Systems of Inequalities

The Central Idea

Consider

The first inequality includes points on or above the line

The second includes points on or below

The solution to the system is the overlap:

Where the two boundary lines meet,

and

The point

is part of the boundary of the feasible region.

What Is a System of Inequalities?

A system of inequalities is a collection of inequalities considered simultaneously.

For example,

asks for all ordered pairs satisfying all three conditions.

Systems Use Logical “And”

A solution must satisfy

and

and every additional condition.

In set language, if the individual solution sets are

then the system solution is

A Single Linear Inequality in Two Variables

Consider

The related equation

is the boundary line.

The inequality selects one side of that line.

Because the inequality is strict, points on the boundary are excluded.

Boundary Lines

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

For

the boundary is

This line separates the plane into two half-planes.

Solid and Dashed Boundaries

Use a solid boundary line when equality is included:

Use a dashed boundary line when equality is excluded:

The line style records whether boundary points belong to the solution set.

Half-Planes

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

A linear inequality chooses one of these half-planes, possibly including the boundary.

For example,

includes the half-plane on or above the line

When the Inequality Is Solved for \(y\)

If the inequality is written

shade above the boundary line.

If it is written

Example: Shade Above

Graph

The boundary is

Because

includes equality, draw the line solid.

Shade the region above the line.

Example: Shade Below

Graph

The boundary is

Because the inequality is strict, draw a dashed line.

Shade below it.

Test Points

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

The origin

is often convenient.

Substitute it into the inequality.

If the inequality is true, shade the side containing the test point.

Example: Test the Origin

Graph

The boundary is

Test

Then

This is true.

So shade the side containing the origin.

When the Origin Is on the Boundary

For

the origin lies on the boundary line

So \((0,0)\) cannot distinguish the two sides.

Choose another test point, such as

Systems as Overlapping Half-Planes

Suppose

Graph the first permitted half-plane.

Graph the second.

The solution is the region where the shaded portions overlap.

The Feasible Region

In applications, the solution region of a system of inequalities is often called the feasible region.

Every point in the feasible region satisfies all constraints.

Every point outside violates at least one constraint.

Example: First-Quadrant Constraint

Suppose

The conditions

and

restrict the solution to the first quadrant.

The inequality

keeps points on or below the line

The feasible region is a triangle including its boundary.

Vertices of a Feasible Region

When boundary lines intersect, their intersection points often form vertices of the feasible region.

For

the vertices are

Why Vertices Matter

Vertices identify important boundary combinations.

Later, in linear optimization, maximum or minimum values of a linear objective over a bounded polygonal feasible region occur at vertices.

For now, vertices are useful landmarks for understanding the geometry of constraints.

Strict Systems Can Exclude Their Boundaries

Consider

The region lies inside the same triangular shape as before, but none of the three boundary lines is included.

The graph uses dashed boundaries.

Mixed Boundary Types

A system may mix strict and inclusive inequalities:

Then:

• \(x+y=6\) is solid;
• \(x=0\) is dashed;
• \(y=0\) is solid.

Checking a Point in a System

Consider

Test

First:

True.

Second:

True.

Third:

True.

Therefore,

is a solution of the system.

A Point That Fails One Condition

Test

in the same system.

Although

and

we have

false.

So the point is not in the feasible region.

No-Solution Systems

Consider

The first inequality requires points above the higher parallel line.

The second requires points below the lower parallel line.

No point can satisfy both.

Therefore,

Systems with an Unbounded Region

Consider

The solution lies above both boundary lines.

The feasible region extends upward indefinitely.

So the solution region is unbounded.

Bounded Versus Unbounded Regions

A feasible region is bounded if it fits inside some finite enclosure.

It is unbounded if it extends indefinitely in at least one direction.

Both kinds occur naturally in systems of linear inequalities.

Redundant Inequalities

Consider

If

then

is automatically true.

The fourth inequality adds no new restriction.

It is redundant.

Contradictory Constraints

Consider

No real number satisfies both conditions.

So the system has

as its solution set.

One-Variable Systems of Inequalities

Systems of inequalities also occur in one variable.

For example,

The solution is the overlap:

“And” Means Intersection

For

and

the solution is

The same intersection idea extends from number lines to coordinate planes.

Systems with Vertical Boundaries

Consider

The boundary lines are

and

The solution is the vertical strip between them, including both boundaries.

Systems with Horizontal Boundaries

Consider

The solution is the horizontal strip

Standard-Form Inequalities

Many constraints appear naturally as

For example,

may represent a resource limit.

There is no need to solve for \(y\) unless doing so makes graphing or interpretation easier.

Graphing a Standard-Form Inequality by Intercepts

Consider

The boundary line is

The intercepts are

and

Draw the solid line through those points.

Test the origin:

True.

So shade the side containing the origin.

Graphing After Solving for \(y\)

The same inequality

can be rewritten as

Now the direction “shade below” is immediately visible.

Be Careful with Negative Division

Consider

Solve for \(y\):

Multiply by \(-1\) and reverse the inequality:

Forgetting the reversal would shade the wrong side.

Applied Example: Production Constraint

Suppose product \(x\) requires \(2\) machine-hours and product \(y\) requires \(4\) machine-hours.

At most \(40\) machine-hours are available.

The capacity condition is

If production quantities cannot be negative, also require

Production Feasible Region

The system is

The boundary

has intercepts

and

The feasible region is the first-quadrant triangle on or below that line.

Adding a Second Resource Constraint

Suppose the same products also require labor:

Now the system becomes

The feasible region is the overlap of all four conditions.

Boundary Intersections Become Candidate Vertices

To find where

and

meet, solve the system of equations.

From

we get

Substitute:

Then

So

The boundary lines intersect at

Verify a Candidate Vertex

Check

in the original inequalities:

and

Both are satisfied exactly.

Because

and

the point is feasible.

Not Every Boundary Intersection Is Feasible

With several constraints, two boundary lines may intersect at a point that violates a third inequality.

Therefore, every candidate vertex should be checked against the entire system.

Budget Example

Suppose item \(x\) costs \$4 and item \(y\) costs \$7.

A buyer can spend at most \$70:

If quantities must be nonnegative:

If the items are indivisible, we may additionally require

Continuous Region Versus Discrete Feasible Points

The inequality system may describe a continuous polygonal region over real numbers.

But if \(x\) and \(y\) count objects, only integer lattice points inside that region are practically meaningful.

This distinction becomes important in integer programming.

Minimum Requirements

Suppose a blend must contain at least \(20\) units of one nutrient:

This inequality selects the side on or beyond the threshold line.

Systems may therefore mix upper bounds and lower bounds.

Example with Upper and Lower Bounds

Consider

The solution lies between the parallel lines

and

inside the first quadrant.

Feasible Bands

When one quantity must stay between lower and upper limits, the solution often forms a band.

For example,

means both

and

Systems Can Describe Geometry

A rectangle can be represented by

For example,

describes a closed rectangular region.

Triangular Regions

A triangular region can be formed by three nonparallel boundary inequalities.

For example,

The three boundary lines form a triangle.

Polygonal Regions

Several linear inequalities can combine to form polygons with many sides.

Each active boundary contributes part of the feasible region's edge.

This geometry is fundamental to later linear programming.

Inactive or Redundant Boundaries

An inequality may have a boundary line that never forms part of the final feasible region.

If another constraint is stricter everywhere relevant, the weaker boundary is inactive.

This is the geometric version of redundancy.

Systems of Strict Inequalities Can Be Open Regions

A system such as

describes the interior of a triangle without its sides.

Topologically, the feasible region is open relative to those boundaries.

Inclusive Systems Can Be Closed Regions

The system

includes all three boundary segments.

The triangular feasible region is closed and bounded.

Checking Feasibility Algebraically

A point belongs to the feasible region if substitution makes every inequality true.

This gives an exact algebraic check independent of the graph.

Graphing as a Representation of Logical Intersection

Each shaded half-plane represents one solution set.

The final overlap represents logical conjunction:

The graph is therefore a visual representation of set intersection and logic.

Systems of Inequalities and Optimization Preview

Suppose a business has a feasible production region.

A later problem may ask which feasible point maximizes profit or minimizes cost.

That adds an objective function to the constraint system.

The present lesson focuses only on describing the feasible region itself.

Objective Functions Are Separate from Constraints

For example, constraints may be

while profit might be

The inequalities define what is allowed.

The objective function measures how desirable each allowed point is.

Systems of Inequalities with More Variables

In three variables, a linear inequality such as

describes one side of a plane in three-dimensional space.

A system of such inequalities describes an intersection of half-spaces.

Higher-Dimensional Feasible Regions

In many variables, linear inequalities define higher-dimensional regions called polyhedra or, more generally, convex feasible sets.

The two-dimensional shaded polygon is the simplest visual case of a much broader idea.

Convexity Preview

Linear inequality feasible regions are convex.

Informally, this means that if two points are feasible, then the entire line segment connecting them is also feasible.

This property becomes important later in optimization and geometry.

Why Convexity Holds

Suppose both points satisfy

A weighted average of the two points also satisfies the same linear inequality.

Because intersections of convex sets remain convex, a system of linear inequalities also has a convex feasible region.

Test-Point Method for a Full System

To graph

graph each boundary:

Then test a convenient point for each inequality separately.

Finally, keep only the overlapping shaded region.

Boundary Intersections Help Organize the Graph

For the system

the boundary lines intersect where

and

From

we have

Substitute:

So the boundaries meet at

Parallel Boundary Systems

Consider

The boundaries are parallel.

The feasible region is the strip between them, including both lines.

Parallel Constraints Can Also Be Inconsistent

Consider

Now the required upper region starts above the required lower region.

No overlap exists.

The system is infeasible.

Coincident Boundaries

Consider

The inequalities are equivalent.

The second constraint adds no new information.

The system has the same solution set as either inequality alone.

Reversed Equivalent Boundaries

Consider

The second inequality simplifies to

Both can be true only when

The system solution collapses from a two-dimensional region to a line.

A System Can Have Lower-Dimensional Solutions

Although many systems of inequalities produce regions with area, some produce:

• a line;
• a line segment;
• a single point;
• the empty set.

The dimension of the solution set depends on how the constraints interact.

Single-Point Feasible Region Example

Consider

The only possible values are

So the feasible region is the single point

A Reliable Single-Inequality Graphing Workflow

1. Replace the inequality with equality to find the boundary line.
2. Graph the boundary accurately.
3. Use a solid line for \(\le\) or \(\ge\).
4. Use a dashed line for \(<\) or \(>\).
5. Use the solved-for-\(y\) direction or a test point to choose the correct side.

A Reliable System-Graphing Workflow

1. Graph each boundary line.
2. Mark each boundary as solid or dashed.
3. Shade the permitted half-plane for each inequality.
4. Identify the overlap of all shaded regions.
5. Find important boundary intersections.
6. Check representative points algebraically.

A Reliable Feasible-Region Workflow

1. List every constraint.
2. Add nonnegativity or domain restrictions when required.
3. Graph all boundaries.
4. Find the common overlap.
5. Identify candidate vertices.
6. Verify each candidate against every inequality.
7. Determine whether the region is bounded, unbounded, empty, or lower-dimensional.

A Reliable Applied-Constraint Workflow

1. Define the variables and units.
2. Translate phrases such as at most, at least, no more than, and no less than.
3. Construct one inequality per independent constraint.
4. Add physical restrictions such as \(x\ge0\) and \(y\ge0\).
5. Graph or solve for the feasible region.
6. Interpret the geometry in context.

Common Translation Phrases

Common Errors

Shading the Union Instead of the Intersection

A system requires every inequality to hold, so use the overlapping region.

Using a Solid Line for a Strict Inequality

\(<\) and \(>\) exclude the boundary and require dashed lines.

Using a Dashed Line for an Inclusive Inequality

\(\le\) and \(\ge\) include the boundary and require solid lines.

Forgetting to Reverse the Symbol When Solving for \(y\)

Dividing by a negative number reverses the inequality.

Testing a Point on the Boundary

A boundary point cannot distinguish which side satisfies a strict inequality. Choose a point off the line.

Ignoring Nonnegativity

Applied quantities such as production, time, and inventory often require \(x\ge0\) and \(y\ge0\).

Assuming Every Boundary Intersection Is Feasible

A candidate vertex must satisfy every inequality in the system.

Missing Redundant Constraints

Some inequalities add no new restriction and may never form part of the final boundary.

Assuming Every Feasible Region Is Bounded

Many systems produce unbounded strips, wedges, or half-planes.

Ignoring Discrete Domains

If variables count objects, only integer points may be meaningful even though the graphical region is continuous.

Practice

1. Check a Point

Does

satisfy

2. Boundary Type

Should the boundary of

be solid or dashed?

3. Shade Above or Below

For

which side of the boundary is shaded?

4. Test-Point Method

Does the origin satisfy

5. One-Variable System

Solve

6. No-Solution System

Determine the solution set:

7. First-Quadrant Feasible Region

What are the vertices of

8. Standard-Form Constraint

A product \(x\) uses \(2\) hours and product \(y\) uses \(5\) hours. At most \(50\) hours are available. Write the system including nonnegativity.

9. Boundary Intersection

Find the intersection of

and

10. Redundant Constraint

In

which inequality is redundant?

11. Strip Between Parallel Lines

Describe the solution of

12. Collapsed Region

Solve

Key Ideas

• A system of inequalities requires all inequality conditions to hold simultaneously.
• The solution set is the intersection of the individual solution sets.
• A linear inequality in two variables describes a half-plane separated by a boundary line.
• Use solid boundaries for \(\le\) and \(\ge\), and dashed boundaries for \(<\) and \(>\).
• Test points help determine which side of a boundary satisfies an inequality.
• The overlap of all permitted half-planes is the feasible region.
• Feasible regions may be bounded, unbounded, empty, or lower-dimensional.
• Boundary intersections often form vertices of polygonal feasible regions.
• Every candidate vertex must satisfy every original constraint.
• Redundant inequalities add no new restriction.
• Applied systems commonly include resource limits, minimum requirements, budgets, and nonnegativity constraints.
• Continuous graphical regions may need to be restricted to integer points when variables count discrete objects.
• Systems of linear inequalities form the foundation of feasible-region methods and linear optimization.

Where This Leads

Systems of inequalities extend simultaneous reasoning from isolated intersection points to entire regions of allowable values. Instead of asking where several equations are exactly equal, we ask where several upper bounds, lower bounds, thresholds, and domain restrictions all hold together.

From here, Systems and Matrix Methods can move deeper into substitution and elimination, augmented matrices, row operations, Gaussian elimination, matrix equations, determinants, rank, and eventually optimization problems where a linear objective is evaluated over a feasible region defined by systems of inequalities.