Answer
The solution is $(-\infty, 2 ]$.
Work Step by Step
Recall that:
the square root of $k^2$ is $|k|$ (for all real numbers $k$).
Thus, the problem could be rewritten as
$\sqrt{(x-3)^2} = \sqrt{(x-2)^2} + 1$
by squaring the two sides, we get
$(x^2 -6x+9) = (x^2 -4x +4) +(1) + 2 |x-2|$
Combining like terms gives:
$-2x+4= -2(x-2) = 2 |x-2|$
Thus, we have
$-(x-2)= |x-2|$
From the definition of the absolute value function, we get $x-2 \le 0$, and then $x \le 2$. Hence, the solution as a half infinite interval is $(-\infty, 2]$.
