The square of a negative number must be positive. We can analyze each situation using this fact.

Multiplying or dividing the square of a negative number by a negative number results in multiplying or dividing a positive number by a negative number; in both situations, the result is negative.

Subtracting  the square of a negative number from a negative number results in subtracting a positive number from a negative number; the result is a negative number.

As seen in this example, however, adding the square of a negative number to a negative number might yield a positive result, so it is the correct choice:

We analyze all of the situations.

Multiplying three integers, two of which are positive and one of which is negative:

The product of the two positive numbers must be positive; the product of this and a negative number must be negative.

Subtracting the sum of two positive integers from a negative integer:

The sum of two positive integers must be positive. Therefore, we are subtracting a positive number from a negative number, which must be negative.

Dividing a negative integer by the square of a negative integer:

The square of a negative integer is positive. This is therefore the quotient of two numbers of unlike sign, which must be negative.

Adding three integers, two of which are negative and one of which is positive:

As can be seen from this example, the result can be nonnegative:

This is the correct choice.

The absolute value of a nonnegative number is the number itself; the absolute value of a negative number can be obtained by removing the negative symbol. Therefore, 

 and .

This makes the statements  and  both equivalent to the statement , which is false.

This also makes the statement  equivalent to , which is the only true statement of the three.

All four numbers are negative, so the greatest of the four will be the one with the least absolute value - that is, the one which is the least when the negative symbol is removed. Therefore, we can solve this by finding the greatest number of the set 

.

This can be done by going from left to right and examining the digits. We see that 

,

so, since the least positive number has the greatest opposite, the smallest of the original numbers is .

The initial elevation of the plane is 282 feet below sea level, which is  feet in elevation. Ascent is an increase in elevation and descent is a decrease, so, to the elevation, in turn, add 12,879 feet, subtract 1,276 feet, and add 4,874 feet. 

This expression is

.

Since absolute value bars are grouping symbols, we first raise  to the power of 3. An odd power of a negative number is negative, so we do this by raising 5 to the third power, then affixing a negative symbol.

The absolute value of a negative number is the corresponding positive, so

.

We are looking for the difference between the current elevation of the plane and the initial elevation, so we subtract initial elevation  from current elevation . This is

.

 is the correct choice.

The absolute value of a negative number can be determined by removing the negative symbol, so . Of the three numbers of the set , only 12 is greater than or equal to 12 (it is equal), so only the first statement is correct.

The initial elevation of the plane is 282 feet below sea level, which is  feet in elevation. Ascent is an increase in elevation and descent is a decrease, so, to the elevation, in turn, add 12,879 feet, add 4,276 feet, and subtract 1,874 feet. 

This expression is

.