Feasible Regions

The Central Idea

Consider the inequality

This does not describe one point or one line.

The equation

forms the boundary line.

The inequality

includes every point on one side of that line, together with the line itself.

Now add another constraint:

Only points satisfying both conditions remain.

The common region is called the feasible region.

From Intervals to Regions

In one variable,

describes a ray on the number line.

In two variables,

describes a two-dimensional region.

The underlying logic is the same:

• a boundary divides the space;
• the inequality selects one side;
• several inequalities are combined by intersection.

What Does Feasible Mean?

A point is feasible if it satisfies every constraint in the problem.

A point is infeasible if it violates even one required constraint.

For example, suppose

and

The point

is feasible because

and

But

is infeasible because

is false.

Boundary Equations

To graph a linear inequality, first replace the inequality symbol with equality.

For

the boundary equation is

This line separates the plane into two half-planes.

The original inequality selects one of them.

Half-Planes

A line divides the coordinate plane into two regions called half-planes.

For the line

one half-plane contains points satisfying

and the other contains points satisfying

The line itself contains the points where

Inclusive and Strict Boundaries

If the inequality uses

or

the boundary line belongs to the solution set.

If the inequality uses

or

the boundary line is excluded.

This is the two-dimensional analogue of closed and open endpoints on a number line.

Solid and Dashed Boundaries

When graphing by hand, an included boundary is traditionally drawn as a solid line.

An excluded boundary is traditionally drawn as a dashed line.

Thus:

uses a solid boundary line, while

uses a dashed boundary line.

Determine Which Side to Keep

Once the boundary line is drawn, we must determine which side satisfies the inequality.

One method is to solve for \(y\).

For example,

becomes

This means points whose \(y\)-coordinates lie on or below the line

Testing a Point

When the correct side is not immediately obvious, choose a test point not lying on the boundary.

The origin

is often convenient.

For

substitute \((0,0)\):

This is true, so the half-plane containing the origin is the solution side.

When the Origin Is Not a Good Test Point

If the boundary passes through

then the origin cannot distinguish the two sides.

For example,

has boundary

which contains the origin.

Choose another point such as

Then

is true, so the side containing \((0,1)\) is selected.

Several Constraints Mean Intersection

Suppose a problem requires

and

Each inequality defines a half-plane.

The feasible region is the intersection of all three.

A point must lie in every selected half-plane simultaneously.

Nonnegativity Constraints

Applied models often include

and

These are called nonnegativity constraints.

They restrict the solution to the first quadrant when \(x\) and \(y\) represent quantities that cannot be negative, such as:

• products;
• hours;
• people;
• distance;
• materials;
• money;
• or other counts and amounts.

Example: Three Constraints

Consider

The first inequality keeps points on or to the right of the \(y\)-axis.

The second keeps points on or above the \(x\)-axis.

The third keeps points on or below the line

The intersection is a triangular region in the first quadrant.

Boundary Intersections

The corners of a polygonal feasible region occur where boundary lines intersect.

For

and

the intersection points are

and

These are called vertices or corner points of the feasible region.

Vertices Come from Active Constraints

At a vertex, two or more boundary constraints are usually satisfied with equality.

For example, at

we have

and

These constraints are active at that point because the point lies exactly on their boundaries.

Inactive Constraints

A constraint may be satisfied without being exactly on its boundary.

At

the constraint

is satisfied because

It is not active there because equality does not hold.

Bounded Feasible Regions

A feasible region is bounded if it fits inside some finite part of the plane.

For example,

forms a bounded triangle.

Neither \(x\) nor \(y\) can grow without limit while remaining feasible.

Unbounded Feasible Regions

A feasible region is unbounded if it extends indefinitely in at least one direction.

For example,

contains points arbitrarily far from the origin.

Unbounded does not mean “no solution.” It means the solution set has no finite outer boundary in at least one direction.

Empty Feasible Regions

Sometimes constraints are incompatible.

For example,

and

cannot both be satisfied.

The feasible region is therefore

This is the two-dimensional analogue of contradictory inequalities on the number line.

A Feasible Region Can Be a Line

Suppose

and

Both constraints hold only when

So the feasible set collapses from a two-dimensional region to a one-dimensional line.

A Feasible Region Can Be a Point

Enough independent constraints can reduce the feasible set to one point.

For example,

and

force

The feasible region is then the singleton set

Dimension Can Collapse

A feasible set does not have to fill an area.

Depending on the constraints, it may be:

• a two-dimensional region;
• a line segment;
• a ray;
• a full line;
• a single point;
• or the empty set.

Constraints can reduce the available degrees of freedom.

Testing Whether a Point Is Feasible

Suppose the constraints are

and

Test

We have

and

Therefore, \((2,2)\) is feasible.

One Failed Constraint Is Enough

Using the same system, test

The first three conditions give

Once one required constraint fails, the point is infeasible.

There is no need for it to fail every constraint.

Finding Vertices Algebraically

Suppose two boundary lines are

and

Solve the two equations simultaneously.

From the first,

Substitute into the second:

Simplify:

Then

So the boundary lines intersect at

Not Every Boundary Intersection Is Feasible

When several constraints are present, the intersection of two boundary lines is only a candidate vertex.

It must still satisfy every other constraint.

A line intersection outside the common shaded region is not a vertex of the feasible region.

Candidate Vertices and Verification

A reliable vertex workflow is:

1. identify relevant pairs of boundary lines;
2. solve each pair for its intersection;
3. substitute the candidate point into every constraint;
4. keep only the feasible intersections.

This parallels the distinction between candidate solutions and verified solutions in equation solving.

Strict Inequalities and Missing Edges

Consider

and

The triangular interior is feasible, but points on

are excluded.

The region therefore has an open edge along that boundary.

Mixed Boundary Inclusion

A feasible region can contain some boundary segments but exclude others.

For example,

excludes the \(y\)-axis, includes the \(x\)-axis, and includes the sloping line

Boundary inclusion must be tracked constraint by constraint.

Linear Constraints

A linear constraint in two variables has a form such as

Its boundary is the line

The feasible side is one of the two half-planes determined by that line.

Several linear constraints produce intersections of half-planes.

Polygonal Feasible Regions

When a finite collection of linear inequalities forms a bounded feasible set, the boundary is often a polygon.

Its sides lie along boundary lines, and its vertices occur where active boundaries meet.

This geometric structure becomes important in optimization.

Why Feasible Regions Matter in Optimization

Suppose a business can produce quantities \(x\) and \(y\), but labor, material, budget, and capacity impose constraints.

The feasible region contains every production plan that obeys all restrictions.

An optimization problem then asks:

Among all feasible points, which one makes an objective such as profit largest or cost smallest?

Optimization comes after feasibility.

Constraints Before Objectives

A point with excellent profit is irrelevant if it violates required constraints.

Therefore, modeling often proceeds in this order:

1. define variables;
2. write constraints;
3. determine the feasible region;
4. then evaluate an objective over that region.

Application: Production Limits

Suppose a shop makes products \(A\) and \(B\).

Let

and

Suppose labor imposes

and material imposes

Also, production quantities cannot be negative:

The feasible region contains all production combinations that satisfy every limitation.

Context May Require Integer Points

The graphical feasible region for production quantities may contain noninteger points such as

But if \(x\) and \(y\) count indivisible objects, only lattice points with integer coordinates may be physically meaningful.

The continuous feasible region is still useful, but context may restrict the actual choices to

Continuous Versus Discrete Feasibility

If quantities can vary continuously, every point in the region may be allowed.

If quantities must be integers, only selected points in the region are feasible.

This distinction leads toward integer programming and discrete optimization later.

Application: Budget and Minimum Requirements

Suppose two activities cost \$4 and \$6 per unit, with a budget of at most \$60:

Suppose at least \(5\) total units are required:

And suppose

The feasible region is the set of nonnegative combinations satisfying both the budget ceiling and the minimum-output requirement.

Redundant Constraints

A constraint is redundant if removing it does not change the feasible region.

For example, suppose

and

The last constraint adds nothing because every point satisfying

already satisfies

Dominating Constraints

When two constraints limit the same direction, one may be stronger than the other.

For example,

is stronger than

If both are required, the second is redundant.

Recognizing redundancy can simplify a model before graphing or solving.

Constraints Can Encode Geometry

The inequalities

describe a rectangle.

The inequalities

describe a triangle.

Different systems of inequalities create different geometric regions.

Nonlinear Feasible Regions

Feasible regions are not limited to straight-line boundaries.

For example,

describes the disk inside and on a circle of radius \(5\).

The constraint

describes the region on or above a parabola.

Linear inequalities produce polygonal regions, while nonlinear inequalities can produce curved boundaries.

Boundary Logic Still Generalizes

For a nonlinear inequality such as

the boundary equation is

The inequality then selects one side of that boundary: the interior of the circle.

The same boundary-and-region reasoning still applies.

Graphical Verification

A graph can help verify whether an algebraically described region makes sense.

Useful checks include:

• Does each boundary line match its equation?
• Is the correct side selected?
• Are strict boundaries excluded?
• Does the common region satisfy every constraint?
• Are claimed vertices actually on the relevant boundaries?

Algebraic Verification

A graph is useful, but exact verification comes from substitution.

If a point appears feasible, substitute it into every original inequality.

If a vertex is found graphically, solve the relevant boundary equations algebraically when an exact coordinate is needed.

Graphical and symbolic methods strengthen one another.

Worked Example: Basic Feasible Region

Describe the region satisfying

The boundaries are

and

The feasible region is the triangle in the first quadrant bounded by these lines.

Its vertices are

and

Worked Example: Two Sloping Constraints

Consider

The sloping boundary lines intersect at

The axis intersections relevant to the feasible region include

and

Together with

these points form the vertices of a bounded polygonal feasible region.

Worked Example: Incompatible Constraints

Consider

and

The first requires points on one side of

while the second requires points on the opposite side of the parallel line

No point satisfies both.

Therefore,

Worked Example: Unbounded Region

Consider

The first two constraints keep the region in the first quadrant.

The third keeps points on or above the line

The solution extends indefinitely upward and rightward.

So the feasible region is unbounded.

A Reliable Graphing Workflow

1. Write each inequality clearly.
2. Replace each inequality symbol with equality to obtain its boundary.
3. Graph each boundary.
4. Use a solid boundary for \(\le\) or \(\ge\), and a dashed boundary for \(<\) or \(>\).
5. Determine the correct side of each boundary.
6. Keep only the overlap of all required regions.
7. Identify vertices or special boundary points when useful.
8. Verify representative points algebraically.

A Reliable Feasibility Workflow

1. Define the variables and their domains.
2. Translate each restriction into an inequality.
3. Include contextual restrictions such as nonnegativity or integrality.
4. Construct the common feasible region.
5. Check whether the region is empty, bounded, unbounded, or lower-dimensional.
6. Verify candidate points and vertices against every original constraint.

Common Errors

Graphing Only the Boundary

An inequality usually represents a region, not merely the line where equality holds.

Shading the Wrong Side

Use a test point or solve for \(y\) to determine which half-plane satisfies the inequality.

Using a Solid Boundary for a Strict Inequality

Strict inequalities exclude the boundary.

Forgetting That All Constraints Must Hold

The feasible region is an intersection, not a union, when every condition is required.

Assuming Every Line Intersection Is a Vertex

Candidate intersections must satisfy all other constraints.

Ignoring Contextual Restrictions

Negative or fractional values may be mathematically feasible but physically meaningless in a particular model.

Confusing Unbounded with Empty

An unbounded region has infinitely many feasible points; an empty region has none.

Practice

1. Test a Point

Does

satisfy

2. Identify a Boundary

What is the boundary line of

3. Included or Excluded?

Is the boundary of

part of the feasible region?

4. Bounded or Unbounded?

Classify the feasible region:

5. Empty Region

What is the feasible set of

and

6. Find a Boundary Intersection

Find the intersection of

and

7. Contextual Feasibility

Suppose \(x\) and \(y\) count whole products. Is

a feasible production plan even if it satisfies all continuous inequalities?

Key Ideas

• A feasible region is the set of all points satisfying every required constraint.
• A linear inequality in two variables defines a half-plane bounded by a line.
• The boundary is included for \(\le\) and \(\ge\), and excluded for \(<\) and \(>\).
• Several required inequalities combine by intersection.
• Test points help determine which side of a boundary satisfies an inequality.
• Nonnegativity constraints often restrict applied models to the first quadrant.
• Boundary intersections can produce candidate vertices, but every candidate must satisfy all constraints.
• Feasible regions may be bounded, unbounded, lower-dimensional, a single point, or empty.
• Redundant constraints do not change the feasible region.
• Context may require additional restrictions such as integer coordinates.
• Linear constraints produce polygonal feasible regions; nonlinear constraints can produce curved regions.
• Feasibility is determined before optimization: only feasible points are legitimate candidates for an objective.

Where This Leads

Feasible regions extend the idea of solution sets from intervals on a number line to regions in the coordinate plane.

This prepares the way for systems of inequalities, linear programming, optimization, constraint modeling, convex regions, and later multidimensional problems in which equations and inequalities define the space of allowable solutions.