The provision of the bottom-end of the engineering range is the only additional information that provides us a fixed endpoint from which we can build off of by supplementing with the ranges provided in the question, to give us the full range between both engineering and achitecture graduates. See the diagram provided to understand how this can be done.

Even if the mean and medians were provided, these additional values give us no information on the endpoints of the salaries, and the question only asks for the range. 

Let's draw a number line. 

Since a number line is straight and contains the numbers consecutively, we just subtract \(\displaystyle 1\) from \(\displaystyle 8\) to get \(\displaystyle 7\). 

Open circles mean the values are excluded from the set.

The number line shows the set is between \(\displaystyle 1\) and \(\displaystyle 8\) exclusive.

The only value in that set would be \(\displaystyle 6.5\). 

Because a number line contains both positive and negative integers, we need to consider both possibilities. 

\(\displaystyle 30^4\) is \(\displaystyle 810000\) and that value is the same as \(\displaystyle (-30)^4\). Therefore we eliminate the \(\displaystyle 29, 30\) choice because \(\displaystyle 1000000\) will always be greater than those values raised to the \(\displaystyle 4th\) power.

Next \(\displaystyle 31^4\) is \(\displaystyle 923521\). We elminate both the positive and negative range of \(\displaystyle 30, 31\). If we look at the difference between \(\displaystyle 31^4\) and \(\displaystyle 30^4\), it's over \(\displaystyle 100000\).

Then, we should guess that \(\displaystyle 32^4\) will definitely be greater than \(\displaystyle 1000000\) so therefore answer is \(\displaystyle -31, -32\).

Remember, a negative value raised to an even power will always have a positive value. 

Since perimeter of equilateral triangle is \(\displaystyle 9\sqrt{6}\) and we have three equal sides, we just divide that vaue by \(\displaystyle 3\) to get \(\displaystyle 3\sqrt{6}\). To find height, we can set-up a proportion. 

The height is opposite the angle \(\displaystyle 60\). Side opposite \(\displaystyle 60\) is \(\displaystyle \sqrt{3}\) and the side of equilateral triangle which is opposite \(\displaystyle 90\) is \(\displaystyle 2\).

\(\displaystyle \frac{3\sqrt{6}}{2}=\frac{h}{\sqrt{3}}\) Cross multiply.

\(\displaystyle 3\sqrt{18}=2h\) Divide both sides by \(\displaystyle 2\)

\(\displaystyle \frac{3\sqrt{18}}{2}=h\) 

Let's simplify by factoring out \(\displaystyle \sqrt{9}\) to get a final answer of \(\displaystyle \frac{9\sqrt{2}}{2}\). 

The distance between two points is always positive. We calculate lx₂ - x₁l, which will give us the distance between the points.

|14- (-4)| = |14+4| = |18| = 18

To begin, you must simplify so that you "isolate" \(\displaystyle x\), (i.e. at least eliminate any coefficients from it). To do this, divide all of the members of the inequality by \(\displaystyle 2\):

\(\displaystyle 13 \le x < 32\)

Now, this inequality represents all of the numbers between 13 and 32.  However, it does include \(\displaystyle 13\) (hence, getting a closed circle for that value) and does not include \(\displaystyle 32\) (hence, getting an open circle for that value). Therefore, it looks like:

Since the inequality represents one range of values between two end points (both of which are included, given the sign being "less than or equal"), you know that whatever you answer, it must be convertible to the form:

\(\displaystyle -2 \leq x \leq 5\)

Now, you know that it is impossible to get this out of the choices that have no absolute values involved in them. Therefore, the only options that make sense are the two having absolute values; however, here you should choose only the ones that have a \(\displaystyle \leq\), for only that will yield a range like this. Thus, we can try both of our options.

The wrong answer is simplified in this manner:

\(\displaystyle |3x - 12| \leq 6\)

\(\displaystyle -6\leq 3x - 12 \leq 6\)

\(\displaystyle 6\leq 3x \leq 18\)

And you can stop right here, for you know you will never have \(\displaystyle -2\) for the left terminus.

The other option is simplified in this manner:

\(\displaystyle |4x - 6| \leq 14\)

\(\displaystyle -14 \leq 4x - 6 \leq 14\)

\(\displaystyle -8 \leq 4x \leq 20\)

\(\displaystyle -2 \leq x \leq 5\)

This is just what you need!

It is not necessary to solve this problem by multiplying terms out. Notice that between quantities A and B, the last three terms switch places for the two large numbers. as such they can be rewritten:

Quantity A:

\(\displaystyle (310789)*(210237)\)

\(\displaystyle (310000+789)*(210000+237)\)

\(\displaystyle (310000)(210000)+(789)(210000)+(237)(310000)+(789)(237)\)

Quantity B:

\(\displaystyle (310237)*(210789)\)

\(\displaystyle (310000+237)*(210000+789)\)

\(\displaystyle (310000)(210000)+(237)(210000)+(789)(310000)+(789)(237)\)

Both quantities A and B share the exact same terms, save for two:

Quantity A: \(\displaystyle (789)(210000)+(237)(310000)\)

Quantity B: \(\displaystyle (237)(210000)+(789)(310000)\)

From visual inspection, it is clear that B is larger.

Since \(\displaystyle x\) is always positive, and \(\displaystyle |x|>|y|\), it follows that \(\displaystyle y>-x\) for all possible values.

For the A, it is possible to choose values that make the statement false, for example \(\displaystyle y=-0.9\) and \(\displaystyle x=1.1\).

C is always false.