Question

Are the equations `|x - 3| = 7` and `|x| - 3 = 7` equivalent?

Answer 1

No, they aren't.

Explanation:

The two would be equivalent if `|x-3|=|x|-3`.

So, all I need to do if giving a counter example: pick `x=0`. On one hand, you have `|0-3|=|-3|=3`. On the other hand, `|0|-3=-3`. In general, you see that `|x-3|` is always positive, because it is an absolute value, while `|x|-3` can be negative, if `x \in (-3,3)`.

So, the two equalities aren't equivalent:

`|x-3|=7` is true for `x=10` or `x=-4`, while

`|x|-3=7 -> |x|=10` is true for `x=\pm10`.

Answer 2

Yes, they are equivalent.

Explanation:

The clue is in the equals sign!

An equals sign is absolute. The value of what is on its left is 'exactly' the same as the value on its right.

As number 7 is on the right in both cases, then by definition, what is on the left also has the same intrinsic value

In that `7=7` so `|x-3| = |x|-3`

In other words, the value of 7 imposes restrictions on both of `|x-3|` and `|x|-3`.

The thing is: The domain of each of the x's will not be identical

Consider `|x_1-3|=7` The only feasible values for `x_1` are {-4 , +10}

Consider `|x_2|-3=7` The only feasible value for `x_2` are {-10 , +10}

However, in both cases of `x_1" and "x_2` the left hand side still has the intrinsic value of 7

`color(brown)((|x-3| =7 ))color(green)(-=)color(blue)( ( |x|-3=7))`