Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$
$\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}$
If the two equal values are $3$ and $x+2$, then $x=1$. Also, $y-4\le 3$ because $3$ is the common value. Solving for $y$, we get $y \le 7$. Therefore the portion of the line $x=1$ where $y \le 7$ is part of $S$. This is a ray with an endpoint of $(1, 7)$.
Similar to the process above, we assume that the two equal values are $3$ and $y-4$. Solving the equation $3=y-4$ then $y=7$. Also, $x+2\le 3$ because 3 is the common value. Solving for $x$, we get $x\le1$. Therefore the portion of the line $y=7$ where $x\le 1$ is also part of $S$. This is another ray with the same endpoint as the above ray: $(1, 7)$.
If $x+2$ and $y-4$ are the two equal values, then $x+2=y-4$. Solving the equation for $y$, we get $y=x+6$. Also $3\le y-4$ because $y-4$ is one way to express the common value. Solving for $y$, we get $y\ge 7$. We also know $3\le x+2$, so $x\ge 1$.Therefore the portion of the line $y=x+6$ where $y\ge 7$ is part of $S$ like the other two rays. The lowest possible value that can be achieved is also $(1, 7)$.
Since $S$ is made up of three rays with common endpoint $(1, 7)$, the answer is $\boxed{\textbf{(E) }\text{three rays with a common endpoint}}$