Problem 96 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

If a system of linear inequalities has no solution, it implies that there is no point or region in the coordinate plane that simultaneously satisfies all the inequalities in the system.

Step 1: Understanding Linear Inequalities

Linear inequalities describe an area of the coordinate plane that satisfies all solutions to an equation. These are represented with a line and a shaded region either above, below, or on the line. When multiple linear inequalities are considered together, they form a system which has solutions common to all the inequalities.

Step 2: Discussing the No Solution Case

For a system of linear inequalities to have no solution, it means that there is no point in the coordinate plane that would satisfy all the inequalities at once. This situation typically arises when the shaded regions of the inequalities don't intersect, that is, there is no common area.

Step 3: Example of No Solution Case

A simple example could be the system: \(y > x + 1\) and \(y < x - 1\). The first inequality shades the region above the line \(y=x+1\) and the second inequality shades the region below the line \(y=x-1\). It's visibly clear that there is no overlapping or common shaded area between the two inequalities. So there is no solution to this system of inequalities.