MHB How can we rewrite a modulus function?

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The modulus function |x-3|<10 can be rewritten as -10<x-3<10. However, the expression |x-3|+|x+1|+|x|<10 cannot be simplified in the same manner due to its complexity. Instead, it can be expressed as a union of inequalities. The correct representation for this case is (x < -8/3) ∪ (x > 4). Understanding the limitations of rewriting modulus functions is crucial for accurate mathematical expressions.
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We can rewrite |x-3|<10 in the following way.

-10<x-3<10

But can rewrite |x-3|+|x+1|+|x|<10 in the following way?

-10<x-3+x+1+x<10.

If we cannot, will anybody please explain why we cannot?
 
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You can't write it that way because it's not in general correct.

You can write it as a union of such rules though.

In this case it appears to be

$(x < -\dfrac 8 3) \cup (4 < x)$
 

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