Question

Let x < 0, 0 < y < 1, z > 1. Which of the following may be false?
(a) (x^2– z^2) has to be positive.
(b) yz can be less than one.
(c) xy can never be zero.
(d) (y^2– z^2) is always negative.

Answer 1

it is (c) xy can never be zero. hope it is useful for you. please mark as brainliest.

Answer 2

Answer:

x² - z² has to be positive

Explanation:

We are given that:

x < 0 ...............> This means that x is a negative number

0 < y < 1 ............> This means that y is a positive proper fraction

z > 1 ............> This means that z is a positive number greater than 1

Now, let's check the options:

First option:

x² - z² has to be positive

This statement is false. If the squaring of x is less than the squaring of z, then the result will be negative.

Example:

Let x be -2 and z be 3

x² - z² = (-2)² - (3)² = 4 - 9 = -5

Therefore, this choice is correct

Second option:

yz can be less than 1

This statement is correct. Since y is a proper fraction (less than 1), therefore, multiplying it by an integer can give a result less than 1

Example:

Assume y = 1/3 and z = 2

yz = (1/3) * 2 = 2/3 ...............> less than one

Therefore, this choice is incorrect

Third choice:

xy can never be zero

This statement is correct. For a product to be zero, either x or y has to be zero. Since neither of them is zero, therefore, the product can never be zero

Therefore, this choice is incorrect

Fourth choice:

y² - z² is always negative

This statement is correct. y is a proper fraction whose value is less than 1, therefore, squaring it will give a smaller value

On the other hand, z is a positive number greater than 1, therefore, squaring it will give a larger value.

This means that, subtracting the square of z from the square of y will always give a negative answer.

Therefore, this choice is incorrect

Hope this helps :)