In Exercises 73–74, use the graph of the rational function to sol...

Question

In Exercises 73–74, use the graph of the rational function to solve each inequality.

Graph of a rational function with asymptotes and a point at (-9/5, 0).

1/4(x + 2) > - 3/4(x - 2)

Graph of a rational function f(x) = (x + 1)/(x^2 - 4) with labeled axes.

Accepted Answer

Hey everyone in this problem Using the following graph, we're asked to solve the given inequality And the inequality we're given is 1/5 X plus three is greater than negative 4/5 X minus three. Alright, so let's just rewrite what we're giving. Okay, we have 1/5 times X plus three. Okay. And just pay attention to the brackets here. Okay? That X plus three is all in the denominator And this is greater than -4/5. Next ministry. Alright. So we're given this inequality and we want to use the graph to solve it. And let's look at this graph is the graph of five, X plus nine, divided by five times X squared minus nine. Alright, So if we can manipulate our inequality to get a function that looks like the function that is in this graph, then we'll be able to use the graph to solve our inequality. So let's start by moving everything to one side. You'll notice that we have an X plus three term on the left and x minus three term on the right. If we were able to find a common denominator of X plus three X minus three, that's gonna be x squared minus nine. Okay? Like this function on the left. So let's move everything to the left first and then try to find a common denominator and work that way. So we have 1/5, X plus three plus 4/5 X minus three. Okay. And when we're working with inequalities when we're adding and subtracting it's the same as when we're working with an equal sign with an equation we can add and we can subtract from both sides. Um And we just leave the inequality as this All right. So now let's try to find that common denominator in this left most term. The denominator is five times X plus three. On the right term. We have five times X minus three. So for the left term let's multiply by X -3. Okay. We can multiply in the numerator and denominator and that won't change the equation. Okay so we get x minus 3/5 X plus three X minus three. And then for the second term we're gonna do the same thing. But this time we're gonna multiply by the X plus three because we don't have that in the denominator. So we have four X plus 3/5 X plus three X minus three. Alright, now we have this common denominator of five times X plus three times x minus three. We can go ahead and write these together. So we get x minus three plus four X plus three. All over five, X plus three, X -3 is greater than zero. Now remember we're trying to get this inequality to look something like this function from our graph. So let's try to expand simplify a little if we can in the numerator we get x minus three plus four X. Last 12. And in the denominator we have five. Okay. And then X plus three times x minus three. We're gonna get x squared minus nine. Okay, this is a difference of squares. And you'll see that this denominator is a denominator of the function in the graph. So that's good. We are getting closer and now we just need to simplify that numerator. Ok so we have four X plus X. We have five X. We have minus three plus 12. That's gonna be plus nine. Okay. And then our denominator five times X squared minus nine. Okay. And if we look at this this is the function from the graph This function that I'm circling in blue five X plus 9/5 times X squared minus nine. Is this function we found here in our inequality. So we want this function to be greater than zero. Hm We want The function greater than zero. So we want to find okay the grass above the X axis. If the graph is above the X axis that means that that function is positive. Those are positive Y values. So we want to find the graph when the graph is above the X axis. Alright so let's go look at the graph. Now. Now we know that we're looking for above the X axis. Where do we have above the X axis? We have above the X axis. Kind of here. We have above the X axis over here as well. Things have kind of copied out those functions. Now if we're looking strictly at the X values where we're above the X axis or where our function is positive. Hey, what are those? Well, we see that we go from this point -9/5. Okay, over to here before we switch back over. So -1, -3. So this interval here that we've identified Is going to be negative three Up to -9/5. Now I've used round brackets to indicate that we aren't including the end points. Why aren't we including imports while at negative 9/5 our function is equal to zero. Our inequality is greater than zero. It's not greater than or equal to so we don't include that endpoint. Okay. And then on the other side are function looks like it's going off to positive infinity or to negative infinity at three. So the function doesn't look like it's defined at -3. So we're not going to include that point either. All right then on the right hand side we have our function here positive. Okay. 12 at about three X equals three and onwards to infinity. So we're gonna take the union or function from three to positive infinity. Okay. And again, for the same reasons we're not including the endpoint. Alright, so that's it. That's where that function is above the X axis, which means that that function is positive. Okay. And we want our function to be positive to solve our inequality. So these are all of the X values that solver inequality. And we look at our answer choices. We see that this matches with answer choice C. Okay. We have the union of the interval from negative three to negative 9/5 with the interval from three to positive infinity. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) > - 3/4(x - 2)

Key Concepts

Rational Functions

Inequalities

Asymptotes

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