Activity
Inequalities and Systems of Inequalities part 2: Linear inequalities
Explore linear inequalities in part 2 of our series! Students learn to graph and interpret solutions using accessible tools like Desmos and tactile supports—making algebra concepts clear and fun.
In this second post of the Inequalities series (Activity Set 5), students explore how to represent and solve linear inequalities using both tactile tools and digital technology. Building on the foundational concepts from Part 1, this lesson introduces shading regions on a graph to show solutions, distinguishing between solid and dashed lines, and interpreting inequality symbols. Students practice plotting and analyzing linear inequalities using the pegboard and the Desmos Graphing Calculator with screen reader support—developing spatial reasoning and tech fluency along the way.
Linear inequalities are math statements that compare two expressions using inequality symbols (<, >, ≤, ≥)—and they involve variables raised only to the first power (like x or y). They’re just like linear equations, but instead of saying things are equal, they show a range of possible answers.
For example, the inequality y < 2x + 3, y means that y can be any value that is less than what you get when you plug in a number for x and do 2x + 3.
You can graph linear inequalities on a coordinate grid, usually shading one side of the line to show all the solutions!
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 2: Linear inequalities.
Activity 2: Graphing linear inequalities
In this activity, students will graph linear inequalities and determine solutions for linear inequalities.
Objectives
- Graph linear inequalities on the pegboard and using the Desmos graphing calculator.
- Determine the location of shading and specific points in relation to the graphed inequality.
Materials needed
- xy-coordinate plane pegboard or APH graph board
- materials for graphing (e.g., pegs, rubber bands, push pins)
- PC laptop or desktop with JAWS
- Internet browser
Lesson/activity sequence
At the end of Activity 1 in this set, the question was posed: What happens if you add slope to a horizontal line? How does that impact the inequality?
Slope is a number that tells you how steep a line is on a graph. It shows how much the line goes up or down as you move to the right.
For example, if a line goes up 2 spaces for every 1 space it goes to the right, the slope is 2. If it goes down 3 spaces for every 1 space to the right, the slope is –3.
Let’s consider the last inequality in the list from the previous activity, y < 1. The graph of this inequality would be a horizontal, dashed line that intersects the y-axis at (0, 1).
In order to add slope, we need to put this in context of the slope-intercept form, y < mx + b.
The b-value is the y-intercept, so b = 1.
y < mx + 1
The slope of a horizontal line is zero, so m = 0.
y < 0 * x + 1
0 * x = 0, therefore y < 1. That means that the solution is all y-values less than 1.
What if we add slope?
y < mx + b
y < 1x + 1
Editor’s Note: In the inequalityy < mx + b, each part represents something important in a linear equation or inequality:
- y = the output or result (often the dependent variable)
- < = the inequality symbol (this could be <, >, ≤, or ≥)
- m = the slope of the line (how steep it is)
- x = the input (often the independent variable)
- b = the y-intercept (where the line crosses the y-axis)
Now we have a different function. Let’s graph it!
Graphing a linear inequality using the pegboard
Step 1: Place a peg at the y-intercept.
Image 1:
Step 2: Add pegs to the right. The slope is positive one, so that is up one and to the right one. Place a peg and continue placing pegs to the end of the board on the right.
Image 2:
Step 3: From the y-intercept, add pegs to the left. The slope is positive one, so that is down one and to the left one. Place a peg and continue placing pegs to the end of the board on the left. The inequality is less than, so it is a dashed line.
Image 3:
Where is the shading?
Since the sign of inequality is less than, the shading is below the line.
Image 4:
What is the solution set?
{y||y < x + 1}, that is the set of all numbers y such that y is less than x plus one.
What are some solutions of this linear inequality?
Solutions include all ordered pairs that fall in the solution set. Let’s consider some…
(0,0) The origin is a common solution to consider, because it is easy to locate on the coordinate plane and easy to solve for in the inequality.
y < x + 1
0 < 0 + 1
0 < 1
True.
We can also plot it on the pegboard and check to see if it is below the graphed line.
Image 5:
Place a peg at the origin. It is below the graphed line, so it is in the solution set.
(3,3) Is this point in the solution set? Let’s plot the point on the pegboard.
Image 6:
This point is also below the graphed line, so it is in the solution set.
(-4,3) Is this point in the solution set? Let’s plot the point on the pegboard.
Image 7:
This point is above the graphed line, so it is not in the solution set. Let’s prove that using the inequality. Substitute the x- and y-values from the ordered pair into the inequality.
y < x + 1
3 < -4 + 1
3 < -3
False. This is not in the solution set.
(2,3) Is this point in the solution set? Let’s plot the point on the pegboard.
Image 8:
This point is on the line. Does that mean that it is included in the solution set or not? Let’s test it by substituting the x- and y-values from the ordered pair into the inequality.
y < x + 1
3 < 2 + 1
3 < 3
False. Three is not less than three. Because the sign of inequality is less than and not less than or equal to, any ordered pair on the dashed line will not be a solution to the inequality.
Graphing a linear inequality using Desmos
Let’s practice graphing the same linear inequality using Desmos. Make sure JAWS is running and open Desmos in the web browser.
Step 1: Input y < x + 1 in expression one.
Step 2: Audio trace the inequality.
Press h to hear the graphed line.
Determine the y-intercept and the zero using the points of interest keys (k or i).
Press Alt + s to hear a graph summary and determine the location of the shading.
Image 9:
Determining solutions to the inequality using Desmos
Solutions include all ordered pairs that fall in the solution set. We considered some possible solutions when graphing the inequality on the pegboard. Using the pegboard (or graph board), it is pretty easy to determine whether or not a solution is in the shaded region.
Unfortunately, Desmos does not tell you whether or not a point is in the shaded region. For example, if we plot (0,0) in Desmos, the point is graphed along with the inequality, but there is no auditory indication that it is in the shaded region.
One method for using Desmos to show solutions is to add a note to the expression list that proves the point is in the solution set using the inequality. To go to the next line in a note, press Shift + Enter.
Image 10:
Sharing this with a teacher will demonstrate understanding of the concept and is, perhaps, the easiest method.
A second method for determining solutions using the calculator is to use systems, and it incorporates many of the skills learned in these activity sets. Let’s consider the case of (0,0):
Step 1: Input x = 0 in expression two. This graphs a vertical line on the y-axis.
Image 11:
Step 2: Audio trace expression two. Use the points of interest keys (k or i) to locate the intersection with expression one. It is located at the point (0,1).
Image 12:
Step 3: You have graphed expression one, y < x + 1, and know that the shading is below the line. There are two directions you can go on this vertical line, up or down. To move in these directions, use the Right Arrow to go up and the Left Arrow to go down. We want to go down, below the intersection with expression one, which is in the shaded region. Press the left arrow several times. Remember that you can press Control to stop the speech and press left arrow again to orient yourself on the line. Try to locate the point (0, 0).
Image 13:
You have navigated straight down a vertical line from the intersection with y < x + 1, into the shaded area, to determine that the point (0,0) is in the solution set.
This method uses only the calculator to determine if a point is in the solution set. However, it does require solid understanding of Desmos navigation and the characteristics of linear inequalities and systems.
Try this out with the other points, (3,3), (-4,3), and (2, 3). See if you can determine if they are in the solution set using only the calculator. Feel free to use this Desmos graph as an example.
In this activity we have graphed linear inequalities using a pegboard and Desmos, as well as determined ordered pairs in the solution set. The next activity will introduce systems of inequalities and determining solutions to systems of inequalities.
Resources
Activity set 1, Graphing Concepts series:
- Graphing concepts activity 1: Attributes and coordinates
- Graphing concepts activity 2: Plotting points
- Graphing concepts activity 3: Graphing a line
Activity set 2, Graph Board to Desmos series:
- Post 1: Setting up Jaws for Desmos
- Post 2: Plotting points and navigating Audio Trace
- Post 3: Representing functions in Desmos
- Input functions into the expression list
- Create tables from functions
- Navigate tables
- Editing expressions
- Sharing a graph
Activity set 3, Linear Functions and Problem Solving series (y-intercept and slope) series:
- Linear functions and problem solving 1: y-intercept and slope on a graph board
- Linear functions and problem solving 2: y-intercept and slope using Desmos
- Linear functions and problem solving 3: y-intercept and slope problem solving
Activity set 4, Undetermined points and systems of linear equations series:
- Undetermined points and systems of linear equations: What’s the Point activity 1
- Undetermined points and systems of linear equations 2: Finding the intersection
- Undetermined points and systems of linear equations 3: Solving a linear equation
Activity set 5: Inequalities and systems of inequalities series:
- Inequalities and systems of inequalities part 1: Simple inequalities
- Inequalities and systems of inequalities part 2: Linear inequalities
- Inequalities and systems of inequalities part 3: Systems of inequalities
Additional math resources by TEAM Initiative:
- Digital math: Introducing skill level checklists
- Digital math: Workflows from teacher to student and student to teacher
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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