Systems of Inequalities vs Systems of Equations
It should go without saying that the main difference between a system of inequalities and a system of equations is that a system of inequalities consists of at least two inequalities, rather than equations. As we will discuss below, there is also a major difference in how we interpret the solutions to a system of inequalities. One thing remains the same: the variables within the inequalities of the system must have matching definitions. Otherwise, we cannot consider the relationship to be a system. This becomes important when working with systems of inequalities in a situational context. For example, if x represents apples in one inequality, it must represent apples in all of the inequalities in the system.
The Solution Set of a System of Inequalities
Just as a solution to a system of equations satisfies all equations in the system, solutions to a system of inequalities must satisfy all inequalities in the system. Sometimes, a coordinate pair (x, y) will satisfy 1 or 2 of the inequalities in the system, but not all. In these cases, the coordinate pair does not represent a solution to the entire system. One way to think about inequalities is that they represent boundaries. In a system, there are several boundaries, each of which are represented by an inequality. In this sense, a solution to a system of inequalities fits with every boundary defined by the inequalities.
Let's take a visual approach to understand solutions to a system of inequalities. Below is a graph of the following system:
The region highlighted in yellow is referred to the solution region of the system. All coordinate points (x, y) that exist within the solution region satisfy all of the inequalities. In other words, they fit within all boundaries defined by the system.
Systems of Inequalities with No Solution
Sometimes a graph of a system will have overlapping solution regions to individual inequalities in the system, without an overlap of solutions to all inequalities. Recall that solutions to the entire system must fit within all boundaries defined by the system. Thus, when no overlap between all inequalities exist, there is no solution to the system, although we may be tempted to think there are, because some overlaps exist. Take a look at the graph below:
This system actually has no solution. This is because there is no coordinate pair (x, y) that fits within all boundaries to the system. We can identify points that will satisfy two of the three inequalities: for example, there is overlap between the green and red regions, and the blue and red regions. There is even overlap between the blue and green regions that we just can't see on the graph as it is. However, at no point will there be an overlap between all three regions. For this reason, there is no solution to this system.