Solve x-11/1-2x>3

$$\frac { x - 11 } { 1 - 2 x } > 3$$

Solve for x

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

Solution Steps

Denominator 1-2x cannot be zero since division by zero is not defined. There are two cases.

$$1-2x>0 1-2x<0$$

Consider the case when 1-2x is positive. Move 1 to the right hand side.

$$-2x>-1$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x<\frac{1}{2}$$

The initial inequality does not change the direction when multiplied by 1-2x for 1-2x>0.

$$x-11>3\left(1-2x\right)$$

Multiply out the right hand side.

$$x-11>3-6x$$

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

$$x+6x>11+3$$

Combine like terms.

$$7x>14$$

Divide both sides by 7. Since 7 is positive, the inequality direction remains the same.

$$x>2$$

Consider condition x<\frac{1}{2} specified above.

$$x\in \emptyset$$

Now consider the case when 1-2x is negative. Move 1 to the right hand side.

$$-2x<-1$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x>\frac{1}{2}$$

The initial inequality changes the direction when multiplied by 1-2x for 1-2x<0.

$$x-11<3\left(1-2x\right)$$

Multiply out the right hand side.

$$x-11<3-6x$$

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

$$x+6x<11+3$$

Combine like terms.

$$7x<14$$

Divide both sides by 7. Since 7 is positive, the inequality direction remains the same.

$$x<2$$

Consider condition x>\frac{1}{2} specified above.

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

The final solution is the union of the obtained solutions.

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