Any way to solve $|x-8| = |2-x|-6$ algebraically? Everything I've tried has given me $x = 2$ (which is obviously incorrect, since $-6 \neq 6$). 
The actual answer is $x \geq 8$ which I obtained by observing a graph. 
Would love assistance!

For this equation $$|x-8| = |2-x| - 6$$ there are two modulus terms $|x-8|$ and $|2-x|$ so these two terms will behave differently at the points $x=2$ and $x=8$. Now break the terms in the following cases:
Case1:
$x>=8$ , this means 
$-x\leq-8$ or $2-x\leq-6$ which reduces the equation to
$$  x-8 = x-2-6$$
which reduces to an identity and this means $x\geq8$ is one of the solution for the equation.
Case 2:
$x<8$, this means  $-x>-8$ or $2-x>-6$ which reduces the equation to
$$8-x = x-2-6$$ if $$-6<2-x<0$$
or $$2x=16$$ when $$2<x<8$$ giving no solution from this case.
Another equation is 
$$8-x=2-x-6$$ if $$x<2$$ giving
$$8=-4$$ and $$x<2$$ again giving no solution
Conclusion: There is only one possibility that $x\geq8$. Another way of investigating this is by inspection. After proving case 1 and case 2 just put values that are greater or equal 8 in the above equation and you will find that for all $x\geq8$ the equation holds true whereas for the rest of the cases it does not hold. Try putting $x=2$ or any other value. The equation wont hold. 
