When you're working with absolute values and inequalities, recognize that there are two cases that will satisfy the inequality: either the terms within the absolute value directly satisfy the inequality, or they satisfy the "opposite" (flip the sign and multiply by -1). For example, if , then either  or .

For this problem, here is how the two cases will map out:

1) Satisfy the inequality directly:

 means that 

2) Flip the inequality sign and multiply the numeric value by -1:

, so 

This can feel abstract, so it's helpful to test one number for each case to see if it really works. For case 1, try a value greater than 6, so you might try 7.   because 8 > 7, so you have this right.  For case 2, try a number less than -8, such as -9.   because the absolute value of -8 is 8, and 8 is greater than 7.  

When you're working with inequalities and absolute values, make sure that you cover both cases that would satisfy the inequality.

1) The terms within the absolute value directly satisfy the inequality.  Here that's:

, so  and .

2) The "flip the sign and multiply by -1" case. Here that's:

 (note that the inequality sign is flipped, and the value to the right of that sign is multiplied by -1)

, so 

When you're working with absolute values and inequalities, you must always consider both cases that satisfy the inequality. Either the terms within the absolute value satisfy the inequality directly, or they satisfy the "opposite case" in which you reverse the inequality sign and multiply the side opposite the absolute value by -1. (The proof for this: if , then either  or  - in that second case, you flip the sign and multiply 10 by -1)

So let's examine both cases:

1) The direct case

 allows you to add  to both sides and subtract  from both sides to get:

So . At this point you may want to do a quick check to ensure that a number slightly less than -2 does satisfy the inequality provided. If you choose -3, you'll see that it does work: 5 -2(-3) gives you 11, and that does work with the given inequality.

2) The opposite case

Again, you'll add  to both sides to move that term to a positive value on the right of the inequality sign (always try to avoid multiplying/dividing by negatives in inequalities as it is easy to forget to flip the sign), and add  to both sides to isolate the variable. This yields:

So . Again, you may wish to quickly plug in a number that just satisfies this inequality to make sure it works with the original. If you choose 8, then you'd have the absolute value of 5-16, which gives you 11, proving that this inequality does work.

Your solution set is then  or 

Whenever you're working with inequalities and absolute values, you should consider two cases: 1) the expression within the absolute value directly satisfies the inequality, and 2) the expression within the absolute value satisfies the opposite of the inequality (flip the sign and multiply the other expression by -1). Let's consider both cases here:

1) The expression directly satisfies the inequality:

Subtract  from and add  to both sides so that you isolate the variable on the left. That yields:

You may also want to quickly try a number that just barely satisfies this inequality to ensure that it works with the given information. If you try , you'll see that you have:

 which is  which is a "close win" for the inequality, helping to confirm your work here.

2) The expression satisfies the "opposite case" in which you flip the sign and multiply the right-hand expression by -1:

Here, add  and  to both sides to yield:

, which reduces to . Here you may again wish to do a quick test; if you choose something just less than , like perhaps , you can confirm. This way you'd have:

. This means that  which again is a "close win," supporting that your work is correct. The correct answer is then:

Whenever you're dealing with absolute values and inequalities, you should consider both of two cases that form the full solution set. In case 1, the expression within the absolute value directly satisfies the inequality. In case two, the expression within the absolute value satisfies the opposite, where you reverse the inequality sign and negate the expression opposite the absolute value (that explanation sounds complicated, but the example will help). Let's examine both cases for this question.

Case 1 - The expression satisfies the inequality directly.

Here that would mean that . You can then subtract  from both sides and add  to both sides to isolate the variable:

It's always a good idea to check a number that just barely satisfies your new inequality to make sure that your answer works. Here if you try 14, you'll see that the given inequality becomes . That works out to , a 'close win' that tells you that your inequality works.

Case 2 - The expression satisfies the opposite of the inequality.

Here you need to do two things: reverse the inequality sign, and multiply the entire right side of the inequality by -1. That would give you:

Adding  and  to both sides isolates the variable:

And then finish the division:

Again, try a number that just barely satisfies the inequality to check that you got it right. If you use 0, then you'll see that the given inequality becomes 8 > 5, which is another close win.

Therefore, the correct answer is  or .

Whenever you're dealing with absolute values and inequalities, you need to consider two possibilities. In case one, the expression within the absolute value directly satisfies the inequality. In case two, the expression within the absolute value will satisfy the opposite, taken by flipping the inequality sign and negating the entire other side of the inequality (think "a is greater than 1 or less than negative-one" --> case two is that "less than negative..." case). Let's examine both here.

Case 1

Here you can subtract  from both sides and add  to both sides to isolate the variable:

Then divide both sides by  and you have:

It's a good idea to try a number that just barely satisfies your new inequality to ensure that it fits with the given inequality. If you try  here, you'll see that you have a fit:  simplifies to , proving that you have a fit.

Case 2

 (notice that the sign is flipped, and all terms on the right are negated)

Here you can add  to both sides and subtract  from both sides to get:

So 

Notice here that this does not add new information. Any number less than  is already less than  and you've already tried a number between them (1) to ensure that it works. So the full solution set here is just any number less than , making the correct answer .

If the analyst receives her bonus when  , and she received her bonus when the actual number of votes, "a" was 127, we can substitute 127 into the inequality as follows

So, to get rid of the inequality signs, we need a situation where

or where

the first inequality can be simplified to

and the second, to  

So, in order for the analyst to receive her bonus, p must be between 115 and 139, or

In this example, we can eliminate the absolute value signs by creating two paths, one in which we hold the sign, and one in which we flip the sign and multiply one side by -1. In this case, these two options are as follows

and 

So, simplified, we can see that 

and 

so, 

In this case, we'll want to note that it doesn't matter if the home is overvalued by over 10,000 away from 90% of the median home value, or undervalued to this extent. So, we need to utilize absolute values to address that the "distance" between p and 90% of m should be greater than 10,000, as shown in the correct answer.