by **HallsofIvy** » Fri Mar 29, 2019 7:42 am

Another way: after we have [tex]\frac{4x+ 34}{x}< 0[/tex], obviously x cannot be 0 and remember that multiplying both sides of an inequality by a negative number reverses the direction of the inequality while multiplying both sides of an inequality by a positive number does not change it.

So consider two cases:

1) x> 0. Then, multiplying both sides by x, a positive number, 4x+ 34< 0. Subtracting 34 from both sides results in 4x< -34 so x< -17/2. All those values are negative so x> 0 is not true. There are no positive values of x that make this inequality true.

2) x< 0. Then, multiplying both sides by x, a negative number, 4x+ 34> 0. Subtracting 34 from both sides results in 4x> -34 so x> -17/2. Since x must also be negative, this inequality is true for -17/2< x< 0.

("Guest" (who posts an awful lot!) lost the sign going from "4x+ 34= 0" to "x= 17/2".)