Chained Notation
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python.
Read more about this topic: Inequality (mathematics)
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