Solve x+2/5x-4<2

$$\frac { x + 2 } { 5 x - 4 } < 2$$

Solve for x

$$x\in \left(-\infty,\frac{4}{5}\right)\cup \left(\frac{10}{9},\infty\right)$$

Solution Steps

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

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

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

$$5x>4$$

Divide both sides by 5. Since 5 is positive, the inequality direction remains the same.

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

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

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

Multiply out the right hand side.

$$x+2<10x-8$$

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

$$x-10x<-2-8$$

Combine like terms.

$$-9x<-10$$

Divide both sides by -9. Since -9 is negative, the inequality direction is changed.

$$x>\frac{10}{9}$$

Consider condition x>\frac{4}{5} specified above. The result remains the same.

$$x>\frac{10}{9}$$

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

$$5x<4$$

Divide both sides by 5. Since 5 is positive, the inequality direction remains the same.

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

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

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

Multiply out the right hand side.

$$x+2>10x-8$$

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

$$x-10x>-2-8$$

Combine like terms.

$$-9x>-10$$

Divide both sides by -9. Since -9 is negative, the inequality direction is changed.

$$x<\frac{10}{9}$$

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

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

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,\frac{4}{5}\right)\cup \left(\frac{10}{9},\infty\right)$$