`\text{Đáp án:}`
`\text{a)` `=\frac{9}{8}`
`\text{b)` `=\frac{1}{\sqrt{x}-2}`
`\text{c)` `x\le\frac{1}{4}`
`\text{Giải thích các bước giải:}`
`\text{a) Khi x= 49 thì } ` `A=\frac{\sqrt{49}+2}{\sqrt{49+1}}=\frac{7+1}{7+2}=\frac{9}{8}`
`\text{b) }``B=\frac{\sqrt{x}+2}{x-5\sqrt{x}+6}+\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-3}`
`=\frac{\sqrt{x}+2}{(x-2\sqrt{x})-(3\sqrt{x}-6)}+\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-3}`
`=\frac{\sqrt{x}+2}{\sqrt{x}(\sqrt{x}-2)-3(\sqrt{x}-2)}+\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-3}`
`=\frac{\sqrt{x}+2}{(\sqrt{x}-2)\(\sqrt{x}-3)}+\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-3}`
`=\frac{\sqrt{x}+2+(\sqrt{x}+3)(\sqrt{x}-3)-(\sqrt{x}+2)(\sqrt{x}-2)}{(\sqrt{x}-2)\(\sqrt{x}-3)}`
`=\frac{\sqrt{x}+2+x-9-x+4}{(\sqrt{x}-2)(\sqrt{x}+3\)}`
`=\frac{\sqrt{x}-3}{(\sqrt{x}-2)(\sqrt{x}-3)}`
`=\frac{1}{\sqrt{x}-2}`
`\text{c) Ta có:}` `\frac{A}{B}=\frac{\frac{\sqrt{x}+2}{\sqrt{x}+1}}{\frac{1}{\sqrt{x}-2}}=\frac{\sqrt{x}+2}{\sqrt{x}+1}:\frac{1}{\sqrt{x}-2}=\frac{\sqrt{x}+2}{\sqrt{x}+1}\cdot (\sqrt{x}-2)`
`=\frac{(\sqrt{x}+2)(\sqrt{x}-2)}{\sqrt{x}+1}=\frac{x-4}{\sqrt{x}+1}`
`\text{Để}` `\frac{A}{B}\le -\frac{5}{2}`
`\Leftrightarrow \frac{x-4}{\sqrt{x}+1} -\frac{5}{2}`
`\Leftrightarrow \frac{x-4}{\sqrt{x}+1}+\frac{5}{2}\le 0`
`\Leftrightarrow \frac{2(x-4)+5(\sqrt{x}+1)}{2(\sqrt{x}+1)}\le 0 `
`\Leftrightarrow \frac{2x-8+5\sqrt{x}+5}{2(\sqrt{x}+1)}\le 0`
`\Leftrightarrow \frac{2x+5\sqrt{x}-3}{2(\sqrt{x}+1)}\le 0`
`\Leftrightarrow \frac{(2x-\sqrt{x})+(6\sqrt{x}-3)}{2(\sqrt{x}+1)}\le 0`
`\Leftrightarrow \frac{\sqrt{x}(2\sqrt{x}-1)+3(2\sqrt{x}-1)}`
`\Leftrightarrow2(\sqrt{x}+1)\le 0`
`\Leftrightarrow \frac{(2\sqrt{x}-1)(\sqrt{x}+3)}{2(\sqrt{x}+1)}\le 0`
`\text{Vì}` `x\ge 0\Leftrightarrow \sqrt{x}\ge 0\Leftrightarrow 2(\sqrt{x}+1)>0`
`\text{Vậy để }` `\frac{A}{B}\le -\frac{5}{2}\Leftrightarrow \frac{(2\sqrt{x}-1)(\sqrt{x}+3}}{2\(\sqrt{x}+1)}\le 0`
`\text{Mà}` `\sqrt{x}+3>0(x\ge 0)`
`\text{Vậy để }` `(2\sqrt{x}-1)(\sqrt{x}+3)\le 0` `\text{thì:}` `2\sqrt{x}-1\le 0`
`\Leftrightarrow 2\sqrt{x}\le 1`
`\Leftrightarrow \sqrt{x}\le \frac{1}{2}`
`\Leftrightarrow x\le\frac{1}{4}`
`\text{Vậy để}` `A/B\le-5/2` `\text{thì}` `x\le\frac{1}{4}`
