Solutions and the solution set

A solution of an inequality in x is a value of x that makes it a true statement. All the solutions together form the solution set.

Checking values one by one works for small cases. For 45x ≤ 500: x = 10 gives 450 ≤ 500 (true), x = 11 gives 495 ≤ 500 (true), x = 12 gives 540 ≤ 500 (false). So whole numbers of cartons from 0 to 11 are possible, and 11 cartons leave ₹5 over.

Trying values is slow and cannot list infinitely many real solutions, so we solve with rules instead: dividing by 45 gives x ≤ 500/45 = 100/9 ≈ 11.1.

The solution set depends on which numbers x may be. For 45x ≤ 500: if x is a natural number, the set is {1, 2, …, 11}; if x is an integer, it is {…, −2, −1, 0, 1, …, 11}; if x is real, it is every x ≤ 100/9.

Unless a question says otherwise, the variable is taken to be real, and the answer is an interval.

Context can restrict the answer further: a number of cartons must be a whole number and cannot be negative, even though the algebra allows negative reals.