Solve for: /(3 x-5) x-3 > 0

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Expression: $$\frac { 3 x - 5 } { x - 3 } \gt 0$$

For the quotient to be positive, $3x-5$ and $x-3$ have to be both negative or both positive. Consider the case when $3x-5$ and $x-3$ are both negative.

$$3x-5<0$$ $$x-3<0$$

The solution satisfying both inequalities is $x<\frac{5}{3}$.

$$x<\frac{5}{3}$$

Consider the case when $3x-5$ and $x-3$ are both positive.

$$x-3>0$$ $$3x-5>0$$

The solution satisfying both inequalities is $x>3$.

$$x>3$$

The final solution is the union of the obtained solutions.

$$x<\frac{5}{3}\text{; }x>3$$

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