$7\mathrm{.}4.$ Consider the indirect utility function given by

$v(p_1,p_2,m)=\frac{m}{p_1+p_2}$

$($a$)$ What are the demand functions?

$($b$)$ What is the expenditure function?

$($c$)$ What is the direct utility function?

$7\mathrm{.}5.$ A consumer has a direct utility function of the form

$U(x_1,x_2)=u(x_1)+x_2.$

Good $1$ is a discrete good; the only possible levels of consumption of good $1$ are $x_1=0$ and $x_1=1.$ For convenience, assume that $u(0)=0$ and $p_2=1.$

$($a$)$ What kind of preferences does this consumer have?

$($b$)$ The consumer will definitely choose $x_1=1$ if $p_1$ is strictly less than what?

$($c$)$ What is the algebraic form of the indirect utility function associated with this direct utility function?