Answer
$(-\frac{1}{3},0)$
Work Step by Step
Step 1. Rewrite the inequality as $\frac{5x+2}{x}+1\lt0\longrightarrow \frac{5x+2+x}{x}\lt0\longrightarrow \frac{6x+2}{x}\lt0$
Step 2. Identify the boundary points $x=-\frac{1}{3},0$ and separate the number line into intervals $(-\infty, -\frac{1}{3})$, $(-\frac{1}{3},0)$, and $(0,\infty)$
Step 3. Use test points $x=-1,-\frac{1}{6},1$ to get the signs of the left side of the inequality as $+,-,+$
Step 4. Based on the signs, we have the solution as $(-\frac{1}{3},0)$
