Solving Absolute Value Problem: x ≤ 3

[ziaharipur]

Suppose we have this absolute value question | x-3 | = 3 – x
If we solve this question we break it as
X - 3 = 3 – x or -(x - 3) = 3 - x
Now if we solve it we come to know that the part on right is true for all real numbers
And the part on the left is true for only 3

I also have read that if there is a variable on right side of absolute value then we need to verify our solutions.
Now we have two solution one is 3 and the other one is all real numbers. The first solution works but there is a problem with the second one, Suppose we have a real number 4 and we put it in our absolute value equation

| x – 3 | = 3 – x
| 4 – 3 | = 3 – 4
| 1 | = -1
Now when we verify our solution we discard the solution not satisfying the equation as in this case the second solution is not satisfying the solution. So, we have only one solution to this equation and that is 3.

But when I saw the answer of this question in the book I saw that the answer is x < =3

Now I am confuse please help me.

 

[statdad]

Notice that in order for the equation

$$|x -3| = 3 - x$$

to be satisfied, $3 - x$ must be greater-than-or-equal-to zero, since the left side of the equation is. Do you see how that fact leads to the solution?