So I have the following equation where a is a constant
$$|(x_1 - a)| + |(x_2 - a)| \leq 2$$
Can I rewrite this as the following: $$|(x_1 - 2a + x_2)| \leq 2$$
2026-04-08 12:50:26.1775652626
Rewriting a less than or equal to equation with multiple absolute values
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Certainly the first inequality implies the second, by an application of the triangle inequality: if the first holds then $$|(x_1-2a+x_2)| = |(x_1-a)+(x_2-a)| \le |(x_1-a)| + |(x_2-a)| \le 2 $$ And sometimes just one direction of implication is what you actually need in proofs.
But when you reverse the direction of the triangle inequality what you get is not generally true. And therefore, when you reverse the direction of the implication, and ask whether the second inequality implies the first, the answer is "no", as shown in the comments.