Method of characteristics with constant PDE

I was just introduced to method of characteristics for solving PDE's. We solved the wave equation that is inifinitely long using this method. However I am very confused about this method. Here is a question from the book I am trying to do. It says to solve the first order equations for $u(x,t)$ using this method. No initial conditions are given. $$ 2 \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = 0$$

There is an example in the book (Haberman) that solves PDE in this form $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$$

no naturally, for my solution I divide everything by $2$ to get $$ \frac{\partial u}{\partial t} + \frac{3}{2} \frac{\partial u}{\partial x} = 0$$ Then I let $u(x,t) = u(x(t),t)$ and so if I take the derivative of this I get $$\frac{du}{dt}(x(t),t) = \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} \frac{dx}{dt}$$ so it must mean that $\frac{dx}{dt} = \frac{3}{2}$ so this is easy enough so we get a solution $$ x = \frac{3}{2}t + x_0$$

okay but I need this in terms uf $u(x,t)$. I am lost at this point on what to do since the book continues using intial conditions.

Similarly I have a second question that is bothering me even more $$ 2 \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = 1$$

You're on the right track, but so far you haven't used the most important fact about characteristics: that the solution is constant along them. It is correct that for your first question, the characteristics are given by $x(s) = 3/2(s-t_0)+ x_0$. But how do we use this?

Suppose we are given the initial condition $u(x, t_0) = f(x)$. To find the value of the solution at a point $(x,t)$, we have to find a characteristic that passes through $(x,t)$. But this is easy: $$ x(s) = \frac{3}{2} (s-t_0) + x_0 $$ When $s=t$ this is supposed to take the value $x$, so we find $$ x_0 = x - \frac{3}{2}(t-t_0) $$ Plugging it back in, we have $$ x(s) = \frac{3}{2}(s-t_0) + x - \frac{3}{2}(t-t_0) $$ Since $u(x,t)$ is constant along characteristics, we have $$u(x,t) = u(x(s), s) = u(x(t_0), t_0) = f(x(t_0))$$ where we used the initial condition $u(x, t_0) = f(x)$. Using the equation for $x(s)$, we find $$ x(t_0) = x - \frac{3}{2}(t-t_0) $$ Hence the solution is $$ u(x,t) = f(x - \frac{3}{2}(t-t_0)).$$

For your second question, consider the function $u(x(s),s) - s$ and you'll find that it reduces to the first question.