Is the so-called "$ax+b$ group" (the group of affine maps on $\mathbb R$) amenable?
One reprersentation for $ax+b$ group is the set of 2 by 2 matrices for form $$\begin{bmatrix}x & y \\0 & 1\end{bmatrix};\,\,\,x,y\in\mathbb R, x>0$$ endowed with the usual multiplication. The left Haar measure is given by $\frac{dxdy}{x^2}$
Every solvable group is amenable - for a proof see Theorem $4.6.3$ here. Note that the affine group $Aff(\mathbb{R}^n)$ is solvable if and only if $n=1$ (which is the group above). For a discrete subgroup $\Gamma$ in $Aff(\mathbb{R}^n)$ are equivalent: $(1)$ $\Gamma$ is amenable, $(2)$ $\Gamma$ is virtually solvable.
It is solvable, thus it is amenable.