Giải bài 125 trang 60 sách bài tập Toán 6 – Cánh Diều Tập 2

Đề bài

Tìm các số nguyên x sao cho:

a) \(\frac{{ – 2}}{5} < \frac{x}{{15}} < \frac{1}{6}\)

b) \(\frac{1}{2} + \frac{1}{3} – \frac{5}{{12}} < 2x < \frac{{ – 12}}{{31}} + \frac{{ – 136}}{{ – 31}}\)

Lời giải chi tiết

a)

Ta có:

\(\begin{array}{l}\frac{{ – 2}}{5} = \frac{{ – 2.18}}{{5.18}} = \frac{{ – 36}}{{90}}\\\frac{x}{{15}} = \frac{{x.6}}{{15.6}} = \frac{{6x}}{{90}}\\\frac{1}{6} = \frac{{1.15}}{{6.15}} = \frac{{15}}{{90}}\end{array}\)

Do đó:

\(\begin{array}{l}\frac{{ – 2}}{5} < \frac{x}{{15}} < \frac{1}{6} \Leftrightarrow \frac{{ – 36}}{{90}} < \frac{{6x}}{{90}} < \frac{{15}}{{90}}\\ \Rightarrow  – 36 < 6x < 15 \Rightarrow  – 6 < x < 2,5\\ \Rightarrow x = \left\{ { – 5; – 4; – 3; – 2; – 1;0;1;2} \right\}\end{array}\)

b) Ta có:

 \(\begin{array}{l}\frac{1}{2} + \frac{1}{3} – \frac{5}{{12}} = \frac{{1.6}}{{2.6}} + \frac{{1.4}}{{3.4}} – \frac{5}{{12}} = \frac{6}{{12}} + \frac{4}{{12}} – \frac{5}{{12}} = \frac{{6 + 4 – 5}}{{12}} = \frac{5}{{12}}\\\frac{{ – 12}}{{31}} + \frac{{ – 136}}{{ – 31}} = \frac{{ – 12}}{{31}} + \frac{{136}}{{31}} = \frac{{ – 12 + 136}}{{31}} = \frac{{124}}{{31}} = 4\end{array}\)

Do đó:

\(\begin{array}{l}\frac{1}{2} + \frac{1}{3} – \frac{5}{{12}} < 2x < \frac{{ – 12}}{{31}} + \frac{{ – 136}}{{ – 31}}\\ \Leftrightarrow \frac{5}{{12}} < 2x < 4 \Leftrightarrow \frac{5}{{24}} < x < 2\\ \Leftrightarrow x = 1\end{array}\)