Solve rational equation

1 Solve the equation \(\frac { 1 } { x } + \frac { x } { x + 2 } = 1\).

1 Solve the equation. $$\frac { 8 } { x } - \frac { 9 } { x + 1 } = 1$$

1 Solve the equation \(\frac { 5 x } { 2 x + 1 } - \frac { 3 } { x + 1 } = 1\).

3 Solve the equation \(\frac { 4 x } { x + 1 } - \frac { 3 } { 2 x + 1 } = 1\).

1 Solve the equation \(\frac { 2 x } { x + 1 } - \frac { 1 } { x - 1 } = 1\).

5 Solve the equation \(\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3\).

1 Solve the equation \(\frac { 4 x } { x + 1 } - \frac { 3 } { 2 x + 1 } = 1\).

1. (a) Express

$$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 }$$ as a single fraction in its simplest form.
(b) Hence, find the values of \(x\) such that $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 } = \frac { 1 } { 2 } .$$

4. (a) Express $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form.
(b) Hence, show that the equation $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) } = 1$$ has no real roots.