Show that $|a|\leq \int_{-1}^{1}\,|ax+b|\,\text{d}x$

\begin{align} \int_{-1}^1|ax + b| {\rm d}x&=\int_{-1}^1\big|(ax + b)\,\text{sign}(x)\big| {\rm d}x \\&\ge \left|\int_{-1}^1 (ax + b)\,\text{sign}(x) {\rm d}x\right|\\ &= |a|. \end{align}

(I think this is the first time I use the inequality $\displaystyle\int_a^b|f| \ge \left|\int_a^bf\right|$ in this direction!)

Edit by Batominovski. Please undelete this. Your idea works. It just needs a small twist. I fixed that for you. (You can delete this remark. I am ok with that.)

Edit by Aryaman. The above solution is mainly thanks to Batominovski, who fixed a rather silly calculation of mine. Thank you!