What does infinite solution in algebra mean?

Answer 1

Lots of equations or inequalities can have infinitely many solutions. That means that the set of solutions is not finite; if you were to try to list the individual solutions, you would never end.

In inequalities, the situation is quite clear; if the solution to an inequality is of the form:

x > 10

then there are infinitely may solutions (11, 12, 13, 14, 15, ... but also 10.1, 10.2, 10.01, 15.073, etc.); and if you try to write out each solution, you would never get done.

This is also possible with equations, in the case of periodic functions; for example:

sin x = 0

has the solutions (0, 180°, 360°, 540°, ...), or in radians (0, pi, 2 pi, 3 pi, 4 pi, ...)

due to the fact that the sine function is periodic.