If X is between -3 and -1, and Y is between -1 and 1, then X ² - Y² is in between which of the following?

Question

Accepted Answer

Here's a brief explanation:

1.  Determine the range of X squared:
    Given -3 < X < -1.
    Squaring a negative number reverses the inequality signs. So, (-1)^2 < X^2 < (-3)^2, which means 1 < X^2 < 9.

2.  Determine the range of Y squared:
    Given -1 < Y < 1.
    When squaring a number between a negative and a positive value, the minimum squared value is 0 (when Y=0). The maximum squared value occurs at the endpoints. So, 0 <= Y^2 < 1.

3.  Determine the range of X^2 - Y^2:
    To find the minimum value of X^2 - Y^2, we take the minimum of X^2 and subtract the maximum of Y^2.
    Minimum (X^2 - Y^2) = (minimum X^2) - (maximum Y^2) = 1 - (value approaching 1) = a value approaching 0. Since Y^2 is strictly less than 1, X^2 - Y^2 will be strictly greater than 0. So, X^2 - Y^2 > 0.

To find the maximum value of X^2 - Y^2, we take the maximum of X^2 and subtract the minimum of Y^2.
    Maximum (X^2 - Y^2) = (maximum X^2) - (minimum Y^2) = 9 - 0 = 9. Since X^2 is strictly less than 9, X^2 - Y^2 will be strictly less than 9. So, X^2 - Y^2 < 9.

Combining these, we get 0 < X^2 - Y^2 < 9.

4.  Analyze the options:
    A) -9 and 1: Incorrect, the minimum is 0.
    B) -9 and -1: Incorrect, the minimum is 0.
    C) 0 and 8: Incorrect, the maximum is 9.
    D) 0 and 9: Correct, as 0 < X^2 - Y^2 < 9.

The final answer is D.

Answer

-9 and 1

Answer

-9 and -1

Answer

0 and 8

Answer

0 and 9

If X is between -3 and -1, and Y is between -1 and 1, then X ² - Y² is in between which of the following?

Explanation

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