Solve 7-x/2+3x<0

$$\frac { 7 - x } { 2 + 3 x } < 0$$

Solve for x

$$x\in \left(-\infty,-\frac{2}{3}\right)\cup \left(7,\infty\right)$$

Solution Steps

For the quotient to be negative, 7-x and 3x+2 have to be of the opposite signs. Consider the case when 7-x is positive and 3x+2 is negative.

$$7-x>0 3x+2<0$$

The solution satisfying both inequalities is x<-\frac{2}{3}.

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

Consider the case when 3x+2 is positive and 7-x is negative.

$$3x+2>0 7-x<0$$

The solution satisfying both inequalities is x>7.

$$x>7$$

The final solution is the union of the obtained solutions.

$$x<-\frac{2}{3}\text{; }x>7$$