Question

|3-x|= x-3, Its solutions are

Answer 1

x= 3

|3-3|=3-3

|0|=0

0=0hence proved

Answer 2

As the LHS of the equation contains multiple modulus fuctions, we would have to define it one by one.

Case 1 :- x>=3.

Then all the modulus opens with positive sign.

So, |x - 3 - x - 2| = 5

|-3 - 2| = 5

5 = 5.

Hence all the values of x >= 3are the solutions.

Case 2 :- - 2 =< x < 3.

Now, |x - 3| opens as 3 - x (because for x less than 3, x - 3 is negative and its modulus will open with negative sign) and |x + 2| opens as x + 2.

So, |3 - x -x -2| = 5.

|1 - 2x| = 5.

If x >1/2, then 2x - 1 = 5.

2x = 4

X = 2.

It can be an answer as it is between 1/2 and 3.

If x <= 1/2, then 1 - 2x = 5.

2x = -4

X = -2

Case 3 :- x<-2.

|x - 3| opens as 3 - x and |x + 2| opens as -(x + 2).

So, |3 - x - (-(x+2))| = 5

Or, |3 - x + x + 2| = 5

Hence, 5 = 5.

So, all the values of x less than -2 are the solutions.

Therefore, the solution is

x <= -2

x = 2

And x >= 3.