Desmos graph of system of inequalities with 3 shaded areas. Shark (representing JAWS) and the number 3 (representing the third post in this series).
Activity

Inequalities and systems of inequalities part 3: Systems of linear inequalities

Explore systems of inequalities in Part 3! Students graph two inequalities on the same coordinate plane and identify the overlapping solution region using accessible tools like Desmos and tactile supports.

In this third post of the Inequalities and Systems of Inequalities series, students build on previous concepts by exploring systems of linear inequalities. They’ll learn how to graph multiple inequalities on the same coordinate plane and identify the region where their solutions overlap. With hands-on practice using both tactile and digital tools, this activity helps students deepen their understanding of how systems of inequalities can represent real-world situations with multiple constraints.

Note: Some screen readers have trouble reading math equations in this website. For a fully accessible version of this post, download the accessible Word document, Inequalities and systems of inequalities part 3: Systems of linear inequalities.

Activity 3: Systems of linear inequalities

In this activity, students will graph a system of linear inequalities and determine solutions for a system of linear inequalities.

Objectives

Materials needed

Lesson/activity sequence

A system of linear inequalities is simply two or more linear inequalities. The solution to a system of inequalities are all the points located in the intersecting shaded regions. Let’s consider a system that incorporates the inequality we worked with in the previous activity.

y < x + 1

y x

Graphing the system using the pegboard

Step 1: Graph the equation y < x + 1.  

Hopefully, you graphed this in Activity 2 of this set. Plot the y-intercept at (0,1) and place pegs all along the line to create a dashed line.

Step 2: Graph the equation y x.  

There is no b-value in the inequality, which means the b-value is zero. Thus, the y-intercept is at the origin. Place a peg at the origin. 

The slope is negative, -½. From the origin, count down one and to the right two. Place a peg. Continue until you run out of space. 

Return to the origin. Count up one and to the left two. Place a peg. Continue until you run out of space. 

Since the sign of inequality includes an equals, this will be a solid line. Place a rubber band around the pegs.

You should now have both inequalities graphed.

Image 1:

An xy-coordinate plane pegboard with four quadrants. There are pegs creating a line with a slope of 1 that intersects the y-axis at the point (0, 1). There are pegs creating a line with a slope of negative one-half with a y-intercept of zero. The pegs on this line are connected with a rubber band, indicating a solid line.

Step 3: Demonstrate the solution set, where the shaded regions intersect. 

Any value in the solution set has to be true for both inequalities. 

The solution set for y < x + 1  is {y||y < x + 1}.

The solution set for y x is {y||y x}. 

The solution set for the system is {y||x + 1 > y -½x}.

This is all math speak for all the points below y = x + 1 and all the points above and including y x

Locate that and demonstrate shading using the pegboard or graph board.

Image 2:

An xy-coordinate plane pegboard with four quadrants. There are pegs creating a line with a slope of 1 that intersects the y-axis at the point (0, 1). There are pegs creating a line with a slope of negative one-half with a y-intercept of zero. The pegs on this line are connected with a rubber band, indicating a solid line. A hand is placed in the region below the graph of y is less than x plus one and above the graph of y is greater than negative one-half x.

Step 4: Identify a point that is in the solution set for the system of equations.

Plot the point (4,2). 

Image 3:

An xy-coordinate plane pegboard with four quadrants. There are pegs creating a line with a slope of 1 that intersects the y-axis at the point (0, 1). There are pegs creating a line with a slope of negative one-half with a y-intercept of zero. The pegs on this line are connected with a rubber band, indicating a solid line. The point (4, 2) is plotted in the region below the graph of y is less than x plus one and above the graph of y is greater than negative one-half x.

Does this point meet the criteria for our system? It is below y < x + 1 and above y x.

So, yes, it does belong in the solution set. 

Consider other points: (0,0), (3,3), (-4,2), and (2,3).  Plot each point and determine if each belongs in the solution set for the system of inequalities.

Graphing the system of inequalities using Desmos

Let’s practice graphing the same system using Desmos. Make sure JAWS is running, and open Desmos in the web browser.

Step 1: Input the inequalities in the expression list. 

y < x + 1

y x

Image 4:

A Desmos graphing calculator window with the two inequalities graphed. Both inequalities with shading and intersecting regions are shown on the graph paper.

Both inequalities are graphed, and the shading is demonstrated on the graph paper.

Step 2: Audio trace and summarize both expressions. Shading should be below for y < x + 1.

Shading should be above for y x. There is no information given regarding the intersection of the shaded regions.

Step 3: Locate the point (4,2). Input  x = 4 in expression three. This graphs a vertical line that intersects the x-axis at 4.

Image 5:

A Desmos graphing calculator window with the two inequalities graphed. The equation x equals four is graphed in expression three. Both inequalities and the equation are shown on the graph paper.

Step 4: Audio trace expression three. Use the points of interest keys (k or i) to locate the intersection with expression one at (4,5).

Image 6:

A Desmos graphing calculator window with the two inequalities graphed. The equation x equals four is graphed in expression three. Both inequalities and the equation are shown on the graph paper. Audio trace mode is on. The point (4, 5), the intersection of expression one and expression three, is highlighted.

Step 5: Navigate the line x = 4  to locate (4,2). Press left arrow to navigate down the line. You might only find an approximate location. This is okay… close enough!

Image 7:

A Desmos graphing calculator window with the two inequalities graphed. The equation x equals four is graphed in expression three. Both inequalities and the equation are shown on the graph paper. Audio trace mode is on. The point (4, 2.06736) on the graph of x equals 4, in the intersecting region, is highlighted.

Step 6: Use the points of interest keys (k or i) to locate the intersection with expression two at (4, -2).

Image 8:

A Desmos graphing calculator window with the two inequalities graphed. The equation x equals four is graphed in expression three. Both inequalities and the equation are shown on the graph paper. Audio trace mode is on. The point (4, -2), the intersection of expression two and expression three, is highlighted.

Step 7: Navigate the line  x = 4 to locate (4,2). Press right arrow to navigate up the line. You might only find an approximate location. This is okay… close enough!

Image 9:

A Desmos graphing calculator window with the two inequalities graphed. The equation x equals four is graphed in expression three. Both inequalities and the equation are shown on the graph paper. Audio trace mode is on. The point (4, 1.9525) on the graph of x equals 4, in the intersecting region, is highlighted.

This method for determining points in the solution set of a system of inequalities works, and it is a pretty good practice for navigating in Desmos. However, it is not the most efficient. 

It may be more efficient and just as effective to plot the point in the expression list and add a note that demonstrates that the point is true for both inequalities.

Image 10:

A Desmos graphing calculator screen. The inequality y is less than x plus one is in expression one. The inequality y is greater than or equal to negative one-half x is input in expression two. The point (4, 2) is input in expression three, and the label checkbox is checked. There is a note in expression four with both inequalities solved using the x- and y-values from the given point (4, 2). The note shows that the values are true for both inequalities, so the point is in the solution set. The inequalities are graphed on the graph paper. The point (4, 2) is graphed and labeled on the graph paper.

In this activity, we graphed a system of inequalities using the pegboard and the Desmos graphing calculator. In addition, we located a solution that was true for both inequalities in the system.

Resources

Activity set 1, Graphing Concepts series:

Activity set 2, Graph Board to Desmos series:

Activity set 3, Linear Functions and Problem Solving series (y-intercept and slope) series:

Activity set 4, Undetermined points and systems of linear equations series:

Activity set 5: Inequalities and systems of inequalities series:

Additional math resources by TEAM Initiative:

Pegboard post:

This algebra post was created as part of the TEAM Initiative to assist Teachers of Students with Visual Impairments (TSVIs) in teaching students digital math tools and to use tactile resources to support digital concepts. Written by John Rose. If you would like more information about the TEAM Initiative, contact Leslie Thatcher at [email protected].

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