Solve 1-2-x/3x-2<0

$$1 - \frac { 2 - x } { 3 x - 2 } < 0$$

Solve for x

$$x\in \left(\frac{2}{3},1\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3x-2}{3x-2}.

$$\frac{3x-2}{3x-2}-\frac{2-x}{3x-2}<0$$

Since \frac{3x-2}{3x-2} and \frac{2-x}{3x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x-2-\left(2-x\right)}{3x-2}<0$$

Do the multiplications in 3x-2-\left(2-x\right).

$$\frac{3x-2-2+x}{3x-2}<0$$

Combine like terms in 3x-2-2+x.

$$\frac{4x-4}{3x-2}<0$$

For the quotient to be negative, 4x-4 and 3x-2 have to be of the opposite signs. Consider the case when 4x-4 is positive and 3x-2 is negative.

$$4x-4>0 3x-2<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 3x-2 is positive and 4x-4 is negative.

$$3x-2>0 4x-4<0$$

The solution satisfying both inequalities is x\in \left(\frac{2}{3},1\right).

$$x\in \left(\frac{2}{3},1\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{2}{3},1\right)$$