Prove that for all $x\in\mathbb{R}\enspace \lvert x\rvert + \lvert x-6 \rvert\geq 6$

Question

I am beginning proofs in analysis. I am reading Kane's book, but I am not sure if this proof counts as a proof in analysis. I have tried proving by contradiction, but I failed. I also tried using the triangle inequality ($\lvert x - 6\rvert\leq\lvert x\rvert + \vert\text{-}6\rvert$), but it just does not follow from it. The farthest I got was showing that $\lvert 2x -6\rvert\leq \rvert x\lvert + \rvert x - 6\lvert$ by letting $y = x-6$ and using the triangle inequality. If anybody has any idea how to proceed, I would greatly appreciate some guidance. Thanks.

Answer 1

Use the triangle inequality $$|a|+|b|\geq |a+b|$$ and the fact that $|-a|=|a|$.