The book Numerical Analysis by Richard L. Burden has the following exercise:
Suppose $ p^{*} $ must approximate $ p $ with relative error at most $ 10^{-3} $. Find the largest interval in which $ p^{*} $ must lie for each value of $ p $.
a) 150 $\quad$ b) 900 $\quad$ c) 1500 $\quad$ d) 90
My answer for c):
$ \dfrac{\vert p-p^{*} \vert}{\vert p \vert} \leq 10^{-3}$.
As $ p =1500 $, $ \dfrac{\vert 1500-p^{*} \vert}{1500} \leq 10^{-3}$. (I put $ \leq $ by the phrase at most).
$ \Rightarrow \vert 1500-p^{*} \vert \leq 1.5$
So my answer is $ [1498.5, 1501.5] $.
Analogously for a), b) and d) my answers are
$ [149.85,150.15] $, $ [899.1, 900.9] $ y $ [89.91, 90.09] $, respectively.
The answers at the end of the book are
$ (149.85, 150.15) $, $ (899.1,900.9) $, $ (1498.5, 1501.5) $ y $ (89.91, 90.09) $, respectively.
Could someone explain to me why p * can not take the extreme values of the intervals?