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Now our expression will look like this:. Now we know that this function passes through coordinates 4, 6 , so let us replace our x and y values for these givens. This means that we can work this backwards and convert our first expression into:. We also know that a log is essentially asking: "To what power does the base need to raised in order to achieve this certain value?

Because we are working with squares, negatives are not a factor—we are looking for whichever point has the largest combination of coordinate point, since a negative square will be a positive. Answer choices A, B, C, and D all produce non-integer values when we take their square root, but answer choice E is the exception.

A rational number is any number that can be expressed as the fraction of two integers, and this is the only option that fits the definition.

We know that our numbers are inclusive, so our first number will be , and will include every number from though That gives us 10 numbers so far. From here, we can see that the first 10 numbers of , , , , , , , and will be included as well, giving us a total of:. Now we also must include every number that ends in 0. For the first not including , which we have already counted! This gives us a total of:. Now, let us add our totals all the numbers with a units digit of 0 and all the numbers with a tens digit of 0 together:.

There are a total of numbers between and , inclusive, so our final probability will be:. Now, we are told that the angles the lines form are congruent. This means that the slopes of the lines will be opposites of one another [Note: perpendicular lines have opposite reciprocal slopes, so do NOT get these concepts confused!

After which, we can test out our answer choices to find the right one. Why not! At a glance, we can see that answer choices H and J would be positive and answer choice K is 0. We can therefore eliminate them all. Let us test it fully to be sure. So we will now get:. For expediency's sake, we can stop here. For more information on this, check out our guide covering absolute value equations. In this case, however, pay attention to the fact that our absolute value must supposedly be less than a negative number.

An absolute value will always be positive as it is a measure of distance and there is no such thing as a negative distance. This means it would be literally impossible to have an absolute value equation be less than In this section, we will look at what these questions have in common and give examples for each type.

In the next section, we will give you all 21 of the most difficult questions as well as answer explanations for each question, including the ones we use as examples here. Some of the reasons why the hardest math questions are the hardest math questions are because the questions do the following:.

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