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Teacher Guide: Graphing Linear Inequalities

Learn to graph inequalities on a coordinate plane using boundary lines and shading.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Graphing Inequalities. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify the boundary line of a linear inequality
  • Determine whether to use a solid or dashed boundary line
  • Use a test point to find the solution region
  • Graph linear inequalities in two variables accurately
  • Interpret solution regions in real-world contexts
Prerequisites
  • • Graphing linear equations using slope-intercept form
  • • Understanding inequality symbols and their meanings
  • • Plotting points on a coordinate plane
  • • Solving one-variable inequalities
Discussion Starters
  • 1. Why do we use a dashed line for some inequalities but a solid line for others?
  • 2. Can a point on the boundary line ever be a solution? When?
  • 3. How would you explain to a friend why we test a point to determine shading?
  • 4. What real-life situations can be modeled by linear inequalities?
Common Misconceptions

Always shade above for 'greater than'

The boundary line is always a solution

Differentiation Ideas

For Struggling Students:

  • • Provide a checklist: 1) Draw line, 2) Solid or dashed?, 3) Test , 4) Shade
  • • Start with vertical and horizontal inequalities like or
  • • Use color coding: one color for solid lines, another for dashed

For On-Level Students:

  • • Graph inequalities given in slope-intercept form
  • • Convert standard form to slope-intercept before graphing
  • • Write inequalities from graphs and real-world scenarios

For Advanced Students:

  • • Graph systems of linear inequalities (find overlapping region)
  • • Explore linear programming with optimization
  • • Write and graph compound inequalities
Standards Alignment
  • HSA.REI.D.12 (CCSS.MATH.CONTENT.HSA.REI.D.12)

    Graph the solutions to a linear inequality in two variables as a half-plane

  • HSA.CED.A.3 (CCSS.MATH.CONTENT.HSA.CED.A.3)

    Represent constraints by inequalities, and interpret solutions as viable or nonviable options

Lesson Resources
  • visualInteractive Inequality Grapher

    Students adjust slope, intercept, and inequality symbol to see how the graph changes

  • activityShading Challenge

    Given a boundary line and test point, determine the correct shading

  • worksheetReal-World Inequalities

    Translate word problems into linear inequalities and graph them

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A linear inequality is like a linear equation, but instead of an equals sign, it uses an inequality symbol: , , , or .
To graph a linear inequality: 1. Draw the boundary line (treat the inequality as an equation) 2. Determine the line style: solid line for or , dashed line for or 3. Shade the correct region: test a point to see which side satisfies the inequality
The solution set is all the points in the shaded region that make the inequality true.

Worked Examples

Graph the inequality on a coordinate plane.

1

Identify the boundary line

Treat it as . Slope , y-intercept → Boundary:

2

Determine the line style

Since the symbol is (not ), points ON the line are NOT included → Use a DASHED line

3

Draw the boundary line

Plot y-intercept , use slope to find , → Dashed line through these points

4

Choose a test point

Use since it's easy to calculate and not on the line → Test point:

5

Substitute into inequality

TRUE! → is a solution

6

Shade the correct region

Since works, shade the side containing → Shade ABOVE the line

Common Mistakes

Using the wrong line style (solid vs. dashed)

Why it's wrong: Students forget that and require a dashed line because points ON the line are not solutions.

Correct: Solid line for and (line included). Dashed line for and (line excluded).

Shading the wrong region

Why it's wrong: Students shade above for 'greater than' without testing, but the inequality might be in standard form.

Correct: ALWAYS test a point! Substitute a simple point like to verify which side to shade.

Forgetting to flip the inequality when multiplying by negative

Why it's wrong: When rearranging to slope-intercept form, dividing by a negative number reverses the inequality.

Correct: If you multiply or divide by a negative number, reverse the inequality symbol: becomes .

Testing a point that lies on the boundary line

Why it's wrong: Points on the line don't help determine which side to shade.

Correct: Choose a test point clearly NOT on the line. Origin is ideal unless the line passes through it.

Why It Matters

Linear inequalities describe real-world constraints and limits:
  • Budgeting: You can spend *at most* 200 dollars on groceries ()
  • Speed limits: You must drive *less than* 65 mph ()
  • Manufacturing: A factory produces *at least* 100 units daily ()
  • Nutrition: Consume *no more than* 2000 calories ()
Understanding how to graph these inequalities helps visualize all possible solutions!

Real World Applications

Budget Constraints

When planning expenses, you often have a maximum amount you can spend.

Example:

If concert tickets cost 75 dollars and food costs 25 dollars per person, and you have 300 dollars total, then where = tickets, = food purchases.

1Try It Yourself

You have 100 dollars to spend on books (15 dollars each) and magazines (5 dollars each).

Write and graph the inequality, then find if buying 4 books and 6 magazines is possible.

Step 1: Write the mathematical expression

Set up:

Manufacturing Limits

Factories have constraints on production capacity and resources.

Example:

A factory can produce at most 500 items per day. If they make chairs and tables, and tables take twice as long, then .

2Try It Yourself

A bakery makes at most 200 items daily. Cakes take 2 units of time; cookies take 1 unit.

Can they make 50 cakes and 80 cookies?

Step 1: Write the mathematical expression

Inequality:

Key Takeaways

  • 1Linear inequalities use , , , or instead of an equals sign
  • 2The boundary line is drawn by treating the inequality as an equation
  • 3Use a solid line for or (line included) and dashed for or (line excluded)
  • 4Test a point not on the line to determine which side to shade
  • 5The solution set is all points in the shaded region

Frequently Asked Questions

How do I know which side to shade?

Pick a test point not on the line ( is easiest). Substitute into the original inequality. If TRUE, shade that side. If FALSE, shade the opposite side.

Why is the line sometimes dashed?

A dashed line means points ON the line are NOT solutions. This happens with strict inequalities ( or ). Solid lines ( or ) include points on the line.

What if the origin is on the boundary line?

Choose a different test point like or . Any point clearly not on the line works.

Glossary

Linear inequality
An inequality involving linear expressions, using , , , or
Boundary line
The line you get when you replace the inequality symbol with an equals sign
Solution set
All the ordered pairs that make the inequality true
Test point
A point used to determine which side of the boundary line to shade
Half-plane
The region of the coordinate plane on one side of a line