Today we will be taking a look at absolute value inequality problems.
Absolute value inequality problems are problems such as these:
The absolute value of three x minus two is less than four.
For problems like these, we might be asked to find and graph the domain of our variable. In this case, our variable is x.
So what exactly does this mean? Find and graph the domain of x.
Well, the domain of x is any value that x can be while still maintaining the integrity of this inequality.
So how can we find the domain of x?
We can use the phrase: “More is or”
And “less is nest” to help us always solve these problems correctly.
So what exactly does this phrase mean?
This phrase is talking about the location of the absolute value symbol with relation to the inequality.
Is it on the more than side, or the less than side?
In this case, our absolute value symbol is on the less than side of the inequality, so we will proceed to solve it using the “less is nest” portion of the phrase.
For less is nest, we will rewrite the absolute value inequality without the absolute value symbol.
From here, we will begin to build our nest.
To do so, write this same inequality symbol on the left side of the, move the 4 to the left side and add a negative before it.
And there we have it, we have built up our nest.
x will be a value in between these boundaries.
To find our solution, isolate x by itself. The first step we will have to do is to add two to each side of the inequality.
We will get that 3x lies in between negative two and positive six.
Our next step will be to divide each side of the inequality by 3.
This tells us that x will lie in between the values -⅔ and 2.
So what does the graph of this look like?
Our graph will be a numberline. We know that x will be any value that lies between negative two-thirds and positive 2. *insert numberline*
Because we use the exclusive operators, we need to draw open circles on the boundaries since x cannot be equal to the boundary values.
We call this a nest since x will be a value that lies in between the boundaries.
Let’s modify our original question.
What happens if we were to flip the inequality.
And this time we will use the inclusive operator, and add a negative in front of 3x.
Since our absolute value symbol is now on the greater than side, we will use the more is or portion of our phrase to solve for the domain.
Our first step with more is or is to rewrite the inequality without the absolute value symbol.
Next, write the word or.
And this time, we will rewrite the same inequality except this time, we will flip the inequality symbol, and then put a negative in front of the number on the other side.
For each of these two inequalites, we need to isolate x.
To solve for x, add 2 to both of these pieces.
On the right hand side we will get that negative 3x is greater than or equal to 6.
And on the left hand side we will get that negative 3x is less than or equal to negative 2.
To isolate x, we will have to divide each of these inequalities by -3 on both sides.
Now remember, there is a very important rule when dealing with inequalities: When we divide or multiply by a negative number, we must make sure that we flip the inequality.
This means that our solutions will be: x is less than or equal to negative two, or, x is greater than or equal to two-thirds.
Let’s graph the solution.
Since we are using the inclusive inequality, we will have to use solid circles for our boundaries. And this is what the graph looks like, it diverges in both directions and x can be something greater than or equal to two-thirds or less than negative two.
So why should we use the phrase “More is or, less is nest”?
This phrase helps us predict what the domain will look like as long as we know what side of the inequality the absolute value symbol is on.
If our absolute value symbol is on the less than side, then we know that x will be nested between two boundaries.
If our absolute value symbol is on the greater than side, then we know that our graph will be split in two directions.
Let’s take a look at this example problem.
Our goal in this case is to match the given graph with the correct inequality.
We could go through each answer choice and try to graph them, but using our phrase, we can quickly eliminate some of the answer choices.
We know that a solution that lies between two points will be a “less is nest” type problem. Since our graph shows us that x lies between two points, then we know that our absolute value symbol will be on the less than side of the inequality operator.
Because of this, we can eliminate answer choice B where our absolute value symbol is on the greater than side.
Since our graph uses solid circles, we can eliminate any answer choices with exclusive inequality operators. In this case, answer choice C can be eliminated.
From here, we can solve one of our remaining answer choices to find out which one is our correct answer.
To solve answer choice A, we will use the less is nest method since the absolute value symbol is on the less than side.
Rewrite the inequality without the absolute value symbol. Next, write the same inequality on the left side of the expression. Finally, write negative 2 on the left side of the inequality.
Now that we have built our nest, we can isolate x by subtracting 1 from all three sides.
This tells us that x lies in between negative 3 and positive 1.
Since our graph shows that x lies between negative 3 and positive 1, this is the correct answer.
So that’s the benefit of using the “more is or, less is nest” phrase. It allows us to eliminate a couple of answer choices, and then we only really have to solve one of our answer choices.
For example, if we solved answer choice A and it turned out to be not the correct answer, then by a process of elimination we know the answer would have been D.
I hope this video has been helpful, and be sure to check out our other content! Thank you for watching.