A point is considered a solution to a system of two inequalities if and only if it satisfies both inequalities simultaneously. This means that when the coordinates of the point are substituted into both inequalities, the resulting statements must both be true.
To elaborate, let’s say we have two inequalities:
1. y >= 2x + 1
2. y <= -x + 4
For a point (x, y) to be a solution, it needs to satisfy:
- The value of y must be greater than or equal to the value derived from the first inequality when the x-coordinate of the point is substituted.
- The value of y must also be less than or equal to the value derived from the second inequality when the same x-coordinate is substituted.
In graphical terms, the solutions to these inequalities are typically represented as shaded regions on a coordinate plane. A point lying within the overlapping shaded area created by both inequalities represents a solution to the system. If the point is outside this area, it does not satisfy one or both inequalities and hence is not a solution.
In summary, only points that are found in the intersection of the solution sets of the two inequalities can be deemed solutions to the system.