Question from MGMAT Num Prop Strat Guide - Q5/page:43

[sudaif]

Can someone please share the process/method of solving an absolute equation of this sort?
Q: when is absolute(x-4) equal to 4-x ?
I realize that one can try positive/negative/fractional values and see what fits the criteria, but I am hoping for a more methodical approach that shows the underlying logic.

[vikrant.bapat]

I think this has a range of answers, but to tackle absolute values, assume two possible values, +ve and -ve. In this example, we need to find out the value of x when we've been given |x-4| = 4-x. This would translate to,

x-4 = + (4-x)
x-4 = - (4-x)

The second doesnt give us anything new.
However, the first equation,
x-4 = 4-x
=> x+x = 4+4
=> 2x = 8
=> x = 4; So now you atleast have one definite value!

Hope i didnt go wrong anywhere...

[sudaif]

that's the wrong answer.
MGMAT Staff - can you pls help?

[Ben Ku]

Can someone please share the process/method of solving an absolute equation of this sort?
Q: when is absolute(x-4) equal to 4-x ?
I realize that one can try positive/negative/fractional values and see what fits the criteria, but I am hoping for a more methodical approach that shows the underlying logic.

When I looked at the problem, I first thought about what x cannot be. Because the absolute value of a number is always positive, then 4 - x ≥ 0, or x ≤ 4. This is our domain. You can check it: any number bigger than 4 will result in 4 - x being negative.

Now we should think about what values of x are solutions to the inequality.

Sudaif was on the right track. You should look for the positive and negative solutions.

| x - 4 | = 4 - x can be either (x - 4) = (4 - x) or (x - 4) = -(4 - x)

The first equation:
(x - 4) = (4 - x)
2x = 8, x = 4. Obviously x = 4 works.

The second equation:
(x - 4) = - (4 - x)
Here, any and every number works. But what limits it? Our domain, x ≤ 4. So therefore this is our solution.

The steps I would take:
(1) Identify the domain (what can the variable NOT be).
(2) Solve the absolute value equation for the positive and negative answers.
(3) See if the answers are within the domain.
(4) Plug in the numbers into the equation again just to confirm they work. Absolute value problems are notorious for having extraneous solutions, so you always have to check them.

Hope that helps.