Solve x+2/x-1-3>0

$$\frac{ x+2 }{ x-1 } -3 > 0$$

Solve for x

$$x\in \left(1,\frac{5}{2}\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-1}{x-1}.

$$\frac{x+2}{x-1}-\frac{3\left(x-1\right)}{x-1}>0$$

Since \frac{x+2}{x-1} and \frac{3\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+2-3\left(x-1\right)}{x-1}>0$$

Do the multiplications in x+2-3\left(x-1\right).

$$\frac{x+2-3x+3}{x-1}>0$$

Combine like terms in x+2-3x+3.

$$\frac{-2x+5}{x-1}>0$$

For the quotient to be positive, 5-2x and x-1 have to be both negative or both positive. Consider the case when 5-2x and x-1 are both negative.

$$5-2x<0 x-1<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 5-2x and x-1 are both positive.

$$x-1>0 5-2x>0$$

The solution satisfying both inequalities is x\in \left(1,\frac{5}{2}\right).

$$x\in \left(1,\frac{5}{2}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(1,\frac{5}{2}\right)$$