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 \((2\lt 3)\).