Solve 7x+1/x+2<0

$$\frac { 7 x + 1 } { x + 2 } < 0$$

Solve for x

$$x\in \left(-2,-\frac{1}{7}\right)$$

Solution Steps

For the quotient to be negative, 7x+1 and x+2 have to be of the opposite signs. Consider the case when 7x+1 is positive and x+2 is negative.

$$7x+1>0 x+2<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+2 is positive and 7x+1 is negative.

$$x+2>0 7x+1<0$$

The solution satisfying both inequalities is x\in \left(-2,-\frac{1}{7}\right).

$$x\in \left(-2,-\frac{1}{7}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-2,-\frac{1}{7}\right)$$