Final answer:
The solution to the given system of equations involves finding the values of x and y that satisfy both expressions. By completing the square in the first equation and solving a simple quadratic equation in the second equation, we can determine the range of x and the possible values of y. The solution is: -√2 + 1 ≤ x ≤ √2 + 1 and y = 1 or y = -1.
Step-by-step explanation:
The solution to the inequality x² + 4y² ≤ 16 and y² + 1 consists of the values of x and y that satisfy both expressions.
We can start by solving the first equation, x² + 4y² ≤ 16, by completing the square.
Completing the square in x² gives us 2(x² - 1)² ≤ 4, which simplifies to (x² - 1)² ≤ 2.
To solve this inequality, we take the square root of both sides, considering both the positive and negative roots: x² - 1 ≤ √2 and -(x² - 1) ≤ √2.
Simplifying these inequalities gives us -√2 + 1 ≤ x ≤ √2 + 1.
Now, let's solve the second equation, y² + 1, which is a simple quadratic equation with a vertex at (0,1).
The solutions to this equation are y = 1 and y = -1.
Combining both sets of solutions, we have the solution to the given system of equations as:
-√2 + 1 ≤ x ≤ √2 + 1 and y = 1 or y = -1.