If $7+3x\ge 5-\frac{x}{2}$ and $2x+3\le 5-2x$; then $x$ can take which of the following values?

Let's solve the first inequality:
$$7+3x\ge 5-\frac{x}{2}$$
Multiply the entire inequality by 2 to eliminate the fraction:
$$14+6x\ge 10-x$$
$$6x+x\ge 10-14$$
$$7x\ge -4$$
$$x\ge -\frac{4}{7}\approx -0.571$$
Now let's solve the second inequality:
$$2x+3\le 5-2x$$
$$2x+2x\le 5-3$$
$$4x\le 2$$
$$x\le \frac{2}{4}$$
$$x\le \frac{1}{2}=0.5$$
Combining both inequalities, we get the valid range for $x$:
$$-\frac{4}{7}\le x\le 0.5$$
Among the given options (0, 1, 2, -1), only $0$ lies within this range.