Graphing Linear Inequalities

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

Advanced25 minLesson
Printable worksheet

Definition

A linear inequality is like a linear equation, but instead of an equals sign, it uses an inequality symbol: <<, >>, ≤\leq, or ≥\geq.
y<2x+3y≥−x+1y < 2x + 3 \quad\quad y \geq -x + 1
To graph a linear inequality:
  1. 1.Draw the boundary line (treat the inequality as an equation)
  2. 2.Determine the line style: solid line for ≤\leq or ≥\geq, dashed line for << or >>
  3. 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.

Try it now

What type of line do you use for y>2x+1y > 2x + 1?

Worked Examples

Graph the inequality y>x−2y > x - 2 on a coordinate plane.
1

Identify the boundary line

Treat it as y=x−2y = x - 2. Slope m=1m = 1, y-intercept b=−2b = -2 → Boundary: y=x−2y = x - 2
2

Determine the line style

Since the symbol is >> (not ≥\geq), points ON the line are NOT included → Use a DASHED line
3

Draw the boundary line

Plot y-intercept (0,−2)(0, -2), use slope to find (1,−1)(1, -1), (2,0)(2, 0) → Dashed line through these points
4

Choose a test point

Use (0,0)(0, 0) since it's easy to calculate and not on the line → Test point: (0,0)(0, 0)
5

Substitute into inequality

0>0−2⇒0>−20 > 0 - 2 \Rightarrow 0 > -2 TRUE! → (0,0)(0, 0) is a solution
6

Shade the correct region

Since (0,0)(0, 0) works, shade the side containing (0,0)(0, 0) → 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 ≤\leq and ≥\geq (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 (0,0)(0, 0) 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: −2y>6-2y > 6 becomes y<−3y < -3.

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 (0,0)(0, 0) is ideal unless the line passes through it.

Interactive Visual

Function graph

Choose the inequality sign:

y > m · x + b
-6-4-20246-6-4-20246xy(0, 0)

Tap or hover over the graph to read off values.

y = mx + bSolution regionBoundary not included (< or >)

Function graph

-6-4-20246-505xy(0, 0)
Solution regionBoundary included (≤ or ≥)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

x - 2

m=1, b=-2

Expression Calculator

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History

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Practice Problems

15 problems
Problem 1 of 15
Easy
What type of line do you use for y>2x+1y > 2x + 1?

Why It Matters

Linear inequalities describe real-world constraints and limits:
  • Budgeting: You can spend *at most* 200 dollars on groceries (x+y≤200x + y \leq 200)
  • Speed limits: You must drive *less than* 65 mph (s<65s < 65)
  • Manufacturing: A factory produces *at least* 100 units daily (p≥100p \geq 100)
  • Nutrition: Consume *no more than* 2000 calories (c≤2000c \leq 2000)
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 75t+25f≤30075t + 25f \leq 300 where tt = tickets, ff = 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: Choose the right answer

Which inequality describes the budget?

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 c+2t≤500c + 2t \leq 500.

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: Choose the right answer

Which inequality describes the limit of 200 units of time?

Key Takeaways

  • 1Linear inequalities use <<, >>, ≤\leq, or ≥\geq instead of an equals sign
  • 2The boundary line is drawn by treating the inequality as an equation
  • 3Use a solid line for ≤\leq or ≥\geq (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

Pick a test point not on the line ((0,0)(0,0) is easiest). Substitute into the original inequality. If TRUE, shade that side. If FALSE, shade the opposite side.
Pick a test point not on the line ((0,0)(0,0) is easiest). Substitute into the original inequality. If TRUE, shade that side. If FALSE, shade the opposite side.
A dashed line means points ON the line are NOT solutions. This happens with strict inequalities (<< or >>). Solid lines (≤\leq or ≥\geq) include points on the line.
Choose a different test point like (1,0)(1, 0) or (0,1)(0, 1). Any point clearly not on the line works.

Glossary

Linear inequality
An inequality involving linear expressions, using <<, >>, ≤\leq, or ≥\geq
Boundary line
The line you get when you replace the inequality symbol with an equals sign
Solution set
All the ordered pairs (x,y)(x, y) 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
Today's challengeOne problem, three tries