Question

[Ngnndka]

Giúp mình với

Answer 1

[Đoàn Đức Hà]

\(A=\left(1-\frac{\sqrt{x}}{1+\sqrt{x}}\right)\div\left(\frac{\sqrt{x}+3}{\sqrt{x}-2}+\frac{\sqrt{x}+2}{3-\sqrt{x}}+\frac{\sqrt{x}+2}{x-5\sqrt{x}+6}\right)\)(ĐK: \(x\ge0,x\ne4,x\ne9\))

\(=\frac{1+\sqrt{x}-\sqrt{x}}{1+\sqrt{x}}\div\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)+\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{1}{1+\sqrt{x}}\times\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}{\sqrt{x}-3}=\frac{\sqrt{x}-2}{\sqrt{x}+1}\)

\(A< \frac{1}{2}\Leftrightarrow\frac{\sqrt{x}-2}{\sqrt{x}+1}< \frac{1}{2}\)(\(x\ge0,x\ne4,x\ne9\)) 

\(\Leftrightarrow\frac{2\sqrt{x}-4-\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}+1\right)}< 0\)

\(\Leftrightarrow\sqrt{x}-5< 0\)

\(\Leftrightarrow x< 25\).

Vậy \(0\le x< 25,x\ne4,x\ne9\)thì \(A< \frac{1}{2}\).

\(A=\frac{\sqrt{x}-2}{\sqrt{x}+1}=\frac{\sqrt{x}+1-3}{\sqrt{x}+1}=1-\frac{3}{\sqrt{x}+1}\inℤ\Leftrightarrow\frac{3}{\sqrt{x}+1}\inℤ\)

suy ra \(\sqrt{x}+1\inƯ\left(3\right)=\left\{1,3\right\}\)vì \(x\inℤ\)

\(\Rightarrow x\in\left\{0,4\right\}\)

Đối chiếu điều kiện ta chỉ có \(x=0\)thỏa mãn.