Dear Uncle Colin,

I’m pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help?

– Nasty Absolute Value Inequalities Ending Rongly

Hi, NAVIER, and thanks for your message!

These are nasty, but can be made simpler by rearranging and sketching.

I would begin by getting the absolute value on one side and everything else on the other: $x^2 – x – 2 < \left| 5x-3\right|$.

The left-hand side is a quadratic…