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Giải bài tập 3 trang 34 SGK Toán 9 tập 1 - Cánh diều

a. Cho (a > b > 0). Chứng minh: (frac{1}{a} < frac{1}{b}). b. Áp dụng kết quả trên, hãy so sánh: (frac{{2022}}{{2023}}) và (frac{{2023}}{{2024}}).

Tổng hợp Đề thi vào 10 có đáp án và lời giải

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Đề bài

 

🃏a. Cho \(a > b > 0\). Chứng minh: \(\frac{1}{a} < \frac{1}{b}\).

🎶b. Áp dụng kết quả trên, hãy so sánh: \(\frac{{2022}}{{2023}}\) và \(\frac{{2023}}{{2024}}\).

a. Cho \(a > b > 0\). Chứng minh: \(\frac{1}{a} < \frac{1}{b}\). b. Áp dụng kết quả trên, hãy so sánh: \(\frac{{2022}}{{2023}}\) và \(\frac{{2023}}{{2024}}\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Xét hiệu để chứng minh. Sau đó dùng kết quả vừa chứng minh để so sánh.

Lời giải chi tiết

a. Do \(a > b\) nên \(b - a < 0\). Do \(a > b > 0\) nên \(ab > 0\). Xét hiệu \(\frac{1}{a} - \frac{1}{b} = \frac{{b - a}}{{ab}}\). Do \(\left\{ \begin{array}{l}b - a < 0\\ab > 0\end{array} \right.\) nên \(\frac{{b - a}}{{ab}} < 0\). Vậy \(\frac{1}{a} < \frac{1}{b}\). b. Ta có: \(\frac{{2022}}{{2023}} = 1 - \frac{1}{{2023}};\,\frac{{2023}}{{2024}} = 1 - \frac{1}{{2024}}\) Theo kết quả vừa chứng minh ta có: \(2024 > 2023\) nên \(\frac{1}{{2023}} > \frac{1}{{2024}}\) suy ra \( - \frac{1}{{2023}} <  - \frac{1}{{2024}}\) nên \(1 - \frac{1}{{2023}} < 1 - \frac{1}{{2024}}\). Vậy \(\frac{{2022}}{{2023}} < \frac{{2023}}{{2024}}\).

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