Solve 10x-2/4-5x>5

$$\frac{ 10x-2 }{ 4-5x } > 5$$

Solve for x

$$x\in \left(\frac{22}{35},\frac{4}{5}\right)$$

Solution Steps

Denominator 4-5x cannot be zero since division by zero is not defined. There are two cases.

$$4-5x>0 4-5x<0$$

Consider the case when 4-5x is positive. Move 4 to the right hand side.

$$-5x>-4$$

Divide both sides by -5. Since -5 is negative, the inequality direction is changed.

$$x<\frac{4}{5}$$

The initial inequality does not change the direction when multiplied by 4-5x for 4-5x>0.

$$10x-2>5\left(4-5x\right)$$

Multiply out the right hand side.

$$10x-2>20-25x$$

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

$$10x+25x>2+20$$

Combine like terms.

$$35x>22$$

Divide both sides by 35. Since 35 is positive, the inequality direction remains the same.

$$x>\frac{22}{35}$$

Consider condition x<\frac{4}{5} specified above.

$$x\in \left(\frac{22}{35},\frac{4}{5}\right)$$

Now consider the case when 4-5x is negative. Move 4 to the right hand side.

$$-5x<-4$$

Divide both sides by -5. Since -5 is negative, the inequality direction is changed.

$$x>\frac{4}{5}$$

The initial inequality changes the direction when multiplied by 4-5x for 4-5x<0.

$$10x-2<5\left(4-5x\right)$$

Multiply out the right hand side.

$$10x-2<20-25x$$

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

$$10x+25x<2+20$$

Combine like terms.

$$35x<22$$

Divide both sides by 35. Since 35 is positive, the inequality direction remains the same.

$$x<\frac{22}{35}$$

Consider condition x>\frac{4}{5} specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{22}{35},\frac{4}{5}\right)$$