A system of inequalities may have no solution, but is there a case where there might only be one solution?
sure. Two intervals that overlap at a single point.
x+y >= 3
2x-3y <= 1
y <= x/2
The single point which satisfies all three equations is (2,1)
Thank you!!
Yes, there can be cases where a system of inequalities has only one solution. This occurs when the inequalities intersect at a single point on the coordinate plane. In other words, the region that satisfies all the inequalities overlaps at exactly one point.
Yes, a system of inequalities can indeed have just one solution. This occurs when the inequalities converge to a single point or region in the coordinate plane. Let's go through the process of solving a system of inequalities to understand this better.
To find the solution(s) to a system of inequalities, we typically follow these steps:
1. Graph each inequality separately on a coordinate plane.
2. Identify the overlapping region(s) where the shaded areas of the individual graphs intersect. This is called the solution region.
3. Determine the coordinates or conditions that satisfy all the inequalities within the solution region.
4. Express the solution in the appropriate form (e.g., an equation, interval notation, inequalities).
In cases where the solution region forms a single point or a limited area, we would have only one solution for the system of inequalities.
For example, consider the following system of inequalities:
1. x ≥ 2
2. y ≤ 4
Graphing each inequality on a coordinate plane, you would shade the entire region to the right of the vertical line x = 2 (a half-plane to the right) and the entire region below the horizontal line y = 4 (a half-plane below) respectively.
As you observe the shading, you'll notice that the two shaded regions overlap only at the point (2, 4). This means that (2, 4) is the common solution to both inequalities, representing the one and only solution of this system.
Keep in mind that not all systems of inequalities will have a single solution. Some may have infinitely many, while others might have no solution at all. The number and type of solutions depend on the specific inequalities involved and their intersecting regions on the coordinate plane.