NP Chapter 3 P 75 Exercise 5

[ray_serrano]

The explanation says that were are changing |x - 4| to -(x-4) because x -4 <=0. I don't see how we know x -4 <= 0.

[RonPurewal]

The explanation says that were are changing |x - 4| to -(x-4) because x -4 <=0. I don't see how we know x -4 <= 0.

if x - 4 were positive, the absolute value of (x - 4) would still be x - 4. in that case, -(x - 4) = 4 - x would be a negative quantity, so the situation in the problem would be impossible.
since (x - 4) positive leads to an absurd conclusion, we can safely conclude that (x - 4) is not positive.

if the absolute value works out to -(x - 4), that implies that -(x - 4) is itself a non-negative quantity (otherwise it couldn't be the absolute value of anything).

[mauheilbron]

I found this answer worded a little confusingly (sorry!)

The way I see it the absolute value on the left side basically implies |x-4| >= 0. Since |x-4| = 4 - x then 4 - x >= 0 (if a = b and a > 0 then b > 0.) Solving for x, x<= 4.

hope that helps and wasn't completely unnecessary.

[esledge]

Since |x-4| = 4 - x then 4 - x >= 0 [moderator note: zero or pos, because by definition, an absolute value can't be neg] Solving for x, x<= 4.

I think that's a great way to look at it!