Rational inequalities

[The Velociraptors]

Hi, I am attaching my work for a question below:

So I know this is wrong because apparently the hole is also on the number line. But I‘m confused on why this is so because the location of the hole is (2,10/9), and I thought the number line only accounts for coordinates on the x axis. Also, can you clarify if I’m supposed to take the holes into account when I’m finding the degree of the rational function? Is the degree of this rational function 4?

 

[Dr.Peterson]

apparently the hole is also on the number line

If you were graphing, the hole would not be on the x-axis (which you could think of as a number line); but in solving the inequality, that value has to be excluded because the LHS isn't defined there. So you do need to include it in graphing the solution.

Don't confuse these two problems!

Also, can you clarify if I’m supposed to take the holes into account when I’m finding the degree of the rational function? Is the degree of this rational function 4?

How do you define the degree of a rational function? That term applies to polynomials. And how would it affect your answer?

A quick search shows at least two ways to define it in this context (I don't think either is standard): the greater of the degrees of numerator and denominator, or the difference of those degrees. So you need to tell us what definition is used in your course.

 

[Steven G]

You must skip x=-2. That is a given.

Now (x+5)^2 and (x-2)(x-2) = 1 > 0 (if x is not 2) are always positive. So for the fraction to be positive you will need x(x-3)>0

Just solve x(x-3)>0 and make sure not to include x=2.

 

[The Velociraptors]

If you were graphing, the hole would not be on the x-axis (which you could think of as a number line); but in solving the inequality, that value has to be excluded because the LHS isn't defined there. So you do need to include it in graphing the solution.

Don't confuse these two problems!

How do you define the degree of a rational function? That term applies to polynomials. And how would it affect your answer?

A quick search shows at least two ways to define it in this context (I don't think either is standard): the greater of the degrees of numerator and denominator, or the difference of those degrees. So you need to tell us what definition is used in your course.

I thought I need the degree of the function to determine if the end behavior follows the right side or if it is the opposite of it. Basically, I thought I needed it to determine the sign of each region.

We talked about the greater of the degrees of the numerator and denominator. Is there such thing as an overall degree of the rational function?

So is the understanding that a rational number line accounts for coordinates on the x-axis wrong?
This differs from polynomial inequalities, right?

 

[The Velociraptors]

You must skip x=-2. That is a given.

Now (x+5)^2 and (x-2)(x-2) = 1 > 0 (if x is not 2) are always positive. So for the fraction to be positive you will need x(x-3)>0

Just solve x(x-3)>0 and make sure not to include x=2.

So am I only supposed to use the degree of the numerator disregarding the holes and vertical asymptote to find the degree of the function?

 

[Dr.Peterson]

We talked about the greater of the degrees of the numerator and denominator. Is there such thing as an overall degree of the rational function?

Not in my own experience, though as I said, I find some people using the concept.

The question is, why do you ask? What are you going to use it for? Were you taught something about the graph that is determined by the "degree"? Or are you just curious?

So is the understanding that a rational number line accounts for coordinates on the x-axis wrong?
This differs from polynomial inequalities, right?

I don't think you mean to ask about rational numbers, which are irrelevant, but just about "the number line as used in this context".

But as I said, the way you use a number line in solving an inequality (either polynomial or rational) is different from the x-axis in graphing a function (either polynomial or rational or other). In the former, it is just a way of keeping track of values of x for which the expression is positive, negative, zero, or undefined. The last category does not show up on the x-axis of a graph, but has to be accounted for in solving the inequality.

 

[The Velociraptors]

Not in my own experience, though as I said, I find some people using the concept.

The question is, why do you ask? What are you going to use it for? Were you taught something about the graph that is determined by the "degree"? Or are you just curious?


I don't think you mean to ask about rational numbers, which are irrelevant, but just about "the number line as used in this context".

But as I said, the way you use a number line in solving an inequality (either polynomial or rational) is different from the x-axis in graphing a function (either polynomial or rational or other). In the former, it is just a way of keeping track of values of x for which the expression is positive, negative, zero, or undefined. The last category does not show up on the x-axis of a graph, but has to be accounted for in solving the inequality.

I‘m asking because, for example, knowing the degree in polynomial inequalities help with the end behavior. If that’s not the same in the number line used in the rational inequality context, then how do I determine whether each region is greater than or less than 0, or the sign of each region?

Perhaps I am thinking too much about polynomial inequalities, but on a polynomial inequality graph, is’t each value a zero? That’s why I thought I was supposed to only account for the zero’s.

 

[pka]

Hi, I am attaching my work for a question below:

So I know this is wrong because apparently the hole is also on the number line. But I‘m confused on why this is so because the location of the hole is (2,10/9), and I thought the number line only accounts for coordinates on the x axis. Also, can you clarify if I’m supposed to take the holes into account when I’m finding the degree of the rational function? Is the degree of this rational function 4?

View attachment 29941

Have a look at this link. Look at the graph of the question then scroll down to the interval notation.

 

[Steven G]

You need end behavior to graph the function. You just need to decide when the function is greater than or equal to 0.

x can not be 2 or -5.

The denominator is always positive (if we leave out 5). The sign of the numerator will determine the sign of the fraction. The numerator will be positive if x and (x-3) are both positive or they are both negative. The fraction equals 0 when x=0 or x=3.

It remains to determine if/when x and (x-3) are both positive or if/when x and (x-3) are both negative.