The problem in the image asks for the set of $x$-values that satisfy the equation $[x] = -2$, where $[x]$ represents the floor function (the greatest integer less than or equal to $x$).

Understanding the Floor Function:

The floor function $[x] = -2$ means that $x$ is any real number such that its floor is equal to $-2$. This will happen when $-2 \leq x < -1$.

This is because:

• If $x = -2$, the floor of $x$ is $-2$.
• If $x = -1.5$, the floor of $x$ is still $-2$.
• But if $x = -1$, the floor of $x$ becomes $-1$, not $-2$.

Thus, the interval for $x$ is $-2 \leq x < -1$.

Correct Answer:

The correct option is (B) $-2 \leq x < -1$.

Would you like further details or clarifications on this? Here are 5 questions related to this concept:

1. What is the definition of the floor function?
2. How does the floor function behave for negative values of $x$?
3. What is the difference between the floor function and the ceiling function?
4. Can the floor function be applied to non-integer values?
5. How do you graph the floor function?

Tip: The floor function rounds a number down to the nearest integer, regardless of whether the number is positive or negative.