$2x + 3\tan x = \frac{7\pi}{6}$ and $\tan x = -\frac{1}{\sqrt{3}}$
Step-by-Step Solution
Key Concept: Graphical analysis of trigonometric functions helps identify the number of solutions by counting intersection points
Both the curves $y = \tan x$ and $y = 2\cos x$ are plotted on the same graph in the interval $[0, 2\pi]$. Examining the intersection points carefully, the two graphs meet at exactly three distinct points in this interval. These intersection points correspond to the solutions of the combined equation.
Correct Answer: 3