Solve each rational inequality. Give the solution set in interval...

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. (2x - 3)/(x + 1) > 4

Accepted Answer

Hey, everyone in this problem for the given inequality we're asked to solve and to write the solution set in interval notation. The inequality we're given is four X minus six divided by X plus five is greater than two. We have four answer choices. Option A is the open interval from negative 5 to 8. Option B is a union of two open intervals. The first is from negative infinity to negative five and the second is from eight to positive infinity. Option C is the union of two intervals as well. The first is just the point negative five. And the second is the open interval from eight to positive infinity. And option D is that there is no solution. So we're gonna start by writing what we were given four X minus six, divided by X plus five is greater than two. Now, when we're given an inequality like this, what we wanna do is get everything to one side numbers and variables and have just zero on the other side. OK. It makes it a lot easier to compare an inequality if we're comparing it to zero. So there's a couple ways we could do this OK. And the first, your first instinct might be to multiply both sides of this equation by X plus five. Now, you could do that, but you have to be a little bit careful. OK. So there are two things you need to do if you do that. The first is you need to add the restriction that XX cannot be equal to negative five. OK? If it is, then the denominator is zero, we can't divide by zero, we would have an undefined expression on the left hand side. So that would not be a solution is you have to add that restriction in the second thing to consider is whether X plus five is positive or negative. And if you have an inequality and you multiply by a negative value, you have to switch the sign of the inequality. So when you multiply, you'd have to have two cases. One for X plus five being greater than or equal to zero and one for being less than OK. So that is one approach and you can do that approach. And I encourage you to try that approach. If you want to, what we're gonna do instead is we're gonna subtract two from both sides. So we don't have to consider two separate cases. So we're gonna get four X minus six divided by X plus five mines too is greater than zero. Now, we've got everything on one side but we want to write it as a single rational function. So we wanna have terms that are multiplied together or divided. OK? Nothing or we have this minus two. So in order to do that, we need to find a common denominator, we already have a denominator of X plus five in the first term. So let's stick with that. We're gonna multiply the second term by X plus five in the numerator and the denominator to keep it equivalent, we get four X minus six minus two, multiplied by X plus five, all divided by X plus five. And this is greater than zero. Now, simplifying the numerator, we have four X minus six minus two, X minus 10. And we have to distribute that negative two to both terms in the brackets. All divided by X plus five is greater than zero four, X minus two. X is going to give us two X negative six minus 10, gives us negative 16. So we get two X minus 16 divided by X plus five is greater than zero. We can divide the entire expression left and right hand side by two. Just to simplify that numerator and we can write it as X minus eight divided by X plus five is greater than zero, divided by two, which is just zero. So now we have a single rational function on the left hand side compared to zero on the right hand side, exactly what we want. And from this stage, what we wanna do is evaluate our points of interest. OK? And the points of interest are going to be where either the numerator or the denominator is equal to zero. So if the numerator is equal to zero, X minus eight equals zero, tells us that X is equal to eight. And if the denominator is equal to zero, X plus five is equal to zero, which tells us that X is equal to negative five. We have these two X values of interest X equals eight and X equals negative five. And why we wanna look at where they're equal to zero is because we're comparing them with the zero. OK. So if we look at where they're equal to zero, then on either side, they could switch from being positive to negative and that switch is gonna occur when they go through zero. So now we can use those points to break up our number line into separate intervals and evaluate each one to see whether it's positive or negative, which will either agree or disagree with our inequality. All right. So let's get started on that. We're gonna draw a number line and you don't need to draw a number line when you solve these problems. But I think sometimes it's helpful to visualize the intervals and keep our work organized and we're gonna put our two points on here. So we have our point negative five, we have our point positive eight and our X values along our number one. We're gonna start on the far left and the interval is when X is less than negative five. So we can choose any X value as a test point. So say X is equal to negative 10. If we use X is equal to negative 10, X minus eight is going to be negative, it's gonna be less than zero, X plus five is also going to be negative or less than zero. And so the left hand side of inequality, which we're gonna call L H X is gonna be a negative divided by a negative which gives us a positive value. So the left hand side is going to be greater than zero, which satisfies our inequality. That's exactly what we want. And we want the left hand side greater than zero. That's what we have. And so this interval, we're gonna give it a green check mark to indicate that it satisfies our solution or sorry, it satisfies our inequality. And so it is a solution. Now moving to the right, we have the endpoint of this interval which is X equals negative five. Can we have X equals negative five? Is that going to be part of our solution? The answer is no. If we have X equals negative five, we already talked about this with our restriction, the denominator is equal to zero. The left hand side becomes undefined. And so that does not satisfy our inequality. So what we do there is we draw an open circle to indicate that we are not going to include that point. And it's just gonna help us remember which points to include when we go to write this in interval notation, moving to our next interval from X equals negative five, all the way up to positive eight. Again, you can choose any test point within that interval. They'll say something like X equals zero. If X is equal to zero, X minus eight is going to be negative or less than zero, X plus five is going to be positive or greater than zero. So the left hand side will be a negative divided by a positive which gives us a negative value. The left hand side is less than zero, which does not satisfy our inequality. So we're gonna give a red X to this interval. It's not gonna be part of our solution side moving to the right, we get the point X equals eight. OK. If X is equal to eight, then the left hand side is equal to zero. But we have a strict inequality here. We want the left-hand side to be greater than zero, not greater than or equal to zero. OK. So if X equals eight, the inequality is not satisfied, this is not part of our solution. So just like for X equals negative five, we draw an open circle and now we're on to our final interval where X is greater than eight. Again, you can choose any test point if you'd like say X is equal to 10, X minus eight is going to be positive or greater than zero, X plus five is also gonna be positive or greater than zero. And so a positive value divided by a positive value is gonna give you a left hand side that is greater than zero and that satisfies our inequality. That's exactly what we want. So we get another green check mark on this interval. We're gonna give ourselves some more room and we're just gonna write this out an interval notation like the question asked. And so we have the union of two intervals and we found two intervals that satisfy our inequality. The first was from negative infinity all the way up to negative five. We're gonna use round brackets to indicate that we are not including those end points. Hey, and we're taking the union of that interval with the second interval we found which is from positive eight all the way up to infinity. OK? So this is the complete solution set for this problem. If we go back to our answer choices, we can see that this corresponds with answer choice. B. Thanks everyone for watching. I hope this video helps see you in the next one.

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. (2x - 3)/(x + 1) > 4

Key Concepts

Rational Inequalities

Interval Notation

Critical Points

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