A point is considered a solution to a system of two inequalities if it satisfies both inequalities simultaneously. In simpler terms, for a point (x, y) to be a solution:
- It must be located within the region defined by the solutions of both inequalities.
- The point must comply with the conditions set by each individual inequality in the system.
- If the inequalities are written in the form of:
y > mx + b
y < nx + d
The point (x, y) must satisfy both
y > mx + b and y < nx + d.
To illustrate:
- Example 1: For the inequalities
y > 2x + 1andy < -x + 3, the point (1, 3) can be analyzed: - Substituting into the first inequality:
3 > 2(1) + 1→3 > 3(not true) - Substituting into the second inequality:
3 < -1 + 3→3 < 2(not true) - This point does not satisfy both inequalities, hence it is not a solution.
- Example 2: For the same inequalities above, consider the point (0, 2):
- For the first inequality:
2 > 2(0) + 1→2 > 1(true) - For the second inequality:
2 < -0 + 3→2 < 3(true) - Since (0, 2) satisfies both inequalities, it is considered a solution to the system.
In conclusion, checking whether the point satisfies each inequality will determine if it is part of the solution set for the given system of inequalities.