Câu 10: Trang 34 sách VNEN 9 tập 1
Cho biểu thức: P = $\frac{3(x + \sqrt{x} - 3)}{x + \sqrt{x} - 2}$ + $\frac{\sqrt{x} + 3}{\sqrt{x} + 2}$ - $\frac{\sqrt{x} - 2}{\sqrt{x} - 1}$
a) Rút gọn biểu thức P
b) Tìm x để P < $\frac{15}{4}$.
Bài Làm:
a) P = $\frac{3(x + \sqrt{x} - 3)}{x + \sqrt{x} - 2}$ + $\frac{\sqrt{x} + 3}{\sqrt{x} + 2}$ - $\frac{\sqrt{x} - 2}{\sqrt{x} - 1}$
= $\frac{3(x + \sqrt{x} - 3)}{(\sqrt{x} + 2)(\sqrt{x} - 1)}$ + $\frac{\sqrt{x} + 3}{\sqrt{x} + 2}$ - $\frac{\sqrt{x} - 2}{\sqrt{x} - 1}$
= $\frac{3(x + \sqrt{x} - 3)}{(\sqrt{x} + 2)(\sqrt{x} - 1)}$ + $\frac{(\sqrt{x} + 3)(\sqrt{x} - 1)}{\sqrt{x} + 2}$ - $\frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{\sqrt{x} - 1}$
= $\frac{3(x + \sqrt{x} - 3) + (\sqrt{x} + 3)(\sqrt{x} - 1) - (\sqrt{x} - 2)(\sqrt{x} + 2) }{(\sqrt{x} + 2)(\sqrt{x} - 1)}$
= $\frac{3x + 3\sqrt{x} - 9 + x - sqrt{x} + 3\sqrt{x} - 3 - x + 4}{(\sqrt{x} + 2)(\sqrt{x} - 1)}$
= $\frac{3x + 5\sqrt{x} - 8}{(\sqrt{x} + 2)(\sqrt{x} - 1)}$
= $\frac{(3\sqrt{x} - 5)(\sqrt{x} - 1)}{(\sqrt{x} + 2)(\sqrt{x} - 1)}$
= $\frac{3\sqrt{x} + 8}{\sqrt{x} + 2}$
b) P < $\frac{15}{4}$ $\Leftrightarrow $ $\frac{3\sqrt{x} + 8}{\sqrt{x} + 2}$ < $\frac{15}{4}$
$\Leftrightarrow $ 12$\sqrt{x}$ + 32 < 15$\sqrt{x}$ + 30
$\Leftrightarrow $ 3$\sqrt{x}$ > 2
$\Leftrightarrow $ $\sqrt{x}$ > $\frac{2}{3}$
$\Leftrightarrow $ x > $\frac{4}{9}$
Vậy x > $\frac{4}{9}$.