Systems of Inequalities

A system of inequalities consists of two or more inequalities that are considered together. The solution to the system is the set of values that satisfy all of the inequalities simultaneously. Systems of inequalities are typically used to represent constraints in optimization problems, geometric regions, or various real-world scenarios.

For example:

• x + y ≤ 5
• x − y ≥ 1

The solution to this system is the set of values for x and y that satisfy both inequalities. This solution can be visualized as a region on a graph where the shaded area represents all possible points that satisfy the system.

General Systems of Inequalities

A general system of inequalities in one n variable with n inequalities is of the following form:

a₁₁x₁ + a₁₂x₂ + a₁₃x₃ ........ + a_1nx_n < b₁

a₂₁x₁ + a₂₂x₂ + a₂₃x₃ ........ + a_2nx_n < b₂

_.

_.

_.

a_n1x₁ + a_n2x₂ + a_n3x₃ ........ + a_nnx_n < b_n

Graphical Representation of Systems of Inequalities

System of inequalities is a group of multiple inequalities. First, solve each inequality and plot the graph for each inequality. The intersection of the graph of all the inequalities represents the graph for systems of inequalities.

Consider an example,

Example: Plot graph for systems of inequalities

• 2x + 3y ≤ 6
• x ≤ 3
• y ≤ 2

Solution:

Graph for 2x + 3y ≤ 6

Shaded region of the graph represents 2x + 3y ≤ 6

👁 Graph for 2x + 3y ≤ 6

Graph for x ≤ 3

Shaded region represents x ≤ 3

👁 Graph for x ≤ 3

Graph for y ≤ 2

Shaded region represents y ≤ 2

👁 Graph for y ≤ 2

Graph for given system of inequalities

Shaded region represents given system of inequalities.

👁 Graph for System of Inequalities

Read More,

• Linear Inequalities
• Quadratic Inequalities