Solve 4x+1/x-3<2

$$\frac { 4 x + 1 } { x - 3 } < 2$$

Solve for x

$$x\in \left(-\frac{7}{2},3\right)$$

Solution Steps

Denominator x-3 cannot be zero since division by zero is not defined. There are two cases.

$$x-3>0 x-3<0$$

Consider the case when x-3 is positive. Move -3 to the right hand side.

$$x>3$$

The initial inequality does not change the direction when multiplied by x-3 for x-3>0.

$$4x+1<2\left(x-3\right)$$

Multiply out the right hand side.

$$4x+1<2x-6$$

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

$$4x-2x<-1-6$$

Combine like terms.

$$2x<-7$$

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

$$x<-\frac{7}{2}$$

Consider condition x>3 specified above.

$$x\in \emptyset$$

Now consider the case when x-3 is negative. Move -3 to the right hand side.

$$x<3$$

The initial inequality changes the direction when multiplied by x-3 for x-3<0.

$$4x+1>2\left(x-3\right)$$

Multiply out the right hand side.

$$4x+1>2x-6$$

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

$$4x-2x>-1-6$$

Combine like terms.

$$2x>-7$$

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

$$x>-\frac{7}{2}$$

Consider condition x<3 specified above.

$$x\in \left(-\frac{7}{2},3\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{7}{2},3\right)$$