I4.2 Let $[x]$ be the largest integer not greater than $x$. For example, $[2.5]=2$.
If $b$ satisfies the system of equations $\left\{\begin{aligned} A x^{2}-4 & =0 \\ 3+2(x+[x]) & =0\end{aligned}\right.$, find the value of $b$.
$\left\{\begin{aligned} 16 x^{2}-4 & =0 \\ 3+2(x+[x]) & =0\end{aligned}\right.$ from the first equation $x=\frac{1}{2}$ or $-\frac{1}{2}$.
Substitute $x=\frac{1}{2}$ into the second equation: LHS $=3+2\left(\frac{1}{2}+0\right)=4 \neq$ RHS
Substitute $x=-\frac{1}{2}$ into the second equation: LHS $=3+2\left(-\frac{1}{2}-1\right)=0=$ RHS
$$
\therefore b=-\frac{1}{2}
$$