Q.Solve the following for $x$, where $|x|$ is the modulus function, $[x]$ is the greatest integer function, $\{x\}$ is the fractional part function: $|x^2-9|+|x^2-4|=5$.

Let $u=x^2\ge0$; the equation is $|u-9|+|u-4|=5$, with critical points $u=4,9$.

Case $u<4$: $(9-u)+(4-u)=13-2u=5 \implies u=4$ — but this contradicts $u<4$, so no solution strictly here (it's the boundary, covered next).

Case $4\le u\le9$: $(9-u)+(u-4)=5=5$ — an identity, true for every $u$ in this whole range. …

Substitute $u=x^2$ to reduce to a single-variable modulus equation, split into three cases by the critical points $4,9$, and note the middle case collapses to an always-true identity …

Reporting only a couple of specific $x$-values instead of recognising the middle case is an identity that make …