How can we rewrite a modulus function?

Nousher

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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?
 
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?
The following expression:

| (x-3) + (x+1) + (x) |<10

can be rewritten as:

-10<x-3+x+1+x<10 ......... Do you see the difference?
 
|a+ b| is NOT the same as |a|+ |b|! For example, if a= 3 and b= -1 then 3- 1= 2 so |3- 1|= 2 but |3|+ |-2|= 3+ 2= 5.
(In general, \(\displaystyle |a+ b|\le |a|+ |b|\).)

-10< x- 3+ x+ 1+ x< 10 is the same as -10< 3x- 2< 10 so that -8< 3x< 12, -8/3< x< 4.
x lies between -2 and 2/3 and +4.

But |x- 3|+ |x+ 1|+ |x| is not the same as |x- 4+ x+ 1+ x|!

If x< -1 all three of x- 3, x+ 1, and are x are negative so this is
-10< -(x- 3)- (x+ 1)- x= -3x+ 2< 10. -12< -3x< 8. 4> x> -8/3.
Since we are requiring that x< -1, this is -8/2< x< -1

If -1< x< 0 the x+ 1 is positive but x- 3 and x are still negative.
Now we have -10< -(x- 3)+(x+ 1)- x= -x+ 4< 10. -14< -x< 6, 14> X> -6.
Since we are requiring that -1< x< 0, which is inside -6< x< 14, this case gives -1< x< 0.

If 0< x< 3, both x+ 1 and x are positive but x- 3 is still negative.
Now we have -10< -(x- 3)+ (x+ 1)+ x= x+ 4< 10, -14< x< 6.
Since we are requiring 0< x< 3, which is inside -14< x< 6, this case gives 0< x< 3.

If x> 3 all three of x- 3, x+ 1, and x are positive,
Now we have -10< (x- 3)+ (x+ 1)+x= 3x- 2< 10, -8< 3x< 12, -8/3< x< 4.
Since we are requiring x> 3, this case gives 3< x< 4.
 
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