my work below:
step 1: -6r-6 < or = to -4r+8
step 2: -6+4r < or = to 8+6
step 3: -2r < or = to 14/2
step 4: -2r div -2r < or = to 7
Answer: r > or = -7
Your work is fine with a few minor technicalities. From going to step 2 to step 3, your inequality would not be true as it seems like you 'randomly' divided by 2 but I presume you're just getting ahead of yourself and was meant for step 3 to step 4.
Also, I think you meant to divide both sides by -2 not -2r as you would end up with: which isn't necessary.
(Didn't divide by 2 yet. Remember: What you do to one side you do to the other)