Graphing Linear Inequalities
Linear inequalities y < mx + b or y > mx + b divide the plane into two half-planes. The boundary line y = mx + b is dashed (strict) or solid (inclusive). Shade the half-plane that satisfies the inequality.
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y < mx + b shades below the line; y > mx + b shades above. Systems of inequalities define polygonal feasible regions. Test point (0,0) works when the line does not pass through origin.
Ready to run the numbers?
Why: Inequality graphs model constraints in linear programming, optimization, and feasibility regions.
How: Plot the boundary line, pick a test point, shade the side where the inequality holds. For systems, shade the intersection.
Run the calculator when you are ready.
📐 Examples — Click to Load
Inequality 1: y op mx + b
Inequality 2: y op m₂x + b₂
Intercepts & Key Points
Above vs Below Distribution
📐 Calculation Steps
📊 Chart Interpretation
Bar chart: Compares y-intercept, x-intercept (if any), and key points. Use these to sketch the boundary line.
Doughnut chart: Shows the proportion of the plane above vs below the line. The shaded region satisfies the inequality.
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Dashed line = strict (< or >); solid = inclusive (≤ or ≥)
Feasible region = intersection of all inequality half-planes
📋 Key Takeaways
- • Solid vs dashed lines: Use solid for ≤ and ≥ (boundary included), dashed for < and > (boundary excluded)
- • Test point method: Substitute (0,0) into the inequality. If true, shade toward origin; if false, shade away
- • Feasible regions: The intersection of shaded half-planes in a system gives the feasible region
- • Linear programming: Systems of linear inequalities define constraints; optimal solutions lie at vertices
💡 Did You Know?
📖 How It Works
To graph y < 2x + 1: (1) Draw the boundary line y = 2x + 1 as dashed (strict inequality). (2) Pick a test point like (0,0). Is 0 < 1? Yes. (3) Shade the half-plane containing (0,0).
Intercepts
y-intercept: set x=0 → (0, b). x-intercept: set y=0 → (-b/m, 0) if m≠0.
System of Two Inequalities
Graph each inequality. The feasible region is where both shadings overlap. Vertices occur at line intersections.
Converting from General Form
Given Ax + By ≤ C: solve for y to get y ≤ (-A/B)x + C/B (flip the inequality if B < 0). Then m = -A/B and b = C/B.
📝 Worked Examples
Example 1: y < 2x + 1
Boundary: y = 2x + 1 (dashed). Test (0,0): 0 < 1 ✓. Shade below the line. The solution set is all points in the half-plane below the line.
Example 2: y ≥ -x + 3
Boundary: y = -x + 3 (solid). Test (0,0): 0 ≥ 3 ✗. Shade away from origin, i.e., above the line. The line is included.
Example 3: Budget 2x + 3y ≤ 12
Rewrite: y ≤ -⅔x + 4. Boundary solid. Test (0,0): 0 ≤ 4 ✓. Shade below. Feasible region: all affordable (x,y) combinations.
Example 4: System y > x and y < -x + 4
Two lines: y = x and y = -x + 4. Shade above the first, below the second. The overlap is a triangular feasible region.
🌐 Real-World Applications
Linear inequalities model constraints in business, engineering, and science. Budget limits, resource caps, and production capacity all translate to half-planes.
🔗 Related Concepts
Linear inequalities extend linear equations. The solution set is a half-plane instead of a line. Systems of inequalities define polygonal regions used in linear programming to maximize or minimize objective functions subject to constraints.
📊 Table: Inequality Symbols
| Symbol | Line Type | Shading |
|---|---|---|
| < | Dashed | Below line |
| > | Dashed | Above line |
| ≤ | Solid | Below line |
| ≥ | Solid | Above line |
🎯 Expert Tips
💡 Test Point (0,0)
Always try (0,0) first. If the line passes through origin, use (1,0) or (0,1) instead.
💡 Rewrite for y
Convert 2x + 3y ≤ 12 to y ≤ -⅔x + 4 to use slope-intercept form for graphing.
💡 Feasible Region
In LP, the optimum lies at a vertex of the feasible polygon. Check each vertex.
💡 Shading Direction
y < mx+b → below; y > mx+b → above. Use test point to confirm.
❓ FAQ
When do I use a dashed vs solid line?
Dashed for < and > (boundary not included). Solid for ≤ and ≥ (boundary included).
What is the test point method?
Pick a point not on the line (usually (0,0)). Substitute into the inequality. If true, shade the side containing that point.
What is a feasible region?
The set of points satisfying all inequalities in a system. In linear programming, it is a convex polygon.
How do I graph 2x + 3y ≤ 12?
Solve for y: y ≤ -⅔x + 4. Graph the line y = -⅔x + 4 (solid). Test (0,0): 0 ≤ 4. Shade below.
What is the connection to linear programming?
Inequalities define constraints. The objective function is optimized at vertices of the feasible region.
Can the feasible region be empty?
Yes. If the half-planes do not overlap, no point satisfies all inequalities.
What if the line passes through (0,0)?
Use a different test point such as (1,0) or (0,1). Any point not on the line works.
⚠️ Common Mistakes
1. Forgetting to flip when dividing by negative: When solving for y from Ax + By ≤ C, if B < 0, the inequality flips.
2. Wrong test point: Never use a point on the line. If (0,0) is on the line, use (1,0) or (0,1).
3. Dashed vs solid: Strict < and > use dashed; ≤ and ≥ use solid. Mixing these loses points!
4. Shading wrong side: Always verify with the test point. One wrong shade = wrong answer.
🔢 Quick Reference
📋 Step-by-Step Procedure
- Identify the boundary line: y = mx + b. Plot the y-intercept (0, b) and use slope m to find another point.
- Draw the line: solid for ≤ or ≥, dashed for < or >.
- Choose a test point not on the line. (0,0) is convenient unless it lies on the line.
- Substitute the test point into the inequality. If true, shade the side containing the point; if false, shade the other side.
- For systems: repeat for each inequality, then the feasible region is the overlap of all shadings.
The Bar chart shows intercepts to help you plot the line. The Doughnut chart visualizes the above/below split.
Use the clickable examples to load preset scenarios and practice interpreting results.
💡 Remember: Horizontal lines (m=0) like y ≤ 3 shade below; vertical lines require a different approach (x ≤ k shades left). This calculator focuses on non-vertical lines in slope-intercept form.
⚠️ Disclaimer: This calculator graphs linear inequalities in slope-intercept form. For general form Ax + By ≤ C, rewrite as y = (-A/B)x + C/B. Educational use only.
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