One of the key challenges in the field of high-temperature superconductivity is understanding the nature of fermionic quasiparticles. Experiments consistently demonstrate the existence of a second energy scale, distinct from the d-wave superconducting gap, that persists above the transition temperature into the 'pseudogap' phase. One common class of models relates this energy scale to the quasiparticle gap due to a competing order, such as the incommensurate 'checkerboard' order observed in scanning tunneling microscopy (STM) and resonant elastic x-ray scattering (REXS). We develop a minimal phenomenological model that allows us to quantitatively describe STM and REXS experiments and discuss their relation with photoemission spectroscopy. Experimental signatures of the incommensurate order are explained in terms of scattering of short-lived quasiparticles from local impurities. We identify the unknown second energy scale with the inverse lifetime of the quasiparticles, refocusing questions about the nature of the pseudogap phase to the study of the origin of inelastic scattering.