### Instruction: What is the sum of all values of $x$ for which $|x-3| + |3-x| -1 = 3$?### Response: To solve for $x$, I need to get rid of the absolute value signs.
I notice that $|x-3|$ and $|3-x|$ are actually the same expression, since they both measure the distance between $x$ and $3$ on the number line.
So I can rewrite the equation as $2|x-3| -1 = 3$.
To isolate $|x-3|$, I can add $1$ to both sides and then divide by $2$.
I get $|x-3| = 2$.
Now I have two possible cases: either $x-3$ is positive or negative.
If $x-3$ is positive, then $|x-3| = x-3$ and the equation becomes $x-3 = 2$.
Solving for $x$, I get $x = 5$.
If $x-3$ is negative, then $|x-3| = -(x-3)$ and the equation becomes $-(x-3) = 2$.
Solving for $x$, I get $x = 1$.
So the two values of $x$ that satisfy the equation are $x = 5$ and $x = 1$.
To find the sum of these values, I just add them up: $5 + 1 = 6$.