Solve each inequality in Exercises 65–70 and graph the solutio...

Question

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 3/(x +3) > 3/(x - 2)

Accepted Answer

Hey everyone in this problem using a number line were asked to graph the solution set for the following inequality were given four over X plus four is greater than four over x minus five. Alright, so when we have an inequality like this, four over X plus four is greater than four over x minus five. But we want to do first is get everything to one side. Okay? And then just have zero on the other side. This is going to allow us to compare to zero, which is easier than comparing to another number or another function because we can start to look at roots and things like that and whether the function is just positive or negative. Alright, so let's move the four over X -5 to the left hand side. Working with inequalities, kids very similar to working with an equal sign. Just a regular equal equation. Okay, we're moving this over, we're going to subtract um in the inequality stays the same. Okay, The only place where you have issues is if you're dividing by a negative, if you're dividing by a negative you're gonna have to flip that sign. But for just adding and subtracting you treat it just like you would a regular equation. So we get four over X plus four minus four over x minus five. It's greater than zero. Now, it can be a little bit hard when you have two terms like this subtracted to compare them and figure out when they're positive when they're negative. So we want to do is try to get this into um factors, we want to get this into a form where everything is multiplied and divided, it's easier to look at signs whether something is positive or negative that way. So let's start with a common denominator. So if we multiply four By X -5, Okay, we can multiply top and bottom By X -5 and similarly we can multiply for X plus four And then we have X -5. Okay, so we multiply top and bottom of the first term by x minus five. Top and bottom of the second term by X plus four. Okay, so that's just like multiplying by one. Now we can combine these because we have our common denominator we have four times X minus five minus four times X plus four. All divided by X plus four, that's minus five. All right now we have this four. Okay, we can actually go ahead and divide and sorry this is greater than zero. We can divide everything by four. We'd like just to make it A little bit simpler. Okay, so divide by four. This term is going to cancel this term is going to cancel so we're left with X -5 minus X plus four over X plus four X minus five. Okay, and this is greater than and if we're dividing by four on the left hand side we have to do the same to the right, so we get zero divided by four which still gives us zero. Okay, but don't forget that you do have to do that to the right hand side as well, just in case you don't have a zero over there. It's important to remember that. All right, So let's simplify this numerator. We have an X minus X. So the X terms go away. Then we have negative five minus four. So we get negative nine over X plus four and X minus five. And we want to know when this is greater than zero. Okay, so again, I've been mentioning signs. Okay, when when we want to find when this is greater than zero, we want this to be positive. So let's look at these factors. Okay? We have a factor of X plus four in the denominator and a factor of X minus five. Okay, the roots of these factors are gonna be the places where we change signs. Okay, so when we go through zero, we're gonna change signs from either positive to negative or negative to positive. Okay, It's that's what we want to look at. Okay. We want to look at when we're changing those signs so we can figure out when this function is greater than zero. So let's call this function on the left hand side G of X. So we don't have to keep rewriting it and our points of interest are going to be those zeros. Okay, So we have one, X plus four is zero. So we have X is equal to negative four or when x minus five is equal to zero which tells us that X is equal to five. So we have two points of interest. Let's go ahead and draw our number line and see what we get. Zero five. Negative four. Okay so we need to look on intervals. Okay so negative four is a point of interest. So let's look first to the left of negative for yourself some more room And we're gonna look to the left of -4. So if we're to the left of negative four, okay. We're looking at points less than negative four. So choose a point say negative five X plus four. If X is negative five it's going to be negative. Okay, it's going to be less than zero. X minus five. It's also gonna be lessons here. So the denominator X plus four times X minus five. This is going to be a negative times a negative which is actually going to give us a positive, this is going to be bigger than zero. So we have a negative nine in the numerator. We're dividing by a positive number and so our function G of X is going to be negative here. Okay, less than zero. This is not what we want. We want our function G F X to be bigger than zero to satisfy our inequality. So this does not satisfy our inequality. We're not going to include it in our solution. Alright, next interval. Okay we're gonna go from negative four all the way to our next point of interest which is five. Okay, X plus four, choose a point in here, say zero X plus four. If X is zero it's going to be positive bigger than zero, X -5. This is going to be negative, it's gonna be less than zero. So X plus four times X plus five. X minus five. This is going to be a positive times a negative which we know gives us a negative. So that denominator is going to be less than zero. And so when we're looking at G we have negative nine divided by a negative number. So G of X is going to be positive means G of X is going to be bigger than zero. That's what we want. So check mark. We want to include this interval in our solution. So from -4, we're gonna leave open circles for now to five. We want to include this interval in our solution. Alright, now those open circles, do we want to color them in? Okay. Do we want to include these endpoints or not? Well, if we have X equals negative four or we have X equals positive five, we're going to get a zero in the denominator and the function doesn't exist. So we actually don't want to include those points because the function doesn't exist there. Alright, so we leave those open. We don't include the endpoints. Okay. And let's just write this interval out that we've included. So this interval is gonna be from negative four to positive five round brackets indicate that we're not including the end points and we have one interval left. Okay? We have everything to the right of five bigger than five. So let's look at X plus four. Okay, if X is bigger than five and this is positive X -5. If X is bigger than five, this is also going to be positive, bigger than zero. So the denominator X plus four times X minus five is going to be positive. Okay, positive times a positive gives us a positive. And so our function G of X. We have negative nine. So a negative divided by a positive. That's going to be negative less than zero. We want our function to be bigger than zero to satisfy the inequality. So this does not satisfy our inequality and is not going to be included in the solution. Alright? So that's it. Okay. We've done it. We've looked at those intervals we found that the interval from negative 4-5, not including the endpoints, satisfies our inequality and we've drawn our number line. So let's go up to our answer choices and choose the correct one. All right. So if we look at our answer choices. Okay, again we had the interval from negative 4-5. Not including the end points. We drew our number line to include that portion between negative four and five and that's going to be answer a with the number line that looks like the one we drew below. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 3/(x +3) > 3/(x - 2)

Key Concepts

Solving Rational Inequalities

Finding Critical Points and Domain Restrictions

Graphing Solution Sets on the Real Number Line

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