Solve 3x/1-2xgeq4

$$\frac { 3 x } { 1 - 2 x } \geq 4$$

Solve for x

$$x\in [\frac{4}{11},\frac{1}{2})$$

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.

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

Multiply out the right hand side.

$$3x\geq 4-8x$$

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

$$3x+8x\geq 4$$

Combine like terms.

$$11x\geq 4$$

Divide both sides by 11. Since 11 is positive, the inequality direction remains the same.

$$x\geq \frac{4}{11}$$

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

$$x\in [\frac{4}{11},\frac{1}{2})$$

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.

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

Multiply out the right hand side.

$$3x\leq 4-8x$$

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

$$3x+8x\leq 4$$

Combine like terms.

$$11x\leq 4$$

Divide both sides by 11. Since 11 is positive, the inequality direction remains the same.

$$x\leq \frac{4}{11}$$

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

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in [\frac{4}{11},\frac{1}{2})$$