An example of a solution to the linear inequality 2x + 3y ≤ 10 is the ordered pair (1, 2).
To verify this, we substitute x = 1 and y = 2 into the inequality:
2(1) + 3(2) ≤ 10
2 + 6 ≤ 10
8 ≤ 10
Since 8 is indeed less than or equal to 10, the ordered pair (1, 2) satisfies the inequality and is therefore a solution. There are infinitely many other solutions as well. For instance, (0, 0) is also a solution, as 2(0) + 3(0) = 0 ≤ 10.
A linear inequality, unlike a linear equation, has a range of solutions rather than just specific points. Graphically, these solutions represent a region on the coordinate plane, bounded by the line representing the equality part of the inequality.
Here are a few more examples:
- Inequality: x + y > 5
- Solution: (3, 3) because 3 + 3 = 6 > 5
- Inequality: y < 2x - 1
- Solution: (2, 0) because 0 < 2(2) - 1 which simplifies to 0 < 3
- Inequality: -x + 4y ≥ 8
- Solution: (0, 2) because -0 + 4(2) = 8 ≥ 8
In general, any ordered pair (x, y) that makes the inequality a true statement is a solution to that inequality. Finding a solution simply involves substituting the x and y values into the inequality and checking if the resulting statement is true.
