Graphing Linear Inequalities
Learn to graph inequalities on a coordinate plane using boundary lines and shading.
Definition
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Worked Examples
Graph the inequality on a coordinate plane.
Identify the boundary line
Treat it as . Slope , y-intercept → Boundary:
Determine the line style
Since the symbol is (not ), points ON the line are NOT included → Use a DASHED line
Draw the boundary line
Plot y-intercept , use slope to find , → Dashed line through these points
Choose a test point
Use since it's easy to calculate and not on the line → Test point:
Substitute into inequality
TRUE! → is a solution
Shade the correct region
Since works, shade the side containing → Shade ABOVE the line
Answer: Dashed line through with slope 1, shaded 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.
Interactive Visual
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y = 2x + 1
m=2, b=1
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Practice Problems
15 problemsWhat type of line do you use for ?
Why It Matters
- 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 ()
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.
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 .
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
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