We investigate the origin of the X-ray emission in low-luminosity active galactic nuclei (LLAGNs). Yuan and Cui predicted that the X-ray emission should originate from jets rather than an advection-dominated accretion flow (ADAF) when the X-ray luminosity L-X of the source is below a critical value of L-X,L-crit approximate to 10(-6)L(Edd). This prediction implies that the X-ray spectrum in such sources should be fitted by jets rather than ADAFs. Furthermore, below L-X,L-crit the correlation between radio (L-R) and X-ray (L-X) luminosities and the black hole mass (M)-the so-called fundamental plane of black hole activity-should deviate from the general correlation obtained by Merloni et al. and become steeper. The Merloni et al. correlation is described by logL(R) = 0.6logL(X) + 0.78logM + 7.33, while the predicted correlation is logL(R) = 1.23logL(X) + 0.25logM-13.45. We collect data from the literature to check the validity of these two expectations. We find that among the 16 LLAGNs with good X-ray and radio spectra, 13 are consistent with the Yuan and Cui prediction. For the 22 LLAGNs with L-X < L-X,L-crit, the fundamental plane correlation is described by logL(R) 1.22logL(X)+ 0.23logM-12.46, also in excellent agreement with the prediction.