Problem 94 - College Algebra Essentials

Question

What is a solution of a system of linear inequalities?

Step-by-Step Solution

Answer

A solution of a system of linear inequalities is an ordered pair, or set of values, that makes all the inequalities in the system true at the same time.

Step 1: Understanding the concept of linear inequalities

A linear inequality is a mathematical statement that relates a linear expression as either less than, greater than, less than or equal to, greater than or equal to another linear expression. For example, \(2x + 3y \leq 6\) is a linear inequality.

Step 2: Defining a system of linear inequalities

A system of linear inequalities comprises two or more linear inequalities involving the same set of variables. For example, \(\begin{cases} 2x + 3y \leq 6 \ x - y > 1 \end{cases}\) forms a system of two linear inequalities in two variables \(x\) and \(y\).

Step 3: Understanding a solution to a system of linear inequalities

A solution to a system of linear inequalities is any ordered pair of numbers that satisfy all the inequalities in the system simultaneously. In other words, if you plug the values of the solution into every inequality of the system, they should all hold true. This means if you were to graph each inequality on a coordinate plane, the regions where all the inequalities overlap form the solution set.