Math101 · Independent Ontario learning library
Math101learn.math101.caLinear Inequalities in Two Variables
A linear inequality in two variables describes a half-plane of solutions separated by a boundary line. Points on the boundary are included for $\le$ or $\ge$ and excluded for $<$ or $>$.
Half-planes model feasible combinations in budgeting, optimization, design, and systems of constraints. Graphing turns infinitely many solutions into one region.
Intuition and core definition
A linear inequality in two variables describes a half-plane of solutions separated by a boundary line. Points on the boundary are included for $\le$ or $\ge$ and excluded for $<$ or $>$. Every shaded point, not just lattice points, satisfies the inequality.
Notation, language, and conditions
$Ax+By<C$ has boundary $Ax+By=C$. A solid line marks inclusion; a dashed line marks exclusion. In slope-intercept form $y>mx+b$, shading is above the line and $y<mx+b$ shades below, but a test point is safer for general forms.
Why this idea matters
A two-variable linear inequality models an entire feasible half-plane, with its boundary style showing whether equality is permitted.
A dependable method
- Replace the inequality symbol with equality and graph the boundary line.
- Choose solid or dashed style based on endpoint inclusion.
- Select a test point not on the boundary, often $(0,0)$.
- Substitute it into the original inequality.
- Shade the side containing the test point if true, otherwise the opposite side, and verify another point.
Worked example
Representations and interpretation
The equation gives a one-dimensional boundary; the inequality adds a two-dimensional region. An input-output table can generate boundary points, while substitution classifies points on either side.
Reasoning about variations
Multiplying an inequality by $-1$ reverses its symbol but represents the same region: $2x+y\le4$ is equivalent to $-2x-y\ge-4$. The graph should not change even though the written comparison direction does.
Common mistakes
How to check your work
- Substitute one shaded and one unshaded point.
- Verify a boundary point makes the corresponding equation true.
- Rearrange to slope-intercept form and compare the vertical shading description.
Practice
- For $y>3x-2$, is $(1,2)$ a solution?
- Should the boundary of $x-2y<6$ be solid or dashed?
- Does $(2,2)$ lie in the solution half-plane of $x+y\ge1$?
Answers and brief solutions
Show answers
- Yes $2>3(1)-2=1$.
- Dashed The strict symbol excludes equality.
- Yes Since $2+2=4\ge1$, the point lies in the half-plane on or above the boundary $x+y=1$.
Synthesis and transfer
A production constraint such as labour hours can be graphed with other constraints; testing a proposed production point determines feasibility without relying on visual shading alone.
Let $x$ and $y$ be quantities of two products, with labour constraint $2x+3y\le18$. The boundary $2x+3y=18$ uses all available labour, while the side containing $(0,0)$ represents combinations using no more than the limit. Nonnegativity constraints restrict the meaningful region to the first quadrant. A point such as $(3,4)$ is feasible because $6+12=18$ and lies on the solid boundary; $(6,3)$ is not because it requires $21$ units. With several constraints, the feasible set is their intersection, and an optimum—if sought—can be compared at corner points after the region itself is verified.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For $y>3x-2$, is $(1,2)$ a solution?
- $2>3(1)-2=1$.
End of lesson
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