\dfrac{3}{4}x -1 \end{cases}

b

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

Worked Solution
Create a strategy

Use the graph of the system to determine whether each point lies in the solution set to the system.

Apply the idea

The point (-1,-1) is on the solid boundary line in the solution set. It is a solution to the system.

The point (4, -4) is on the dashed boundary line in the solution set. It is not a solution to the system.

The point (2,-1) is not located in the solution set. It is not a solution to the system.

The point (-1,-3) is in the solution set. It is a solution to the system.

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Reflect and check

We can substitute each ordered pair into the given system of inequalities and confirm whether it is a solution by determining if both math statements are true.

Example 2

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
Apply the idea

To sketch the system of inequalities, we can first construct the boundary lines for each inequality, namely y=3 and y=4x+5. When given a strict inequality, we will draw a dashed line. When given a nonstrict inequality we will draw a solid line.

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To determine which side of each inequality will be shaded, we can choose some test points that satisfy each inequality. The test points will indicate which side of the inequality will be shaded. For y \leq 3 we will plot (-2,5) and (-3,-3). For y > 4x+5 we will plot (-2,2) and (4,0).

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The region that will be shaded is the region which satisfies both inequalities. Using the test points, we can see that the shading will occur below the boundary line for y \leq 3 and above y>4x+5.

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Reflect and check

Since each inequality in the system was already written in terms of y, it would have been possible to determine the direction of the shading without first plotting the test points. With some systems of inequalities written in terms of x or in general form, however, it can be less intuitive to know which direction to shade.

b

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

Worked Solution
Create a strategy

Substitute the ordered pair into the inequalities and determine if the system is true.

Apply the idea

\begin{cases} 3 \leq 3 \\ 3 > 4 (-1) + 5\end{cases}

Since both statements in the system are true when substituting (-1,3) in for x and y, (-1,3) is a viable solution to the system.

Reflect and check

We can use the graph of the system to verify that (-1,3) is a solution. Although (-1,3) is on the boundary line of the solution set, it is located on a solid boundary line, meaning it is included in the solution set.

Example 3

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
Create a strategy

Since we know that the minimum accepted score for quantitative reasoning is 14, we can represent this with an inequality showing 14 as the lowest possible solution. A minimum combined score of 29 means the total of the two scores must sum to 29 or more.

Apply the idea

\begin{cases} x \geq 14 \\ x+y \geq 29 \end{cases}

b

Sketch a graph of the system of inequalities.

Worked Solution
Create a strategy

The system can be graphed using intercepts or by converting x+y \geq 29 to slope-intercept form, and graphing using the y-intercept and slope.

Apply the idea
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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
Apply the idea

No. This would mean that the score for the quantitative reasoning test was 15 and the verbal reasoning test 56. By viewing the graph, we can see that this point technically satisfies both inequalities, but we now know that the maximum possible score on the verbal reasoning test is a 50.

d

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

Worked Solution
Create a strategy

We now know that the maximum of the verbal reasoning test was a score of 50, and that y represents the score on the verbal reasoning test.

Apply the idea

Since the score for the verbal reasoning test cannot be above a score of 50, the inequality \\y \leq 50 may be included in the system of inequalities modeling the relationship:

\begin{cases} x \geq 14 \\ y \leq 50 \\ x+y \geq 29 \end{cases}

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.

Outcomes

A.CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A.REI.D.12

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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