Analytic analysis and parametric investigation are employed to study and compare metamaterial properties of two types of composite metasolids. Metasolids are composed of either an elastic inclusion or a rigid core coated by an elastic material, embedded in a stiff matrix. For these types of materials, results related to cylindrical as well as spherical inclusions are presented. Such mono-inclusion two-component and bi-inclusion three-component metasolids have been previously known to exhibit negative mass density near local-resonance frequencies. Through a unified formulation, it is analytically shown how and why adding a rigid mass inside the elastic inclusion to make a bi-inclusion three-component material can dramatically change the homogenized property of the resultant inclusion and increase the tunability of the composite, particularly in terms of local-resonance frequencies and the associated metamaterial-effect frequency bandwidth. In this way, concerning distinctly sound and vibration insulation, a low-frequency metamaterial effect with larger bandwidth can be designed via an inverse problem using a simplified mass-spring model.