Q:

$5-3|x-6|\leq 3x -7$

I have this inequation: 
$$5-3|x-6|\leq 3x -7$$
i solved this this way: 
i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - :  
1) for $x\geq6$ (positive):  
$5-3x+6\leq 3x -7\\  
6x\geq30\\
x\geq5$
2) for $x<6$ (negative):  
$5-3(-x+6)\le3x-7\\
-13\leq-7$
But i dont understand what those $x\geq6$ and $x\geq5$ say to me about $x$. 

A:

We need $x\ge6$ and $x\ge5,$ so $x\ge$max$(6,5)=6$

