Solve 3u+5/u-1>1

$$\frac { 3 u + 5 } { u - 1 } > 1$$

Solve for u

$$u\in \left(-\infty,-3\right)\cup \left(1,\infty\right)$$

Solution Steps

Denominator u-1 cannot be zero since division by zero is not defined. There are two cases.

$$u-1>0 u-1<0$$

Consider the case when u-1 is positive. Move -1 to the right hand side.

$$u>1$$

The initial inequality does not change the direction when multiplied by u-1 for u-1>0.

$$3u+5>u-1$$

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

$$3u-u>-5-1$$

Combine like terms.

$$2u>-6$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$u>-3$$

Consider condition u>1 specified above.

$$u>1$$

Now consider the case when u-1 is negative. Move -1 to the right hand side.

$$u<1$$

The initial inequality changes the direction when multiplied by u-1 for u-1<0.

$$3u+5<u-1$$

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

$$3u-u<-5-1$$

Combine like terms.

$$2u<-6$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$u<-3$$

Consider condition u<1 specified above. The result remains the same.

$$u<-3$$

The final solution is the union of the obtained solutions.

$$u\in \left(-\infty,-3\right)\cup \left(1,\infty\right)$$