Two interpretations of the Lagrange Multiplier
If you would NOT obey the constraint and spend 1\$ more (or more precisely an infinitesimal amount more) your utility would go up by $\lambda$. As such $\lambda$ is what you give up for obeying the constraint, it is the price you pay indirectly and hence it is called the shadow price.

Writing the constraint in reverse has no effect other than changing the sign of $\lambda$, but that depends on the way you set-up your problem anyway (I.e. whether you add the constraint with a plus or a minus).