Find the minimum n such that `|x_1|+|x_2|+ *** +|x_n|=49+|x_1+x_2+ *** +x_n| ` where ` ``x_i<1 ` for all i=1,2,...,n :
The right hand side of the equation is always greater than or equal to 49, so the left hand side must also be greater than or equal to 49.
If `x_i=1,i=1,2,...,n ` then you could have 49 terms; however each `x_i<1 ` so there must be more than 49 terms.
Claim: n=50 is the minimum.
(1) n cannot be 49 or smaller by the argument above.
(2) `x_i=(-1)^n(48/50),1<=i<=48, x_49=-1/2,x_50=1/2 `
Then `sum_(i=1)^n |x_i|=49 ` so the left hand side is 49.
Also `49+| sum_(i=1)^48 (-1)^i -1/2+1/2|=49+0=49 `
Therefore there exists 50 values for `x_i ` such that the equality holds.
Just to answer the side question,
Absolute values can be created with vertical bars (inside the math backquotes of course)
Vertical bar is shift+backslash (the key above enter)
So `|x|` should look something like `|x|`
It automatically adjusts length/height according to the equation inside.