Prove that (1)/(sqrt2-1)+(2)/(sqrt3+1)=>sqrt2+sqrt3

Rationalize first fraction: To simplify the left side of the inequality, we will rationalize the denominators of each fraction. Rationalize the first fraction: (1)/(sqrt2-1) * (sqrt2+1)/(sqrt2+1) = (sqrt2+1)/(2-1) = sqrt2+1.Rationalize second fraction: Rationalize the second fraction: (2)/(sqrt3+1) * (sqrt3-1)/(sqrt3-1) = (2sqrt3-2)/(3-1) = (2sqrt3-2)/2 = sqrt3-1.Combine rationalized fractions: Combine the two rationalized fractions: (sqrt2+1) + (sqrt3-1).Simplify combined expression: Simplify the combined expression: (sqrt2+1) + (sqrt3-1) = sqrt2 + sqrt3.Simplified left side: Now we have the simplified left side of the inequality: sqrt2 + sqrt3. The original inequality to prove was (1)/(sqrt2-1)+(2)/(sqrt3+1) >= sqrt2+sqrt3. After simplification, we have sqrt2 + sqrt3 >= sqrt2 + sqrt3.Prove inequality: Since both sides of the inequality are the same, the inequality holds true.

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Q. Prove that

\frac{1}{\sqrt{2}-1}+\frac{2}{\sqrt{3}+1} \Rightarrow \sqrt{2}+\sqrt{3}

Rationalize first fraction: To simplify the left side of the inequality, we will rationalize the denominators of each fraction. $\newline$ Rationalize the first fraction: $(1)/(\sqrt{2}-1) \times (\sqrt{2}+1)/(\sqrt{2}+1) = (\sqrt{2}+1)/(2-1) = \sqrt{2}+1$ .
Rationalize second fraction: Rationalize the second fraction: (\frac{$2$}{\sqrt{$3$}+$1$}) \cdot (\frac{\sqrt{$3$}$-1$}{\sqrt{$3$}$-1$}) = \frac{$2$\sqrt{$3$}$-2$}{$3$$-1$} = \frac{$2$\sqrt{$3$}$-2$}{$2$} = \sqrt{$3$}\(-1\.
Combine rationalized fractions: Combine the two rationalized fractions: $(\sqrt{2}+1) + (\sqrt{3}-1)$ .
Simplify combined expression: Simplify the combined expression: $(\sqrt{2}+1) + (\sqrt{3}-1) = \sqrt{2} + \sqrt{3}$ .
Simplified left side: Now we have the simplified left side of the inequality: $\sqrt{2} + \sqrt{3}$ . The original inequality to prove was $\frac{1}{\sqrt{2}-1}+\frac{2}{\sqrt{3}+1} \geq \sqrt{2}+\sqrt{3}$ . After simplification, we have $\sqrt{2} + \sqrt{3} \geq \sqrt{2} + \sqrt{3}$ .
Prove inequality: Since both sides of the inequality are the same, the inequality holds true.