Quantifier elimination (QE) is a key task in formal verification algorithms, and the ability to return partial results, such as under-approximations, is beneficial for many QE clients. In Linear Real Arithmetic (LRA), existing QE methods often fail to preserve syntactic convexity, that is, they return a disjunction even for a conjunctive input, or they return a large non-minimal representation. We define the novel concept of Bidirectional Model-Based Projection and a new QE algorithm for LRA (BMBP-QE) that (i) returns a conjunctive over-approximation and a disjunctive under-approximation when interrupted early, (ii) returns a minimal conjunction when the input is conjunctive, and (iii) applies to arbitrary LRA formulae. We show that BMBP-QE outperforms SMT-based QE algorithms, offering improvements in both runtime and result size.