In one of the following graphs in the standard $(x,y)$ coordinate plane, the solution set to the system of inequalities below is shown shaded. Which one?

$$\begin{cases} x+2y\leq 6 \\ 3x^2 \gt 12-3y^2 \end{cases}$$

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We can rewrite the line in slope intercept form:

$$x+2y\leq 6$$ $$2y \leq -x + 6$$ $$y \leq -\frac{1}{2}x+3$$

We are looking for a line with negative slope. It needs to be a bold line since we are looking for less than or equal. We shade below this line since $y\leq ...$

There are two choices that fit these descriptions:

The circle can be rearranged to standard form:

$$3x^2\gt 12-3y^2$$ $$3x^2+3y^2=12$$ $$x^2+y^2=4$$

The circle has a radius of 2, which is less than the $y$-intercept of the line $(2\lt 3)$.