NP Guide, 4th Edition Chapter 3 - Really Simple Question

[bluescale]

Hello,

I have a really simple question, and I'm sure I'm just missing something obvious. It's from the Chapter 3 In Action Problem Set. Question 5 states:

When is |x-4| equal to 4-x?

When I plug in a negative number for for x (such as -6, for example), I see that x can be any number less than 4, so I understand that the answer is x is less than or equal to 4. However, I don't understand how to derive this algebraically. If someone could show me, I'd appreciate it greatly.

Regards,
Faraz

[Ben Ku]

Hi Faraz,

We know that the quantity (x - 4) can be either positive OR negative.

If (x - 4) is positive, then |x - 4| = x - 4.
However, if (x - 4) is negative, then |x - 4| is the opposite of (x - 4) because |x - 4| is now positive. therefore, if (x - 4) is negative, then |x - 4| = - (x - 4) = 4 - x.

So |x - 4| = 4 - x when (x - 4) < 0, or when x < 4.