# How to combine OR linear inequality with absolute value

I have x < -10 OR x > 15

How do I turn it into a single inequality using an absolute value? Like a < |x+b|. What are the rules? I'm not sure if it is right, but I'm expecting an answer something like how factoring quadratic trinomials would be explained, which is really straightforward.

Thank you

• The way I think about this is that $x$ is at most ___ away from the midpoint of $-10$ and $15$ (remember that absolute value tells distance). Commented Oct 8, 2014 at 0:32
• Try doing some examples -- which $x$ satisfy $|x-2| < 1$? Commented Oct 8, 2014 at 0:33
• So it could always be in the form $a<|x-b|$?
– user181677
Commented Oct 8, 2014 at 0:43
• @andybenji btw, your example is AND inequality
– user181677
Commented Oct 8, 2014 at 0:44
• My example wasn't intended to be the exact same as your question. Try out some examples, and see if all the inequalities you can think of can be written in that form. Commented Oct 8, 2014 at 0:49

Hint: You can use the distance from midpoint argument, or use quadratics:$$x < -10 \text{ OR } x> 15 \iff (x+10)(15-x) < 0 \\ \iff (x-\tfrac52)^2> \tfrac{625}4 \iff |x - \tfrac52| > \tfrac{25}2$$