The given system of inequalities is:

$y \leq x + 7$ $y \geq -2x - 1$

We need to determine which point $(x, y)$ satisfies both inequalities.

Steps:

1. The first inequality $y \leq x + 7$ means that for any given value of $x$, the value of $y$ must be less than or equal to $x + 7$.
2. The second inequality $y \geq -2x - 1$ means that for any given value of $x$, the value of $y$ must be greater than or equal to $-2x - 1$.

To find which point $(x, y)$ satisfies both inequalities, you would test different points from the graph of these inequalities in the xy-plane. For instance:

• Choose a point and check if it satisfies both inequalities.

Do you want to explore specific points for this system, or would you like me to visualize the solution with the inequalities plotted?

Also, let me know if you'd like further explanations!

Follow-up Questions:

1. What is the significance of shading in inequality graphs?
2. How do the lines $y = x + 7$ and $y = -2x - 1$ divide the xy-plane?
3. Could a solution point lie exactly on the boundary lines?
4. What happens when the inequalities are strict (e.g., $<$ or $>$) instead of $\leq$ and $\geq$?
5. How can we test a point to determine if it satisfies both inequalities?

Tip:

When graphing inequalities, always remember that a solid line indicates $\leq$ or $\geq$, while a dashed line indicates $<$ or $>$.