It common in the literature to solve the modulus equation like $|x+5|+|x-1|=8$ by dividing into cases when $x<-5$, $-5\leq x<1$ and $x\geq1$.
My question is whether dividing into cases is really necessary, since what we care is just whether the thing inside the absolute value bars is positive or negative, so why can't we just solve each of the four possible combinations of $x+5+x-1$, $x+5-x+1$, $-x-5+x-1$, $-x-5-x+1$. It will also give the correct solutions anyway.
Is there any cases where just by solving the four possible combinations will not give the correct solutions? Or are they just the same method in disguise? If they are the same, why do I never see anyone mention it? I hope my question is understandable. Many thanks!
Best Answer
No. Taking the max of the four possible combinations will always give the right answer, but as you suspect in the question, they're essentially the same method in disguise.
When $x \le -5$, "$-x-5-x+1$" gives the maximum value.
When $-5 \le x \le 1$, "$x+5-x+1$" gives the maximum value.
When $x \ge 1$, "$x+5+x-1$" gives the maximum value.