Why aren't all solutions to an inequality possible in reality?
(TL;DR at the bottom, but the full explanation is better)
Let's answer this using an example.
The equation 2x + 5 > 10 can represent a lemonade stand's sales. What do each of the numbers represent in a real-life situation?
- x represents the amount of lemonade glasses sold
- 2 (dollars) represents the price of each glass of lemonade
- 5 represents the base profit that the stand opened with
- 10 represents the goal for profit
A solution for this equation would be x = -3:
2x + 5 > 10
2(-3) + 5 > 10
6 + 5 > 10
11 > 10 - True
Another solution is x = 11/3:
2x + 5 > 10
2(11/3) + 5 > 10
(22/3) + 5 > 10
(37/3) > 10
(36/3)+(1/3) > 10
12 and (1/3) > 10
These are solutions to our inequality from a mathematical standpoint but are both implausible in real life.
Assuming that reasonable would mean a full glass, -3 and (11/3) are possible answers, but can't be reasonably sold. -3 glasses of lemonade can't be sold, and neither can (11/3) of a glass of lemonade.
Therefore, these values create a situation where "not all solutions represented by inequation are always possible solutions in reality."
TL;DR:
Not all solutions are possible because the value that makes the inequation true is not reasonable or feasible under realistic circumstances, such as a fraction value for people or a negative value for possession.