Multiplication and Division of Signed Mumbers: Identify the greater value

Note that we are dividing a negative number by a positive number:

$-:+=-$

Therefore, the exercise is:

$-?0$

Since we have a negative number, it must be less than zero.

Therefore, the answer is:

$- < 0$

Note that we are dividing a negative number by a negative number, therefore:

$-:-=+$

This means the final exercise looks like this:

$+?0$

Since we got a positive number, it must be greater than zero.

The answer is:

$+ > 0$

Let's first turn our attention to the exercise on the left hand side :

$\frac{-12\frac{1}{8}}{0}=$

Remembering the below formula:

$\frac{x}{0}$

Since no number can be divided by 0 we are able to ascertain that the expression has no meaning.

Note that in the first step we are dividing a positive number by a negative number:

$+:-=-$

Therefore, we the exercise is:

$-:-?0$

Now we are dividing a negative number by a negative number, that is:

$-:-=+$

Therefore, the final exercise will look like this:

$+?0$

Since we have a positive number, it is greater than zero.

Therefore, the answer is:

$+ > 0$

First let's solve the exercise on the left-hand side:

$\frac{0}{-412.5}=$

Here we must remember the formula:

$\frac{0}{x}=0$

In other words, when we divide 0 by any number, the result will always be 0.

Now we have:

$0\text{?}0$

Therefore, the answer is:

$0=0$

Let's solve the exercise from left to right:

$\frac{0}{15}=$

Remember the formula:

$\frac{0}{x}=0$

If we divide zero by any number, the result will always be zero.

Now we are left with the following exercise:

$0:-16?0$

Let's solve the exercise:

$\frac{0}{-16}=$

If we remember the formula above, we should see that the result is zero.

The final exercise will look like this:

$0\text{?}0$

Therefore, the missing sign is:

$0=0$

Note that in the first stage we are dividing a positive number by a negative number:

$+:-=-$

Now the exercise is:

$-:a?0$

Since we don't know whether a is a positive or negative number, we cannot determine the sign.

Note that in the first step we are dividing a negative number by a negative number:

$-:-=+$

This means that the exercise can be written as follows:

$+:-?0$

Now we are dividing a positive number by a negative number:

$+:-=-$

Therefore, the final exercise will look like this:

$-?0$

We are left with a negative number, meaning a number less than zero.

Therefore, the answer is:

$- < 0$

To determine which value is the smallest when $0 > x$, we evaluate each of the given expressions under the assumption that $x$ is negative.

• $-300x$: When a negative number $x$ is multiplied by $-300$, it becomes a large positive number because multiplying two negatives yields a positive. Thus, $-300x$ is a large positive number.
• $-7x$: Similarly, if we multiply $x$ by $-7$, this also results in a positive number, but it will be much smaller than $-300x$ in magnitude because the multiplication factor is much smaller.
• $x + 8$: Adding 8 to a negative number $x$ raises its value towards zero, making it less negative or possibly positive, depending on $x$'s magnitude.
• $x$: As $x$ is a negative number, it remains the smallest among all these expressions. No operation is applied to change its sign or significantly increase its value.

Therefore, among the choices provided, the smallest value is simply $x$ itself because it remains negative and no operation converts it into a positive or larger magnitude value.

The answer is $x$.

To solve this problem, we'll evaluate the expressions based on the condition $0 < x$:

• If $300ax$ is considered, the sign and magnitude depend heavily on $a$. If $a$ is positive, $300ax$ is large positive. If $a$ is negative, it's large negative.
• $\frac{x}{300a}$ is affected by both $x$ and $a$. If $a$ is positive, this expression shrinks, if $a$ is negative, the sign is inverted, and it becomes positive, but its magnitude determination needs $a$.
• $-2x$ results in a definite negative value because $x$ is positive.

Without knowing the sign or value of $a$, we can't definitively compare or compute the sizes of the expressions relative to each other.

Therefore, the solution to the problem is: It is not possible to calculate.