We use the EAGLE (Evolution and Assembly of GaLaxies and their Environments) cosmological simulation to study the distribution of baryons, and far-ultraviolet (O VI), extreme-ultraviolet (Ne VIII) and X-ray (O VII, O VIII, Ne IX, and Fe XVII) line absorbers, around galaxies and haloes of mass $mathrm{M}_{200c}=10^{11}$-$10^{14.5},mathrm{M}_{odot}$ at redshift 0.1. EAGLE predicts that the circumgalactic medium (CGM) contains more metals than the interstellar medium across halo masses. The ions we study here trace the warm-hot, volume-filling phase of the CGM, but are biased towards temperatures corresponding to the collisional ionization peak for each ion, and towards high metallicities. Gas well within the virial radius is mostly collisionally ionized, but around and beyond this radius, and for O VI, photoionization becomes significant. When presenting observables we work with column densities, but quantify their relation with equivalent widths by analysing virtual spectra. Virial-temperature collisional ionization equilibrium ion fractions are good predictors of column density trends with halo mass, but underestimate the diversity of ions in haloes. Halo gas dominates the highest column density absorption for X-ray lines, but lower density gas contributes to strong UV absorption lines from O VI and Ne VIII. Of the O VII (O VIII) absorbers detectable in an Athena X-IFU blind survey, we find that 41 (56) per cent arise from haloes with $mathrm{M}_{200c}=10^{12.0}$-$10^{13.5},mathrm{M}_{odot}$. We predict that the X-IFU will detect O VII (O VIII) in 77 (46) per cent of the sightlines passing $mathrm{M}_{star}=10^{10.5}$-$10^{11.0},mathrm{M}_{odot}$ galaxies within 100 pkpc (59 (82) per cent for $mathrm{M}_{star}>10^{11.0},mathrm{M}_{odot}$). Hence, the X-IFU will probe covering fractions comparable to those detected with the Cosmic Origins Spectrograph for O VI.
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