Sketch the graph of $y = 3^x$. In order to solve the equation $\log_9(5-x) = \frac{x}{2}$, a suitable straight line has to be drawn on the same set of axes as the graph of $y = 3^x$. Find the equation of the straight line. State the number of solutions for the equation $\log_9(5-x) = \frac{x}{2}$.

1. Identify the key features of the graph $y = 3^x$

The base is $3 > 1$, so it is an increasing exponential function.
When $x = 0$, $y = 3^0 = 1$. The $y$-intercept is $(0, 1)$.
As $x \to -\infty$, $y \to 0$. The $x$-axis ($y = 0$) is a horizontal asymptote.

1. Convert the logarithmic equation to exponential form

$\log_9(5-x) = \frac{x}{2}$
$5 - x = 9^{\frac{x}{2}}$

2. Express the exponential term in base 3 to match the given graph

$5 - x = (3^2)^{\frac{x}{2}}$
$5 - x = 3^x$

3. Identify the equation of the straight line

Since the graph is $y = 3^x$, the straight line to draw is $y = 5 - x$.

1. Determine the number of intersections between the curve and the line

The number of solutions to the equation is the number of points of intersection between the curve $y = 3^x$ and the line $y = 5 - x$.
Since $y = 3^x$ is a strictly increasing curve and $y = 5 - x$ is a strictly decreasing line, they intersect at exactly one point.