The question is as follows: $|x+2|<|\frac12x-5|$
The book that I am using mentions that there are 2 useful properties that can be used when solving modulus inequalities, which are $|p|≤q ⇔ -q≤p≤q$ and $|p|≥q ⇔ p≤-q\;or\;p≥q$
However, this question involves 2 moduli which are on both sides. In another example, the book shows that you can use $|p|≥|q| ⇔ p^2≥q^2$
The part where I'm confused at is whether I should use $|p|≤q ⇔ -q≤p≤q$ or $|p|≥|q| ⇔ p^2≥q^2$ to solve this question and how to go forward from there.
Best Answer
Whenever you encounter such dilemma, try both and see what problem you will encounter.
If you use the first method, in a single move, you get rid of one absolute value sign, there is still an absolute value left and there's more work to do. However for the purpose of exploration, you are still encouraged to try it.
If you do the second approach, you get rid of both absolute signs directly.
$$\left( \frac12 x - 5\right)^2-(x+2)^2>0$$
Now, you can use $a^2-b^2=(a-b)(a+b)$ to factorize and solve the problem.