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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?

user64814
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    Have you tried to plot the function $y=|x-3|+|x+1|+|x|$? – ajotatxe May 12 '21 at 17:10
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    $|x -3| < 10$ is actually two equations, depending on whether $(x-3) \geq 0$ or $(x-3) < 0 : ~0 \leq (x-3) < 10$ combined with $0 >3$ and $(3-x) < 10 \implies (x-3) > -10.$ Putting this together, you have that $-10 < (x-3) < 0$ or $0 \leq (x-3) < 10.$ When you have more than one such expression, such as having to contend with $|x-3|, |x+1|,$ and $|x|$, you have (potentially) 8 cases, depending on whether each of the three expressions is or is not negative. These 8 cases will collapse into 4 regions, depending on how $x$ compares with each of ${-1, 0, 3}.$ So you have 4 cases to consider – user2661923 May 12 '21 at 17:40
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    You could use the triangle inequality, $|a+b|\le|a|+|b|$ to write $$|3x-2|=|(x-3)+(x+1)+x|\le|x-3|+|x+1|+|x|<10$$but bear in mind that this new inequality would have a different range of solutions than your starting one. – user170231 May 12 '21 at 17:54
  • @user2661923: I like your comment (+1), though I think you meant $0>x-3$ where you typed $0>3$ – J. W. Tanner May 12 '21 at 18:17
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    @J.W.Tanner Nice catch, thanks for the editing assist. Unfortunately, the comment can't be edited. – user2661923 May 12 '21 at 18:19

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