An equation is shown, where x>0, z<0, and |x| > |z|.

x.y = z

𝑦 < 0
𝑦 > 0
|𝑦| < 1
|𝑦| = 1
|𝑦| > 1

x is positive and z is negative

x has a larger absolute value

xy = z solve for y, y = z/x

y = a negative fraction because x is larger than z.

let's say the answer is -1/2 that would be a possibility given the facts.

y = -1/2
|y| = |-1/2| so |y| = 1/2 since that number is less than 1.
Choose |y|<1

Well, well, well, look at this equation playing with some symbols. Let's see what we can make of it, shall we?

Given that x > 0 and z < 0, and the absolute value of x is greater than the absolute value of z, we need to find the possible values of y.

Now, we know that y < 0, which rules out y > 0 right away. So, adios to that option!

Moving on, we also know that |y| < 1. Ah, |y|, the absolute value of y. Such a mysterious creature, always keeping things in check. In this case, it tells us that y cannot be bigger than 1 or equal to 1. So, say goodbye to |y| = 1!

Finally, |y| > 1, the big leagues! If the absolute value of y is greater than 1, then we have a winner.

So, out of our options, the correct answer is |𝑦| > 1. Looks like we have a winner, folks!

To solve the equation x*y = z, let's consider the given constraints:

1. x > 0: This means that x is a positive number.
2. z < 0: This means that z is a negative number.
3. |x| > |z|: This means that the absolute value of x is greater than the absolute value of z.

Now, let's determine the possible values for y based on the given constraints.

1. 𝑦 < 0:
Since z is negative and y is less than zero, their product (y*z) will be positive. However, the equation states that x*y = z, so x*y will be greater than zero, which contradicts the given constraints. Therefore, 𝑦 cannot be less than zero.

2. 𝑦 > 0:
Since z is negative and y is greater than zero, their product (y*z) will be negative. This matches the equation x*y = z since x is greater than zero. Therefore, 𝑦 can be greater than zero.

3. |𝑦| < 1:
This constraint does not provide any direct information about the possible values of 𝑦. It only states that the absolute value of 𝑦 is less than 1.

4. |𝑦| = 1:
If the absolute value of 𝑦 is equal to 1, then 𝑦 can take the values -1 or 1. Since x is positive, let's substitute both values into the equation to see if a solution exists.

For 𝑦 = -1:
x * (-1) = z
Since x is positive and z is negative, a solution exists.

For 𝑦 = 1:
x * 1 = z
Since x is positive and z is negative, a solution exists.

Therefore, 𝑦 can be either -1 or 1.

5. |𝑦| > 1:
This constraint does not provide any direct information about the possible values of 𝑦, other than the fact that the absolute value of 𝑦 is greater than 1.

In summary, based on the given constraints:

- 𝑦 > 0 satisfies the equation x*y = z.
- |𝑦| = 1 satisfies the equation x*y = z, with 𝑦 being -1 or 1.

Therefore, 𝑦 can be either greater than zero or have an absolute value of 1.

To analyze the equation x * y = z in the given conditions, we need to determine the range of valid values for y.

Given that x > 0 and z < 0, we know that the product of x and y, which is z, will also be less than zero.

To solve for y, we can rearrange the equation as y = z / x. Since we know that z < 0 and x > 0, dividing a negative number by a positive number will yield a negative result.

Therefore, we can conclude that y < 0. This means that the correct answer is 𝑦 < 0.