Solve 1-1/4x/frac{2{5}x-1/2}<0

$$\frac{ 1- \frac{ 1 }{ 4 } x }{ \frac{ 2 }{ 5 } x- \frac{ 1 }{ 2 } } < 0$$

Solve for x

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

Solution Steps

For the quotient to be negative, -\frac{x}{4}+1 and \frac{2x}{5}-\frac{1}{2} have to be of the opposite signs. Consider the case when -\frac{x}{4}+1 is positive and \frac{2x}{5}-\frac{1}{2} is negative.

$$-\frac{x}{4}+1>0 \frac{2x}{5}-\frac{1}{2}<0$$

The solution satisfying both inequalities is x<\frac{5}{4}.

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

Consider the case when \frac{2x}{5}-\frac{1}{2} is positive and -\frac{x}{4}+1 is negative.

$$\frac{2x}{5}-\frac{1}{2}>0 -\frac{x}{4}+1<0$$

The solution satisfying both inequalities is x>4.

$$x>4$$

The final solution is the union of the obtained solutions.

$$x<\frac{5}{4}\text{; }x>4$$