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

Question

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 $x$.

Answer 1

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

Answer 2

Another way: rewrite the expression as $|x-6| \geq 4-x$ and denote $x-6=w$. After a bit of algebra the expression on the right can be written as $-w-2$, hence you need to solve $$ |w| \geq -w-2 $$ Setting $w >0, w=0$ and $w<0$ it is easy to determine that the inequality holds iff $w \geq 0$. Now plug back in the expression for $w$ and you see that $x \geq 6$.