Linear Inequality Systems

Choices That Meet Several Inequalities at Once

A plan may need to stay within a budget and meet a minimum attendance target at the same time. Each condition removes part of the possible choices. A linear inequality system describes the choices that satisfy every condition together.

Linear Inequality System

A linear inequality system is a collection of two or more linear inequalities that must be satisfied simultaneously. For example, an upper bound has the form:

a_1x_1+a_2x_2+\cdots+a_nx_n\leq b

A lower bound uses the opposite comparison:

a_1x_1+a_2x_2+\cdots+a_nx_n\geq b

Where $a_1, a_2, ..., a_n$ are coefficients, $x_1, x_2, ..., x_n$ are variables, $b$ is a constant, and the inequality sign can be $\leq, \geq, <, >$ .

The main difference between linear equations and linear inequalities is the relation between their two sides:

A linear equation states that two expressions are equal ( $=$ )
A linear inequality compares two expressions ( $\leq, \geq, <, >$ )

Linear Inequality Systems with Two Variables

o or more inequalities with two variables (usually $x$ and $y$ ):

\begin{cases} a_1x + b_1y \leq c_1 \\ a_2x + b_2y \leq c_2 \end{cases}

Example:

\begin{cases} 2x + 5y < 10 \\ 6x + 2y > 10 \end{cases}

The solution to this system is the set of ordered pairs $(x,y)$ that satisfy all inequalities simultaneously.

Solving Linear Inequality Systems

To solve a linear inequality system with two variables, we use the graphical method. Each inequality becomes a half-plane, and the solution is the region where all half-planes overlap. The steps below build that region in order.

Draw the Boundary Lines

For each inequality, draw its boundary line by changing the inequality sign to an equals sign.

Example:

For the system:

\begin{cases} 2x + 5y < 10 \\ 6x + 2y > 10 \end{cases}

Draw the lines:

\begin{cases} 2x + 5y = 10 \\ 6x + 2y = 10 \end{cases}

Determine the Solution Regions

To determine the solution region for each inequality:

Take a test point (e.g., the origin $(0,0)$ if it's not on the line)
Substitute it into the inequality
If the result is true, the region containing the test point is the solution region
If the result is false, the region not containing the test point is the solution region

Example:

For $2x + 5y < 10$ , check point $(0, 0)$ :

2(0) + 5(0) = 0 < 10

(true)

The solution region is the area containing point $(0, 0)$ , which is below the line $2x + 5y = 10$ .

For $6x + 2y > 10$ , check point $(0, 0)$ :

6(0) + 2(0) = 0 > 10

(false)

The solution region is the area not containing point $(0, 0)$ , which is above the line $6x + 2y = 10$ .

Determine the Intersection of Solution Regions

The solution to the linear inequality system is the intersection of all the solution regions, that is, the area satisfied by every constraint at once. A point belongs to the solution only if it satisfies all inequalities together, so testing one corner point is a quick check.

Graphical Representation

When drawing inequality graphs:

For $\leq$ or $\geq$ : use a solid line (the solution region includes points on the line)
For $<$ or $>$ : use a dashed line (the solution region does not include points on the line)
The solution region is shaded to show the solution

Check Your Understanding

\begin{cases} x+y\leq6 \\ 2x-y\geq0 \end{cases}

Determine whether $(2,3)$ and $(1,5)$ belong to the solution set.

Worked Solution

Substitute $(2,3)$ into both inequalities:

2+3=5\leq6

Both statements are true, so $(2,3)$ is a solution.

Now test $(1,5)$ :

1+5=6\leq6

The second statement is false, so $(1,5)$ is not a solution. The boundary lines meet at $(2,4)$ , but that point is only one part of the solution. The full solution is the entire overlapping region, including both solid boundaries.

Modeling Feasible Choices

A linear inequality system describes the set of combinations that are allowed together. A word problem has to be turned into such a system before it can be solved, and the solution is the whole region of allowed combinations.

Modeling Feasible Choices with Inequalities

Linear inequality systems describe the feasible choices in optimization problems, such as:

Production problems with resource constraints
Budget planning with cost constraints
Nutrition planning with calorie constraints

An optimization model also needs an objective, such as minimizing cost or maximizing output. The inequalities alone define which choices are allowed.

Activity Planning

Kiki is organizing an independence day celebration in her neighborhood. From the community fund, there is $\text{Rp}500{,}000.00$ available. For organizing competitions, it costs $\text{Rp}20{,}000.00$ per child. Prizes for winners are budgeted at $\text{Rp}40{,}000.00$ for each type of competition. It's expected that more than $13$ children will participate.

Define:

$x$ = number of participants
$y$ = number of competitions

The mathematical model is:

\begin{cases} 20{,}000x + 40{,}000y \leq 500{,}000 \\ x > 13 \\ x,y \in \mathbb{Z}_{>0} \end{cases}

If we simplify:

\begin{cases} 20x + 40y \leq 500 \\ x > 13 \\ x,y \in \mathbb{Z}_{>0} \end{cases}

The graph shows the continuous feasible region. Because participants and competitions are counted in whole numbers, valid plans are the positive integer points inside that region.

Problem Solving Strategy

To solve linear inequality system problems:

Identify the variables to use
Create a mathematical model based on the given constraints
Solve the system using the graphical method
Interpret the solution in the context of the original problem

Differences Between Linear Equation Systems and Linear Inequality Systems

The two system types differ in what counts as an answer. The table sets the operation sign, the solution form, and the graph side by side.

Aspect	Linear Equation Systems	Linear Inequality Systems
Operation Sign	Uses equals sign ( $=$ )	Uses inequality signs ( $<, >, \leq, \geq$ )
Solution Form	Common points of the equations	The common feasible set of all inequalities
Graphical Representation	Common points of lines	Intersection of shaded half-planes
Solution Methods	Substitution, elimination, determinants	Graphing, feasibility checks, and linear programming when an objective is present
Number of Solutions	Zero, one, or infinitely many	May be empty, a point or boundary segment, or a region with infinitely many real solutions
Drawing	Always uses solid lines	Solid lines for $\leq, \geq$ and dashed lines for $<, >$

Interactive Visualization of Linear Inequality Systems

Each inequality defines an allowed half-plane. With several inequalities, only the overlap of those half-planes satisfies every condition.

The interactive visualization shows that overlap directly.

Example of an Inequality System

This system contains two inequalities, and each inequality produces its own region:

\begin{cases} x + y \leq 10 \\ 15x + 9y \geq 120 \end{cases}

In the visualization below, the orange region shows the solution to $x + y \leq 10$ (all points below or on the line $x + y = 10$ ).

The purple region shows the solution to $15x + 9y \geq 120$ (all points above or on the line $15x + 9y = 120$ ).

The intersection of the two regions is the solution to the linear inequality system.

Linear Inequality System

Visualization of the system of inequalities

x + y \leq 10

and

15x + 9y \geq 120

How to Read the Visualization

In this interactive visualization:

Boundary lines show the equations (e.g., $x + y = 10$ and $15x + 9y = 120$ )
Colored regions show the solution to each inequality
Intersection region (the area satisfying all inequalities) is the solution to the inequality system

The two boundary lines meet at $(5,5)$ . This point forms one corner of the feasible set. The shaded region contains the full set of solutions.

The same reading order applies to a word problem: graph each condition, keep the permitted side, and then take the common region.

Choices That Meet Several Inequalities at Once

Linear Inequality System

A linear inequality system is a collection of two or more linear inequalities that must be satisfied simultaneously. For example, an upper bound has the form:

a_1x_1+a_2x_2+\cdots+a_nx_n\leq b

A lower bound uses the opposite comparison:

a_1x_1+a_2x_2+\cdots+a_nx_n\geq b

Where $a_1, a_2, ..., a_n$ are coefficients, $x_1, x_2, ..., x_n$ are variables, $b$ is a constant, and the inequality sign can be $\leq, \geq, <, >$ .

The main difference between linear equations and linear inequalities is the relation between their two sides:

A linear equation states that two expressions are equal ( $=$ )
A linear inequality compares two expressions ( $\leq, \geq, <, >$ )

Linear Inequality Systems with Two Variables

o or more inequalities with two variables (usually $x$ and $y$ ):

\begin{cases} a_1x + b_1y \leq c_1 \\ a_2x + b_2y \leq c_2 \end{cases}

Example:

\begin{cases} 2x + 5y < 10 \\ 6x + 2y > 10 \end{cases}

The solution to this system is the set of ordered pairs $(x,y)$ that satisfy all inequalities simultaneously.

Solving Linear Inequality Systems

Draw the Boundary Lines

For each inequality, draw its boundary line by changing the inequality sign to an equals sign.

Example:

For the system:

\begin{cases} 2x + 5y < 10 \\ 6x + 2y > 10 \end{cases}

Draw the lines:

\begin{cases} 2x + 5y = 10 \\ 6x + 2y = 10 \end{cases}

Determine the Solution Regions

To determine the solution region for each inequality:

Take a test point (e.g., the origin $(0,0)$ if it's not on the line)
Substitute it into the inequality
If the result is true, the region containing the test point is the solution region
If the result is false, the region not containing the test point is the solution region

Example:

For $2x + 5y < 10$ , check point $(0, 0)$ :

2(0) + 5(0) = 0 < 10

(true)

The solution region is the area containing point $(0, 0)$ , which is below the line $2x + 5y = 10$ .

For $6x + 2y > 10$ , check point $(0, 0)$ :

6(0) + 2(0) = 0 > 10

(false)

The solution region is the area not containing point $(0, 0)$ , which is above the line $6x + 2y = 10$ .

Determine the Intersection of Solution Regions

Graphical Representation

When drawing inequality graphs:

For $\leq$ or $\geq$ : use a solid line (the solution region includes points on the line)
For $<$ or $>$ : use a dashed line (the solution region does not include points on the line)
The solution region is shaded to show the solution

Check Your Understanding

\begin{cases} x+y\leq6 \\ 2x-y\geq0 \end{cases}

Determine whether $(2,3)$ and $(1,5)$ belong to the solution set.

Worked Solution

Substitute $(2,3)$ into both inequalities:

2+3=5\leq6

Both statements are true, so $(2,3)$ is a solution.

Now test $(1,5)$ :

1+5=6\leq6

Modeling Feasible Choices

Modeling Feasible Choices with Inequalities

Linear inequality systems describe the feasible choices in optimization problems, such as:

Production problems with resource constraints
Budget planning with cost constraints
Nutrition planning with calorie constraints

An optimization model also needs an objective, such as minimizing cost or maximizing output. The inequalities alone define which choices are allowed.

Activity Planning

Define:

$x$ = number of participants
$y$ = number of competitions

The mathematical model is:

\begin{cases} 20{,}000x + 40{,}000y \leq 500{,}000 \\ x > 13 \\ x,y \in \mathbb{Z}_{>0} \end{cases}

If we simplify:

\begin{cases} 20x + 40y \leq 500 \\ x > 13 \\ x,y \in \mathbb{Z}_{>0} \end{cases}

The graph shows the continuous feasible region. Because participants and competitions are counted in whole numbers, valid plans are the positive integer points inside that region.

Problem Solving Strategy

To solve linear inequality system problems:

Identify the variables to use
Create a mathematical model based on the given constraints
Solve the system using the graphical method
Interpret the solution in the context of the original problem

Differences Between Linear Equation Systems and Linear Inequality Systems

The two system types differ in what counts as an answer. The table sets the operation sign, the solution form, and the graph side by side.

Aspect	Linear Equation Systems	Linear Inequality Systems
Operation Sign	Uses equals sign ( $=$ )	Uses inequality signs ( $<, >, \leq, \geq$ )
Solution Form	Common points of the equations	The common feasible set of all inequalities
Graphical Representation	Common points of lines	Intersection of shaded half-planes
Solution Methods	Substitution, elimination, determinants	Graphing, feasibility checks, and linear programming when an objective is present
Number of Solutions	Zero, one, or infinitely many	May be empty, a point or boundary segment, or a region with infinitely many real solutions
Drawing	Always uses solid lines	Solid lines for $\leq, \geq$ and dashed lines for $<, >$

Interactive Visualization of Linear Inequality Systems

Each inequality defines an allowed half-plane. With several inequalities, only the overlap of those half-planes satisfies every condition.

The interactive visualization shows that overlap directly.

Example of an Inequality System

This system contains two inequalities, and each inequality produces its own region:

\begin{cases} x + y \leq 10 \\ 15x + 9y \geq 120 \end{cases}

In the visualization below, the orange region shows the solution to $x + y \leq 10$ (all points below or on the line $x + y = 10$ ).

The purple region shows the solution to $15x + 9y \geq 120$ (all points above or on the line $15x + 9y = 120$ ).

The intersection of the two regions is the solution to the linear inequality system.

Linear Inequality System

Visualization of the system of inequalities

x + y \leq 10

and

15x + 9y \geq 120

How to Read the Visualization

In this interactive visualization:

Boundary lines show the equations (e.g., $x + y = 10$ and $15x + 9y = 120$ )
Colored regions show the solution to each inequality
Intersection region (the area satisfying all inequalities) is the solution to the inequality system

The two boundary lines meet at $(5,5)$ . This point forms one corner of the feasible set. The shaded region contains the full set of solutions.

The same reading order applies to a word problem: graph each condition, keep the permitted side, and then take the common region.