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.
In this activity, students will graph a system of linear inequalities and determine solutions for a system of linear inequalities.
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 ≥
-½x
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 ≥
-½x.
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:
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 ≥
-½x is {y||y ≥
-½x}.
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 ≥
-½x.
Locate that and demonstrate shading using the pegboard or graph board.
Image 2:
Step 4: Identify a point that is in the solution set for the system of equations.
Plot the point (4,2).
Image 3:
Does this point meet the criteria for our system? It is below y < x + 1 and above y ≥
-½x.
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.
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 ≥
-½x
Image 4:
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 ≥
-½x. 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:
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:
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:
Step 6: Use the points of interest keys (k or i) to locate the intersection with expression two at (4, -2).
Image 8:
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:
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:
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.
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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