Bài 1: Tìm x ∈ Z biết:
a, $\frac{1}{3}$ + $\frac{-2}{5}$ + $\frac{1}{6}$ + $\frac{-1}{5}$ $\leq$ x < $\frac{-3}{4}$ + $\frac{2}{7}$ + $\frac{-1}{4}$ + $\frac{3}{5}$ + $\frac{5}{7}$ .
($\frac{1}{3}$ + $\frac{1}{6}$)+ ($\frac{-2}{5}$ +$\frac{-1}{5}$ ) $\leq$ x < ($\frac{-3}{4}$ + $\frac{-1}{4}$) + ($\frac{2}{7}$ + $\frac{5}{7}$ ) + $\frac{3}{5}$
$\frac{1}{2}$ + $\frac{-3}{5}$ $\leq$ (-1) + 1 + $\frac{3}{5}$
$\frac{5}{10}$ + $\frac{-6}{10}$ $\leq$ $\frac{3}{5}$
-1 < $\frac{-1}{10}$ $\leq$ $\frac{3}{5}$ < 1
=> x = 0.
b, $\frac{5}{17}$ + $\frac{-4}{19}$ + $\frac{-20}{21}$ + $\frac{12}{17}$ + $\frac{-11}{31}$ $\leq$ $\frac{-3}{7}$ + $\frac{7}{15}$ + $\frac{4}{-7}$ + $\frac{8}{15}$ + $\frac{2}{3}$
Vế trái > $\frac{5}{20}$ + $\frac{-4}{16}$ + $\frac{-21}{21}$ + $\frac{12}{17}$ + $\frac{-11}{31}$
= $\frac{1}{4}$ + $\frac{-1}{4}$ + (-1) + $\frac{12}{17}$ + $\frac{-11}{31}$
= (-1) + $\frac{12}{17}$ - $\frac{-11}{31}$.
Mà $\frac{12}{17}$ > $\frac{1}{2}$ ; $\frac{-11}{31}$ < $\frac{1}{2}$ nên $\frac{12}{17}$ + $\frac{-11}{31}$ > 0.
=> Vế trái > -1.
Vế phải = ($\frac{-3}{7}$ + $\frac{-4}{7}$ ) + ($\frac{7}{15}$ + $\frac{8}{15}$ ) + $\frac{2}{3}$
= $\frac{2}{3}$ < 1.
=> Vế phải < 1.
Thay vào, ta có: -1 < x < 1.
=> x = 0.
