Problem Solving

This is one of the question from the Manhattan guide to Number properties

I do not understand the explanation they have provided.

The way I solved the problem was to set (x-4)=(4-x), to reach the conclusion that x=4 is one solution.
On setting -(x-4)=4-x, we get no solution because both sides cancel each other out.

However, in the Manhattan guide they have provided the answer to be x≤4. On plugging it into the equation |x-4|=4-x, we do find it to be the correct answer, but I still don't understand the way they reached this answer.

Can anyone please help me with this

ChessWriter wrote:

This is one of the question from the Manhattan guide to Number properties

I do not understand the explanation they have provided.

The way I solved the problem was to set (x-4)=(4-x), to reach the conclusion that x=4 is one solution.
On setting -(x-4)=4-x, we get no solution because both sides cancel each other out.

However, in the Manhattan guide they have provided the answer to be x≤4. On plugging it into the equation |x-4|=4-x, we do find it to be the correct answer, but I still don't understand the way they reached this answer.

Can anyone please help me with this

For what values of x does |x-4| = 4-x?

4-x is the DIFFERENCE between 4 and x.
A DIFFERENCE can be negative, 0, or positive.

|x-4| is the DISTANCE between x and 4.
A DISTANCE must be greater than or equal to 0.

For the DIFFERENCE between two values to be equal the DISTANCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
4-x ≥0
4 ≥ x.

Thus, |x-4| = 4-x when x≤4.

GMATGuruNY wrote:

ChessWriter wrote:

This is one of the question from the Manhattan guide to Number properties

I do not understand the explanation they have provided.

The way I solved the problem was to set (x-4)=(4-x), to reach the conclusion that x=4 is one solution.
On setting -(x-4)=4-x, we get no solution because both sides cancel each other out.

However, in the Manhattan guide they have provided the answer to be x≤4. On plugging it into the equation |x-4|=4-x, we do find it to be the correct answer, but I still don't understand the way they reached this answer.

Can anyone please help me with this

For what values of x does |x-4| = 4-x?

4-x is the DIFFERENCE between 4 and x.
A DIFFERENCE can be negative, 0, or positive.

|x-4| is the DISTANCE between x and 4.
A DISTANCE must be greater than or equal to 0.

For the DIFFERENCE between two values to be equal the DISTANCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
4-x ≥0
4 ≥ x.

Thus, |x-4| = 4-x when x≤4.

Thank you Mitch. That was a nice conceptual explanation to the problem. However, is there a way to solve this problem algebraically rather than conceptually?