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{{T}}
[[Category:English]]
[[Category:Systems of Linear Inequalities]]

{{BR}}
= Introduction =

A '''system of linear inequalities''' contains two or more linear inequalities that use the same variables. Your goal is to find all ordered pairs that make '''every inequality in the system true at the same time'''. For two variables, the solutions can be shown on a coordinate plane as the region where the shaded half-planes overlap.

This topic connects [[English:Linear inequality|linear inequalities]], [[English:Linear equation|linear equations]], [[English:Coordinate system|coordinate planes]], [[English:Graph of a function|graphs]], [[English:Systems of linear equations|systems of equations]], and an introduction to [[English:Linear programming|linear programming]]. It is especially useful when a real situation has several limits at once, such as a budget, a capacity, a minimum requirement, or a time restriction.

By the end of this aiMOOC, you should be able to:
# [[English:Graphing linear inequalities|Graph a linear inequality in two variables]]: Draw the boundary line, choose the correct line style, and shade the correct half-plane.
# [[English:Systems of inequalities|Solve a system graphically]]: Identify the overlap that satisfies all inequalities.
# [[English:Verification|Check solution points]]: Substitute coordinates into every inequality and decide whether the point belongs to the solution set.
# [[English:Mathematical modeling|Model constraints]]: Translate real-world limits into linear inequalities and interpret the feasible region.
# [[English:Linear programming|Connect constraints to optimization]]: Recognize how a system of inequalities defines the feasible region for an optimization problem.

[[File:Cartesian-coordinate-system.svg|500px|frameless|center]]

The coordinate plane is the setting for graphing two-variable inequalities. The horizontal axis represents x, the vertical axis represents y, and each point represents an ordered pair.

{{#ev:youtube|https://www.youtube.com/watch?v=unSBFwK881s|500|center}}

{{BR}}
= From Linear Equations to Linear Inequalities =

{{BR}}
== Boundary Lines and Half-Planes ==

A linear equation such as y = 2x + 1 has a line as its graph. A related inequality such as y > 2x + 1 has infinitely many solutions on one side of that line. The line y = 2x + 1 is called the '''boundary line'''. It separates the plane into two '''half-planes'''.

The inequality symbol tells you whether the boundary belongs to the solution set:
# [[English:Strict inequality|Strict inequalities]]: Use a dashed boundary for < or > because points on the boundary do not satisfy the inequality.
# [[English:Inclusive inequality|Inclusive inequalities]]: Use a solid boundary for ≤ or ≥ because points on the boundary do satisfy the inequality.

[[File:Linearineq1.svg|500px|frameless|center]]

A shaded half-plane represents all points that satisfy one inequality. A system requires you to satisfy several inequalities simultaneously.

{{#ev:youtube|https://www.youtube.com/watch?v=EoCeL4SPIcA|500|center}}

{{BR}}
== Choosing Which Side to Shade ==

When an inequality is already written in slope-intercept form, the direction is often easy to read:
# [[English:Greater than|y greater than a line]]: Shade above the boundary.
# [[English:Less than|y less than a line]]: Shade below the boundary.
# [[English:Greater than or equal to|y greater than or equal to a line]]: Shade above and include the boundary.
# [[English:Less than or equal to|y less than or equal to a line]]: Shade below and include the boundary.

For an inequality that is not easy to interpret directly, use a '''test point'''. The point (0, 0) is convenient unless it lies on the boundary. Substitute its coordinates into the inequality. If the resulting statement is true, shade the side containing the test point. If it is false, shade the other side.

Example: For 2x + y < 6, test (0, 0). The statement 2·0 + 0 < 6 is true, so the half-plane containing the origin is part of the solution set.

{{BR}}
== Rearranging Inequalities Safely ==

You can rearrange a linear inequality much like a linear equation. There is one essential exception: '''when you multiply or divide both sides by a negative number, reverse the inequality sign'''.

Example: Start with -2y > 4x - 8. Dividing every term by -2 gives y < -2x + 4. Because you divided by a negative number, > changes to <.

This rule matters when you rewrite an inequality before graphing it.

{{BR}}
= Solving a System by Graphing =

{{BR}}
== The Core Idea: Intersection ==

Suppose a system contains:
y ≥ x - 1
y < -2x + 5

First graph y = x - 1 with a solid line because ≥ includes the boundary. Shade above that line. Then graph y = -2x + 5 with a dashed line because < excludes the boundary. Shade below that line. The solution set is the '''overlap''' of the two shaded regions.

A point belongs to the system only if it satisfies '''both''' inequalities. For example, (0, 0) works because 0 ≥ -1 and 0 < 5.

[[File:Linear Programming Feasible Region.svg|500px|frameless|center]]

The highlighted region in a system is often called the '''feasible region''': the set of points that satisfy every constraint at once.

{{#ev:youtube|https://www.youtube.com/watch?v=5xQqwgS3O4U|500|center}}

{{BR}}
== Step-by-Step Graphing Method ==

Use this method for most systems of two linear inequalities:
# [[English:Boundary line|Draw each boundary line]]: Replace the inequality sign with an equals sign to identify the line.
# [[English:Line style|Choose solid or dashed]]: Use solid for ≤ or ≥ and dashed for < or >.
# [[English:Shading|Shade each solution half-plane]]: Use the direction of y or test a point.
# [[English:Intersection|Find the overlap]]: The region shaded for every inequality is the system's solution set.
# [[English:Check a solution|Verify a point]]: Substitute a point from the overlap into every original inequality.

Keep the original inequalities available while graphing. This reduces errors caused by changing signs or forgetting whether a boundary is included.

{{BR}}
== Systems With Different Types of Solution Sets ==

A system can have several geometric outcomes.

'''A bounded solution region''' is enclosed on all sides. For example, several inequalities can form a triangle or quadrilateral.

'''An unbounded solution region''' extends indefinitely in at least one direction. It can still contain infinitely many valid solutions.

'''No solution''' occurs when the required half-planes do not overlap. For example, y > x + 2 and y ≤ x cannot be true at the same point.

'''A lower-dimensional solution set''' can occur when inclusive constraints force solutions onto a line segment, ray, or single point.

[[File:Feasible region.jpg|500px|frameless|center]]

{{BR}}
= Checking Solutions Algebraically =

A graph gives a visual answer, but substitution gives an exact check.

Consider the system:
x + y ≤ 8
x ≥ 2
y > 1

To test the point (3, 4):
# For x + y ≤ 8, 3 + 4 ≤ 8 is true.
# For x ≥ 2, 3 ≥ 2 is true.
# For y > 1, 4 > 1 is true.

Because all three statements are true, (3, 4) is a solution.

To test the point (1, 4), the first inequality is true and the third is true, but x ≥ 2 is false. Therefore, (1, 4) is not a solution to the system.

This demonstrates an important logical idea: a point must satisfy '''every''' constraint, not just most of them.

{{BR}}
= Modeling Real Constraints =

{{BR}}
== Translating Words Into Inequalities ==

Many systems begin as word problems. Look for quantities that vary, define variables, and translate each restriction.

Suppose a school event sells adult tickets for 8 units of currency and student tickets for 5 units. Let x be the number of adult tickets and y the number of student tickets. If the room holds at most 200 people and the event needs at least 1000 units of ticket revenue, possible constraints are:
x + y ≤ 200
8x + 5y ≥ 1000
x ≥ 0
y ≥ 0

The first inequality is a capacity limit. The second is a minimum revenue requirement. The last two say that ticket counts cannot be negative.

A solution such as x = 100 and y = 60 can be checked against every condition. The system does not describe one answer; it describes all combinations that are allowed.

{{BR}}
== Constraint Language ==

Common phrases often signal specific inequality symbols:
{| class="wikitable"
! Phrase
! Mathematical meaning
|-
| at most
| less than or equal to
|-
| no more than
| less than or equal to
|-
| at least
| greater than or equal to
|-
| no fewer than
| greater than or equal to
|-
| more than
| greater than
|-
| less than
| less than
|}

Always interpret the meaning of the situation rather than relying only on a memorized phrase. Units, context, and whether equality is possible all matter.

{{BR}}
= From Feasible Regions to Optimization =

A system of linear inequalities can act as a set of constraints for a [[English:Linear programming|linear programming]] problem. The overlap of the constraints is the feasible region. An '''objective function''' represents something to maximize or minimize, such as profit, cost, distance, or output.

For a nonempty bounded polygonal feasible region in a two-variable linear program, at least one optimal solution to a linear objective function occurs at a vertex. At Grades 9–10, the important idea is the connection: inequalities describe what is allowed, while the objective function expresses what outcome you want to improve.

[[File:Linopt-feasible-region2.png|500px|frameless|center]]

This diagram adds an objective line and an optimal point to a feasible region. It shows why systems of inequalities are a foundation for optimization.

{{#ev:youtube|https://www.youtube.com/watch?v=htscKOfe7yk|500|center}}

{{BR}}
= Common Errors and How to Avoid Them =

'''Error 1: Using the wrong boundary style.''' Remember that < and > exclude the boundary, while ≤ and ≥ include it.

'''Error 2: Shading the wrong side.''' Use a test point whenever you are uncertain.

'''Error 3: Forgetting to reverse the sign after division by a negative number.''' This can change the entire half-plane.

'''Error 4: Treating one inequality as the whole system.''' The answer is the intersection of all solution sets.

'''Error 5: Trusting the graph without checking.''' A point near a boundary can be hard to read accurately. Substitute coordinates into the original inequalities.

'''Error 6: Ignoring real-world restrictions.''' Quantities such as time, people, or products may need nonnegative or whole-number constraints even when the graph shows more real-number solutions.

{{BR}}
= Worked Example =

A community garden has two types of plots. Let x represent the number of small plots and y the number of large plots. The plan must satisfy:
x + y ≤ 12
2x + 3y ≤ 30
x ≥ 0
y ≥ 0

The first constraint limits the total number of plots. The second could represent a resource limit, such as total labor units. The nonnegative constraints keep the model in the first quadrant.

To graph the system, draw x + y = 12 and 2x + 3y = 30 as solid lines, because both resource limits use ≤. Then include the first-quadrant boundaries x = 0 and y = 0. Shade the points that satisfy all four inequalities. The overlap is a bounded feasible region.

Check the point (6, 4): 6 + 4 ≤ 12 is true, and 2·6 + 3·4 ≤ 30 gives 24 ≤ 30, which is also true. Therefore, (6, 4) lies in the feasible region.

Check the point (8, 4): 8 + 4 ≤ 12 is true, and 2·8 + 3·4 ≤ 30 gives 28 ≤ 30, which is also true, so (8, 4) is also feasible.

Check the point (9, 4): 9 + 4 ≤ 12 is false, so the point is not feasible even though the second constraint gives 30 ≤ 30.

The example shows why each constraint must be checked separately.

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''What part of a graph represents the solution to a system of linear inequalities?'''
(The region where all shadings overlap)
(!Every boundary line)
(!Only the x axis)
(!Only the darkest single boundary)

{{E}}
<br>

{{MC}}

'''Which boundary style is used for a strict inequality such as y > 2x + 1?'''
(A dashed line)
(!A solid line)
(!A double line)
(!No line at all)

{{E}}
<br>

{{MC}}

'''Which boundary style is used for y ≤ 3x - 4?'''
(A solid line)
(!A dashed line)
(!A dotted point)
(!A curved line)

{{E}}
<br>

{{MC}}

'''Why is a test point used when graphing a linear inequality?'''
(To decide which side of the boundary to shade)
(!To calculate the slope of every line)
(!To turn the inequality into an equation)
(!To remove one variable from the system)

{{E}}
<br>

{{MC}}

'''What happens to an inequality sign when both sides are divided by a negative number?'''
(The inequality direction reverses)
(!The inequality becomes an equation)
(!The variables disappear)
(!The inequality direction stays unchanged)

{{E}}
<br>

{{MC}}

'''What must be true of a solution point for a system with three inequalities?'''
(It satisfies all three inequalities)
(!It satisfies exactly one inequality)
(!It lies on every boundary line)
(!It has two positive coordinates)

{{E}}
<br>

{{MC}}

'''What does a feasible region represent?'''
(All points satisfying every constraint)
(!Only points on the x axis)
(!Only the steepest boundary)
(!All points outside the graph)

{{E}}
<br>

{{MC}}

'''What does the constraint x ≥ 0 usually mean in a real world model?'''
(The value of x cannot be negative)
(!The value of x must be zero)
(!The value of x must be one)
(!The value of x has no restriction)

{{E}}
<br>

{{MC}}

'''When does a system of two linear inequalities have no solution?'''
(When the required half planes do not overlap)
(!When both lines are solid)
(!When the origin is not shaded)
(!When both inequalities use x and y)

{{E}}
<br>

{{MC}}

'''Why should you substitute a candidate point into the original inequalities?'''
(To verify that the point satisfies every condition)
(!To change a solid line into a dashed line)
(!To make the coordinate plane larger)
(!To replace the system with one equation)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Boundary line || The line that separates the plane into two half-planes
|-
| Half-plane || One side of a boundary line containing infinitely many points
|-
| Strict inequality || A comparison using less than or greater than that excludes equality
|-
| Test point || A point substituted into an inequality to determine the correct side to shade
|-
| Feasible region || The set of points that satisfies all constraints simultaneously
|-
| Constraint || A mathematical restriction on the possible values of variables
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Dashed boundary'''
| Used when equality is not included
|-
| '''Solid boundary'''
| Used when equality is included
|-
| '''Shade above'''
| Typical direction for y greater than a boundary expression
|-
| '''Shade below'''
| Typical direction for y less than a boundary expression
|-
| '''Overlap region'''
| Points that satisfy every inequality in the system
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Boundary || What line separates the two half-planes of a linear inequality?
|-
| Feasible || What word describes a region whose points satisfy all constraints?
|-
| Overlap || What region is the solution when several shaded sets intersect?
|-
| Dashed || What type of boundary line represents a strict inequality?
|-
| Solid || What type of boundary line represents an inclusive inequality?
|-
| Constraint || What is a restriction in a mathematical model called?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Systems+of+Linear+Inequalities </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
A linear inequality in two variables divides the plane with a { boundary line }. A strict inequality is drawn with a { dashed } boundary. An inclusive inequality is drawn with a { solid } boundary. The solution to one two-variable linear inequality is a { half-plane }. The solution to a system is the { overlap } shared by all inequalities. You can use a { test point } to decide which side of a line satisfies an inequality. In a real-world model, each mathematical restriction is a { constraint }. The set of points satisfying every constraint is the { feasible region }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Boundary line sketch|Boundary line sketch]]: Draw four linear inequalities on coordinate grids, using two strict and two inclusive inequalities, and explain why each boundary is dashed or solid.
# [[English:Test point explanation|Test point explanation]]: Choose one inequality, test the origin or another simple point, and write two or three sentences explaining how the test determines the shaded side.
# [[English:Solution point check|Solution point check]]: Create a system of two inequalities and test three points by substitution, including at least one solution and one non-solution.
# [[English:Vocabulary poster|Vocabulary poster]]: Design a small visual poster that explains boundary line, half-plane, overlap, and feasible region using your own examples.

{{BR}}
=== Standard ===
# [[English:Graph a system|Graph a system]]: Graph a system of three linear inequalities, clearly show the overlap, and label one point that satisfies all three constraints.
# [[English:Real-world constraints|Real-world constraints]]: Write a short scenario involving two limited resources, define variables, and translate at least three restrictions into linear inequalities.
# [[English:Interview about constraints|Interview about constraints]]: Interview a shop owner, coach, teacher, or family member about a situation involving limits, then represent two of those limits as inequalities.
# [[English:Explainer video|Explainer video]]: Record a two-minute tutorial showing how to decide between a dashed and solid boundary and how to find the correct side to shade.

{{BR}}
=== Advanced ===
# [[English:Model and justify|Model and justify]]: Build a complete two-variable model for a school event, small business, or club project, graph the feasible region, and justify every constraint.
# [[English:Compare systems|Compare systems]]: Create one system with a bounded solution region, one with an unbounded region, and one with no solution, then explain the geometric differences.
# [[English:Optimization extension|Optimization extension]]: Add a linear objective function to a feasible region, test its values at the vertices, and explain which feasible point gives the best result.
# [[English:Digital graphing investigation|Digital graphing investigation]]: Use a graphing tool to vary the slope or intercept of one inequality, capture images of three cases, and explain how the feasible region changes.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Reasoning from a graph|Reasoning from a graph]]: Given a graph of three inequalities, identify two solution points and two non-solution points, then justify each choice by referring to the relevant boundaries and shaded regions.
# [[English:Error analysis|Error analysis]]: Analyze an incorrect graph in which a student used a solid line for a strict inequality and shaded the wrong side of another boundary, then explain and correct both errors.
# [[English:Modeling transfer|Modeling transfer]]: Convert a new real-world situation with at least three restrictions into a system of linear inequalities, define the variables and units, and explain what the feasible region means in context.
# [[English:Algebra and graph connection|Algebra and graph connection]]: Rearrange an inequality that requires division by a negative number, graph the result, and explain how reversing the inequality sign affects the solution half-plane.
# [[English:Optimization reasoning|Optimization reasoning]]: Given a bounded feasible region and a linear objective function, compare the objective values at the vertices and defend which feasible solution is best for the stated goal.

<br>
<br>

{{BR}}
= Evidence of Learning =

{| class="wikitable"
! Type of evidence
! What successful learning looks like
|-
| Knowledge
| You accurately explain inequality symbols, boundary lines, half-planes, intersections, constraints, and feasible regions.
|-
| Skills
| You graph individual inequalities, identify overlap regions, use test points, reverse inequality signs correctly when required, and verify solutions by substitution.
|-
| Products
| You produce accurate graphs, written explanations, a real-world model, and at least one visual or digital artifact that communicates your reasoning.
|-
| Transfer
| You recognize new situations with multiple limits, choose suitable variables, build a system of inequalities, and interpret the solution set in context.
|}

<br>
<br>

{{BR}}
= OERs on the Topic =

The following English Wikipedia article provides additional background on linear inequalities, including systems and graphical solution sets.

<iframe> https://en.m.wikipedia.org/wiki/Linear_inequality </iframe>

<br>

{{BR}}
= Linked Learning Areas =

Systems of linear inequalities bring together algebraic manipulation, graph interpretation, logical intersection, and mathematical modeling. They prepare you for later work in optimization, economics, data science, operations research, and other fields where several constraints must be satisfied at the same time.

{| align=center
{{:D-Tab}}
'''[[English:Systems of Linear Inequalities|Systems of Linear Inequalities]]'''
# [[English:Linear inequality|Linear inequality]]
# [[English:Graphing linear functions|Graphing linear functions]]
# [[English:Coordinate plane|Coordinate plane]]
# [[English:Systems of linear equations|Systems of linear equations]]
# [[English:Slope|Slope]]
# [[English:Intercept|Intercept]]
# [[English:Mathematical modeling|Mathematical modeling]]
# [[English:Linear programming|Linear programming]]
|}

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Systems of Linear Inequalities]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Secondary education]]
[[Category:Grades 9-10]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Systems of Linear Inequalities - Versionsgeschichte

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