Problem 96 - College Algebra

Question

What does it mean if a system of linear inequalities has no solution?

Step-by-Step Solution

Answer

When a system of linear inequalities has no solution, there is no common region that satisfies all inequalities in the system. Simply put, there's no intersection of the areas described by these inequalities.

Step 1: Understanding Linear Inequalities

Linear inequalities describe areas in two-dimensional space where the points within these areas satisfy the inequality condition. For instance, for the inequality \(y > 2x + 1\), any point in the plane where the y-coordinate is greater than twice the x-coordinate plus one would satisfy the condition.

Step 2: Comprehending Systems of Linear Inequalities

A system of linear inequalities, on the other hand, is a set of two or more inequalities with the same unknowns. When graphing these inequalities on a coordinate plane, the solution to the system is the intersection area of all inequalities in the system.

Step 3: Understanding No Solution to the System

When a system of linear inequalities has no solution, it means there is no possible region that satisfies all inequalities in the system. In other words, the inequalities don't share a common intersection.