Solve 2x+1/x-5<1

$$\frac { 2 x + 1 } { x - 5 } < 1$$

Solve for x

$$x\in \left(-6,5\right)$$

Solution Steps

Denominator x-5 cannot be zero since division by zero is not defined. There are two cases.

$$x-5>0 x-5<0$$

Consider the case when x-5 is positive. Move -5 to the right hand side.

$$x>5$$

The initial inequality does not change the direction when multiplied by x-5 for x-5>0.

$$2x+1<x-5$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

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

Combine like terms.

$$x<-6$$

Consider condition x>5 specified above.

$$x\in \emptyset$$

Now consider the case when x-5 is negative. Move -5 to the right hand side.

$$x<5$$

The initial inequality changes the direction when multiplied by x-5 for x-5<0.

$$2x+1>x-5$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

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

Combine like terms.

$$x>-6$$

Consider condition x<5 specified above.

$$x\in \left(-6,5\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-6,5\right)$$