Systems of Linear Inequalities

How to find the solution of Linear Inequality Functions

A System of linear inequalities is simply two or more linear inequalities on the same plane. In other words, a system of linear inequalities is just two or more inequalities together.

The easiest way to remember what 'system' means in this context is by answering the following question: 'Does the word system ever refer to just one thing or does system always refer to more than one thing?

The answer is 'more than one thing' Likewise a 'system of linear inequalities' is 'more than one linear inequality'

Interactive System

of Linear Inequalities

Click and drag on the points below and the system of linear inequalities will adjust accordingly.
(Full sized interactive system of linear inequalities)

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Below are the graphs of the linear inequalities: y < x + 1 and y> x

Pictured above is the system of inequalities made up the same two linear inequalities:
y < x + 1
y > x

When we take both of the linear inequalities pictured above and graph them on same Cartesian plane, we get a system of linear inequalities. The solution of this system is the yellow region which is the area of overlap. In other words, the solution of the system is the region where both inequalities are true. The y coordinates of all points in the yellow region are both greater than x+1 as well as less than x.


of a system of Linear Inequalities

The picture below shows a system of linear inequalities.

system of linear inequalities

On the left is the graph of two linear inequalities. What is the solution to this system of linear inequalities?
(Reminder: the solution is the region that both inequalities cover)

Practice Problem

Problem 1

Below is the graph of the following system of inequalities:

  • Y > – X
  • Y > X + 1

From the picture can you determine which area is the solution of this system?

diagram of solution of system of linear equations
Problem 2

On the left is the graph of

  • y = x+1
  • y = -3/2x + 1

Which region on the left is the solution to this system of linear inequalities?

  • y ≥ x+1
  • y ≥ –3/2x+1
(Remember: that simply means which region includes both of the following linear inequalities:
y ≥ x+1 & y ≥ –x+1)

Problem 3

What is the solution to the following system of linear inequalities (whose lines are graphed on the right)
  y ≤ -½X + 2
  y ≥ ½X - 1

The pink area represents the solution to this system of linear inequalities.

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