Step 2: Test point (0, 0). Plug into inequality. True → shade that side. False → shade opposite.
Step 3: Every point in the shaded region is a valid solution.
Enter your inequality below — the graph draws automatically.
What Are Linear Inequalities in Two Variables?
A linear inequality in two variables is a statement that two linear expressions are not equal — one is greater than, less than, greater than or equal to, or less than or equal to the other. In two variables x and y, they take the form:
Unlike a linear equation (which has a single line of solutions), a linear inequality has a half-plane of solutions — an infinite region on one side of a boundary line. Every point (x, y) in that shaded region makes the inequality true.
- Every linear inequality divides the coordinate plane into exactly two half-planes
- The boundary line is the graph of the related equation (replace the inequality with =)
- One half-plane satisfies the inequality; the other does not
- The boundary is included in the solution if the inequality is ≤ or ≥ (non-strict)
- The boundary is excluded if the inequality is < or > (strict)
- Solution sets contain infinitely many ordered pairs (x, y)
Solid vs. Dashed Boundary Line — The Key Visual Rule
The first thing to draw when graphing a linear inequality is the boundary line. But whether that line is solid or dashed tells the reader whether the line itself is part of the solution:
| Symbol | Name | Boundary Line | Is boundary a solution? | Example |
|---|---|---|---|---|
| ≤ | Less than or equal to | Solid line —— | Yes — included | 2x + y ≤ 6 |
| ≥ | Greater than or equal to | Solid line —— | Yes — included | x − 3y ≥ 0 |
| < | Strictly less than | Dashed line - - - | No — excluded | y < 2x + 1 |
| > | Strictly greater than | Dashed line - - - | No — excluded | 3x − y > 4 |
How to Determine Which Side to Shade
After drawing the boundary line, you need to shade the correct half-plane. The fastest method is the test point method:
2. Substitute (0, 0) into the original inequality (not the equation).
3. If the result is TRUE → shade the side that contains (0, 0).
If the result is FALSE → shade the side that does NOT contain (0, 0).
Example: Graph 2x + 3y ≤ 6
Boundary: 2x + 3y = 6 (solid line through (3,0) and (0,2))
Test (0,0): 2(0) + 3(0) = 0 ≤ 6 → TRUE → shade the side containing (0,0) = below-left of the line ✓
An alternative to the test point method is the slope-intercept rule: once the inequality is rearranged to y-form (y < mx+b or y > mx+b), the shading direction is clear — shade below the line for y < or y ≤, and above for y > or y ≥.
5-Step Method: How to Graph a Linear Inequality
Step 1 — Write the boundary equation.
Replace ≤ with = : 3x − 2y = 12
Step 2 — Find two points on the boundary line.
When x=0: −2y = 12 → y = −6 → point (0, −6)
When y=0: 3x = 12 → x = 4 → point (4, 0)
Step 3 — Draw the boundary line.
Connect (0, −6) and (4, 0). Since the symbol is ≤ (non-strict) → draw a solid line.
Step 4 — Test the origin (0, 0).
3(0) − 2(0) = 0 ≤ 12 → TRUE → shade the side containing (0, 0) = above-left of the line.
Step 5 — Shade and label.
Shade the upper-left half-plane. Every point in the shaded region satisfies 3x − 2y ≤ 12.
Quick check: Is (1, 1) a solution? 3(1)−2(1) = 1 ≤ 12 ✓ Is (5, 0) a solution? 3(5)−2(0)=15 ≤ 12? No ✗
Standard Form vs. Slope-Intercept Form
Linear inequalities can be written in two main forms. Each has advantages for graphing:
| Form | General Shape | Best for… | Graphing Tip |
|---|---|---|---|
| Standard Form | ax + by ≤ c | Both intercepts easy to find | Set x=0 for y-int, y=0 for x-int; plot both points |
| Slope-Intercept | y ≤ mx + b | Shading direction immediately obvious | Plot b, use slope rise/run for second point |
| Point-Slope | y−y₁ ≤ m(x−x₁) | Given a point and slope | Convert to slope-intercept first |
To convert Standard Form to Slope-Intercept: isolate y by subtracting ax from both sides and dividing by b. Critical: if b is negative, the inequality sign flips when you divide by b.
−2y ≥ 8 − 4x (subtract 4x from both sides)
y ≤ (8−4x)/(−2) ← FLIP the inequality when dividing by −2
y ≤ −4 + 2x
y ≤ 2x − 4 (final slope-intercept form)
Slope = 2, y-intercept = −4 · Shade below the line (y ≤)
Graphing Systems of Linear Inequalities
When you have two or more inequalities that must all be satisfied simultaneously, you have a system of linear inequalities. The solution is the intersection of all individual solution regions — the area shaded by every inequality at once.
Graph ①: x + y ≤ 6
Boundary: x+y=6 → (0,6) and (6,0). Solid line.
Test (0,0): 0+0=0 ≤ 6 ✓ → shade toward origin (below-left of line).
Graph ②: 2x − y ≥ 0
Boundary: 2x−y=0 → (0,0) and (3,6). Solid line.
Test (1,0): 2(1)−0=2 ≥ 0 ✓ → shade the right side of the line.
System solution: The darker overlapping region where both shadings coincide. Any point in that region satisfies both inequalities.
- If the shadings overlap → the system has infinitely many solutions (the overlap region)
- If the shadings don't overlap → the system has no solution
- The corner points (vertices) of the feasible region are found by solving pairs of boundary equations as a system
- In linear programming, the optimal solution always occurs at a vertex of the feasible region
Special Cases: Horizontal and Vertical Inequalities
When an inequality involves only x or only y, the boundary becomes a vertical or horizontal line — and the shading covers everything on one side:
| Inequality | Boundary | Shaded Region | Example |
|---|---|---|---|
| y ≤ k | Horizontal solid line y = k | Everything at or below y = k | y ≤ 3 → shade below y=3 |
| y > k | Horizontal dashed line y = k | Everything strictly above y = k | y > −1 → shade above y=−1 |
| x ≥ h | Vertical solid line x = h | Everything at or to the right of x = h | x ≥ 2 → shade right of x=2 |
| x < h | Vertical dashed line x = h | Everything strictly to the left of x = h | x < 4 → shade left of x=4 |
Quick Reference: Graphing Linear Inequalities
| Step / Rule | Detail |
|---|---|
| Boundary line type | Solid for ≤, ≥ · Dashed for <, > |
| Find x-intercept | Set y = 0 in boundary equation, solve for x |
| Find y-intercept | Set x = 0 in boundary equation, solve for y |
| Slope-intercept shading | y < mx+b → below; y > mx+b → above |
| Test point (standard) | Use (0,0); if true shade toward origin, else opposite |
| If line passes through origin | Use (1, 0) or (0, 1) as test point instead |
| Dividing by negative b | Inequality sign flips! 2x − 3y > 6 → y < (2x−6)/3 |
| Solution of system | Intersection of all individual shaded regions |
| No solution (system) | Shaded regions don't overlap (empty intersection) |
| Boundary point test | On solid boundary: plug in — should make ≤ or ≥ true with equality |
| Horizontal inequality | y [ineq] k → horizontal boundary; shade above (y>k) or below (y<k) |
| Vertical inequality | x [ineq] h → vertical boundary; shade right (x>h) or left (x<h) |
| Feasible region vertices | Solve pairs of boundary equations simultaneously |
| Number of solutions | Always infinite (a full half-plane) for one inequality |
Real-Life Applications of Linear Inequalities
- Budget planning — "I can spend at most £500 on food and clothing." If food costs £a per unit and clothing £b per unit, then ax + by ≤ 500. The shaded region shows all affordable combinations.
- Linear programming & business optimization — A factory produces two products with limited labor hours and raw materials. Each constraint becomes an inequality. The corner points of the feasible region are where profit is maximized or minimized.
- Nutrition and diet — "Protein intake must be at least 60g per day." A meal plan with food A providing p₁g and food B providing p₂g satisfies: p₁x + p₂y ≥ 60. The solution region contains all valid meal plans.
- Engineering & safety constraints — Bridge load limits, structural stress thresholds, and electrical current ratings are all inequalities. "Total load ≤ 50 tonnes" defines a design region.
- Scheduling & workforce planning — "Employees cannot work more than 40 hours per week." If two shift types have durations h₁ and h₂, then h₁x + h₂y ≤ 40. The inequality defines valid shift combinations.
- Finance and investment — Risk limits: "High-risk investments must not exceed 30% of the portfolio." If total portfolio is P, then x ≤ 0.3P constrains the allocation.
- Environmental regulations — Emission caps on two pollutants from a factory: ax + by ≤ c defines the legally allowable production mix.
- Retail and inventory — Warehouse capacity: x + y ≤ 500 units total. Combined with demand constraints, the feasible region shows optimal stock levels.
- Sports and performance — "Training time on skill A plus skill B must be at most 20 hours per week, and at least 5 hours on each." Multiple inequalities define the feasible training schedule.
Common Mistakes When Graphing Linear Inequalities
A dashed line signals that the boundary is NOT part of the solution. Drawing a solid line suggests points on the boundary are solutions, which is wrong for strict inequalities. Always double-check the symbol before drawing the line.
When converting −3y > 6 to y-form: divide both sides by −3 → y < −2 (sign flips!). Students who forget this shade the completely wrong half-plane — the most common error in this topic.
If the boundary line passes through (0,0), substituting it gives 0 [ineq] 0, which is on the boundary — not in either half-plane. Always pick a different test point like (1, 0), (0, 1), or (1, 1) in this case.
After the test, students sometimes shade the side that does NOT contain the test point even when the test was true. The rule is: if test point gives TRUE → shade that side (the side containing the test point). If FALSE → shade the other side.
In a system, only the OVERLAP of all individual shaded regions is the solution. Some students shade both regions separately and consider any point in either shade as a solution — which is wrong. Only the intersection counts.
Glossary
- Linear Inequality
- A mathematical statement comparing a linear expression to a constant using <, >, ≤, or ≥. In two variables, the solution set is a half-plane.
- Boundary Line
- The line corresponding to the related equation (with = instead of an inequality). It divides the coordinate plane into two half-planes.
- Half-Plane
- One of the two regions that a line divides the coordinate plane into. The solution to a linear inequality is one half-plane (possibly including the boundary).
- Solution Region
- The shaded half-plane containing all ordered pairs (x, y) that satisfy the inequality. Every point in the shaded region is a valid solution.
- Strict Inequality
- An inequality using < or >, which excludes the boundary line. The boundary is drawn as a dashed line.
- Non-Strict Inequality
- An inequality using ≤ or ≥, which includes the boundary line. The boundary is drawn as a solid line.
- Test Point Method
- A technique to determine which half-plane to shade: substitute a known point into the inequality; if true, shade that side; if false, shade the other side.
- System of Linear Inequalities
- Two or more linear inequalities that must be satisfied simultaneously. The solution is the intersection of all individual solution regions — called the feasible region.
- Feasible Region
- The intersection of solution sets for a system of inequalities. In linear programming, the optimal solution is found at a vertex (corner point) of this region.
- Slope-Intercept Form
- y [ineq] mx + b. When in this form, shading direction is immediate: shade above for >/≥ and below for </≤.
- Standard Form
- ax + by [ineq] c. Useful for finding intercepts by setting the other variable to zero.
- Linear Programming
- Optimization technique that uses systems of linear inequalities to define a feasible region, then maximizes or minimizes an objective function within that region.
Frequently Asked Questions
Q: How do you graph a linear inequality?
A: (1) Graph the boundary line (use = instead of the inequality). Make it solid for ≤/≥ or dashed for </>. (2) Test the origin (0,0) in the original inequality. (3) If true, shade toward (0,0). If false, shade the opposite side. Every point in the shaded half-plane is a solution.
Q: What is the difference between a solid and dashed boundary line?
A: A solid line means the boundary is included in the solution (≤, ≥). A dashed line means the boundary is excluded — only the strict interior of the half-plane is a solution (<, >). Think of it as: solid = "on the fence is OK"; dashed = "must be strictly inside."
Q: How do you determine which side to shade?
A: Use the test point method. Choose a point not on the line — usually (0,0). Substitute into the original inequality. If it's true, shade the side containing (0,0). If false, shade the opposite side. Alternatively, for y-form inequalities: shade below for </≤ and above for >/≥.
Q: How do you graph a system of linear inequalities?
A: Graph each inequality separately on the same coordinate plane. Shade each solution region. The system's solution is the overlapping region where all shadings coincide. Find the vertices of this feasible region by solving pairs of boundary equations simultaneously.
Q: What does the shaded region mean in a linear inequality graph?
A: Every point (x, y) in the shaded region satisfies the inequality — it's a valid solution. The region contains infinitely many solution pairs. Points on the solid boundary are also solutions; points on a dashed boundary are not.
Q: Can a linear inequality have infinitely many solutions?
A: Yes. A single linear inequality in two variables always has infinitely many solutions — the entire half-plane is the solution set. Even a system of two inequalities typically has infinitely many solutions (the intersection region), unless the feasible region degenerates to a point or is empty (no solution).
Q: What are linear inequalities used for in real life?
A: They model any situation with a constraint or limit: budget limits (spending ≤ income), production capacity (units ≤ machine hours), nutrition minimums (protein ≥ daily requirement), safety limits (load ≤ max weight), and linear programming optimization in business, engineering, and economics.