Inequalities find application in real-world situations.
Businesses use inequalities to control inventory, plan production lines, produce pricing models, and for shipping/warehousing goods and materials. Look up linear programming or the Simplex method. These methods allow one to maximize profit or minimize cost subject to a wide variety of constraints. (e.g. limited materials, power costs, labor costs, shipping/storing costs, taxes, etc...)
While in some situations, especially in situations involving celestial bodies, for example, the number of stars, distance in space, an infinite solution can work, it isn't the case for much simpler cases.
Consider a warehouse that stocks items with varying prices. While an inequality can be used to find the way the items can be stocked to maximize profit, the warehouse can only have a limited amount of items stocked at once.
Another case is the number of items that a machine can produce. However well the machine is maintained, it will eventually succumb to wear and fall out of use after a time regardless of how long it's been used. At that point, we will have a finite number of items that have been produced by the machine.
Therefore, it is not always reasonable that an infinite number of solutions would work in a real-world situation.