1. Solve for x: 1/(3x - 1) + 2/(x - 3) 2. Rationalize the denominator: 2 / (x+1+√x)

Steps to Solve

1. Solve the first equation Begin by finding a common denominator for the two fractions, which is $(3x-1)(x-3)$.

2. Combine the fractions Multiply the first fraction by $\frac{x-3}{x-3}$ and the second fraction by $\frac{3x-1}{3x-1}$. Combine the numerators: $$ \frac{1}{3x-1} + \frac{2}{x-3} = \frac{1(x-3) + 2(3x-1)}{(3x-1)(x-3)} $$

3. Simplify the numerator Expand and simplify the numerator: $$ \frac{x-3+6x-2}{(3x-1)(x-3)} = \frac{7x-5}{(3x-1)(x-3)} $$

4. Expand the denominator Expand the denominator to get: $$ \frac{7x-5}{3x^2 - 9x - x + 3} = \frac{7x-5}{3x^2 - 10x + 3} $$

5. Rationalize the denominator of the second expression The expression to rationalize is $\frac{2}{(x+1)+\sqrt{x}}$. To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator, which is $(x+1) - \sqrt{x}$.

6. Multiply by the conjugate $$ \frac{2}{(x+1)+\sqrt{x}} \cdot \frac{(x+1)-\sqrt{x}}{(x+1)-\sqrt{x}} $$

7. Simplify the denominator The denominator becomes $((x+1)+\sqrt{x})((x+1)-\sqrt{x}) = (x+1)^2 - (\sqrt{x})^2 = (x^2+2x+1) - x = x^2 + x + 1$.

8. Simplify the expression The expression simplifies to: $$ \frac{2((x+1)-\sqrt{x})}{x^2+x+1} = \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$

9. Final Result Thus the original equations are $$ \frac{1}{3x-1} + \frac{2}{x-3} = \frac{7x-5}{3x^2 - 10x + 3} $$ $$ \frac{2}{(x+1)+\sqrt{x}} = \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$