1. **State the problem:** Solve the inequality $$\frac{\sin x}{\tan x + 1} \geq 0$$ for $x$. 2. **Recall definitions and formulas:** - $\tan x = \frac{\sin x}{\cos x}$. - The inequality involves a fraction, so the sign depends on numerator and denominator. 3. **Rewrite the denominator:** $$\tan x + 1 = \frac{\sin x}{\cos x} + 1 = \frac{\sin x + \cos x}{\cos x}$$ 4. **Rewrite the entire expression:** $$\frac{\sin x}{\tan x + 1} = \frac{\sin x}{\frac{\sin x + \cos x}{\cos x}} = \sin x \cdot \frac{\cos x}{\sin x + \cos x} = \frac{\sin x \cos x}{\sin x + \cos x}$$ 5. **Analyze the inequality:** $$\frac{\sin x \cos x}{\sin x + \cos x} \geq 0$$ The fraction is nonnegative if numerator and denominator have the same sign or numerator is zero. 6. **Find critical points where numerator or denominator is zero:** - Numerator zero when $\sin x = 0$ or $\cos x = 0$. - Denominator zero when $\sin x + \cos x = 0$. 7. **Solve numerator zeros:** - $\sin x = 0$ at $x = k\pi$, $k \in \mathbb{Z}$. - $\cos x = 0$ at $x = \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$. 8. **Solve denominator zero:** $$\sin x + \cos x = 0 \implies \sin x = -\cos x \implies \tan x = -1$$ So, $$x = -\frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}$$ 9. **Determine sign intervals:** The critical points split the real line into intervals. Test each interval for sign of numerator and denominator. 10. **Summary of sign analysis:** - Numerator $\sin x \cos x$ is positive when $\sin x$ and $\cos x$ have the same sign. - Denominator $\sin x + \cos x$ is positive or negative depending on $x$. 11. **Final solution:** The inequality holds where $$\frac{\sin x \cos x}{\sin x + \cos x} \geq 0$$ which means numerator and denominator have the same sign or numerator is zero. 12. **Include points where numerator is zero (fraction equals zero):** $$x = k\pi \quad \text{or} \quad x = \frac{\pi}{2} + k\pi$$ 13. **Exclude points where denominator is zero (undefined):** $$x \neq -\frac{\pi}{4} + k\pi$$ 14. **Express solution set:** $$\{x \mid \frac{\sin x \cos x}{\sin x + \cos x} \geq 0, x \neq -\frac{\pi}{4} + k\pi, k \in \mathbb{Z}\}$$ This can be expressed as intervals between critical points where the fraction is nonnegative. --- **Final answer:** $$\boxed{\frac{\sin x}{\tan x + 1} \geq 0 \iff \frac{\sin x \cos x}{\sin x + \cos x} \geq 0, \quad x \neq -\frac{\pi}{4} + k\pi, k \in \mathbb{Z}}$$