Problem 67 - College Algebra

Question

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

Step-by-Step Solution

Answer

It means that there are no points that satisfy all inequalities in the system at once. This happens when the regions defined by the inequalities do not overlap.

Step 1: Understanding Linear Inequalities

An inequality is like an equation, but instead of an equals sign, it has a sign showing if something is less than or equal to (\( \leq \)), less than (<), greater than (>), or greater than or equal to (\( \geq \)). When these inequalities are linear, they are called Linear Inequalities. A system of linear inequalities is a set of two or more inequalities with the same variables.

Step 2: Graphical Interpretation

Each linear inequality divides the coordinate plane into two half-planes: one that satisfies the inequality, and one that does not. The solution to the system of inequalities is the region where the half-planes of all inequalities in the system overlap. Each point in this region is a solution to the system.

Step 3: Condition for No Solution

If a system of linear inequalities has no solution, it means that there is no area of overlap between the half-planes defined by the inequalities. Therefore, there is no point that satisfied all inequalities at once.