Đề bài

Tìm số ngyên x biết: a) \(\frac{{ - 5}}{7} + 1 + \frac{{30}}{{ - 7}} \le x \le \frac{{ - 1}}{6} + \frac{1}{3} + \frac{5}{6}\) b) \(\frac{{ - 8}}{{13}} + \frac{7}{{17}} + \frac{{21}}{{13}} \le x \le \frac{{ - 9}}{{14}} + 3 + \frac{5}{{ - 14}}\)

Lời giải chi tiết

a)Ta có: \(\begin{array}{l}\frac{{ - 5}}{7} + 1 + \frac{{30}}{{ - 7}} = \left( {\frac{{ - 5}}{7} + \frac{{30}}{{ - 7}}} \right) + 1 = \left( {\frac{{ - 5}}{7} + \frac{{ - 30}}{7}} \right) + 1 = \frac{{ - 35}}{7} + 1 =  - 5 + 1 =  - 4\\\frac{{ - 1}}{6} + \frac{1}{3} + \frac{5}{6} = \left( {\frac{{ - 1}}{6} + \frac{5}{6}} \right) + \frac{1}{3} = \frac{4}{6} + \frac{1}{3} = \frac{2}{3} + \frac{1}{3} = 1\end{array}\)Do đó:\(\frac{{ - 5}}{7} + 1 + \frac{{30}}{{ - 7}} \le x \le \frac{{ - 1}}{6} + \frac{1}{3} + \frac{5}{6} \Leftrightarrow  - 4 \le x \le 1 \Leftrightarrow x \in \left\{ { - 4; - 3; - 2; - 1;0;1} \right\}\)vì x là số nguyên.Vậy \(x \in \left\{ { - 4; - 3; - 2; - 1;0;1} \right\}\)b) Ta có:\(\begin{array}{l}\frac{{ - 8}}{{13}} + \frac{7}{{17}} + \frac{{21}}{{13}} = \left( {\frac{{ - 8}}{{13}} + \frac{{21}}{{13}}} \right) + \frac{7}{{17}} = \frac{{13}}{{13}} + \frac{7}{{17}} = 1 + \frac{7}{{17}} = \frac{{24}}{{17}}\\\frac{{ - 9}}{{14}} + 3 + \frac{5}{{ - 14}} = \left( {\frac{{ - 9}}{{14}} + \frac{5}{{ - 14}}} \right) + 3 = \left( {\frac{{ - 9}}{{14}} + \frac{{ - 5}}{{14}}} \right) + 3 = \frac{{ - 14}}{{14}} + 3 =  - 1 + 3 = 2\end{array}\) Do đó:\(\frac{{ - 8}}{{13}} + \frac{7}{{17}} + \frac{{21}}{{13}} \le x \le \frac{{ - 9}}{{14}} + 3 + \frac{5}{{ - 14}} \Leftrightarrow \frac{{24}}{{17}} \le x \le 2 \Rightarrow x = 2\) vì x là số nguyên.Vậy x = 2.