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

Question

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.

(x^2 -x - 2)/(x^2 - 4x + 3) > 0

Accepted Answer

Hey everyone in this problem Using a number line were asked to graph the solution set for the following inequality were given x squared minus x minus six Over x squared minus six. X plus eight is greater than zero. Okay, so let's just rewrite our inequality X squared minus x minus six over x squared minus six. X plus eight is greater than zero. All right. So when we have a problem like this, what we wanna do is we want to go ahead and factor polynomial. Okay. And the numerator and denominator and work from there. Okay. Sometimes you'll have factors that cancel and make it a little bit simpler. Um But if not it's still going to be easier to compare to um zero on the right hand side. Okay. Alright. So if we factor the numerator We have X -3 times x plus two. And when we factor the denominator We're gonna get X - And X -2. Okay. And that's going to be greater than zero. Alright, so we don't have any factors that cancel but we do have these factors that help us find points of interest. Okay? And help us find points where we switch from being positive to negative. Okay, All of these have roots. Okay, so all of the roots of these factors are going to be our points of interest and that's where this function is going to change science. Okay. It has to go through one of these routes in order to change something. So we have these points of interest. It's gonna be where each of these is equal to zero. So we have negative two. Okay this is the root of X plus two. We have to the root of X -2. We have three. The root of X -3. We have four. The root of X -4. Okay so we have these four points of interest that we're gonna want to look at. Let's just go ahead and call this the left hand side of our inequality. We're gonna call this the function G of X. It's just going to save us from rewriting that whole thing over and over and over. Alright so we have these four points of interest and we want to look at what happens kind of in between them. So let's draw a number line trying to keep that function up. I draw a number line here with all of these numbers of interest. Okay so we're gonna put zero here - -2 one two three, four. And then I'm gonna keep keep going both directions. Okay so our points of interest negative 223 and four have labeled them on the graph and we want to look at what happens in each kind of sector. Okay, so what happens in this left hand side? Okay, when we are below negative two, Well if we're below negative two then X minus three Is going to be less than zero X plus two. Well this is going to be less than zero as well. Mhm. So that means that our numerator, we're gonna call it n we have a negative times a negative it's going to be positive. Now we're gonna do the same for the denominator. X -4. Okay. Well this is gonna be negative X minus two. This is also gonna be negative and again we're taking points less than negative two. So you can choose one in particular if you want to say to you take negative three, that's to the left of negative two and you plug it in and you're gonna get negative negative. So the denominator is going to be positive as well. So that means in this sector G of X. Have a positive divided by a positive is going to be bigger then zero. Okay. That's what we want. We want G Fx to be bigger than zero. That's our inequality that we're solving for. That means that anything to the left of negative two is going to be included. So we're gonna draw it like this in blue to show that it's included in our solution. Now we've left an open circle at -2. We have to decide whether we should color this in or not. What happens at negative two. If we're at negative two. X plus two is going to be zero. Okay so this whole thing is going to be +00 is not greater than zero. If we had greater than or equal to we would include this point but we don't. So this stays as an open circle, negative two is not included. Alright now let's move to the next sector. Okay, so we have from from negative to positive too. Okay, so you can pick a point in this case the easiest point to choose is zero. So we have X -3. Okay, if you put a zero in here this is gonna be negative X-us two. If you put a zero in here this is gonna be positive. Okay so our numerator, we have a negative times a positive. That's gonna be negative less than zero. If we look at our denominator X -4 KF X is zero. This is gonna be negative X -2. Well that's also going to be negative negative times negative is a positive. So our denominator is bigger than zero. So we have a negative divided by a positive. So this sector G of X is going to be less than zero, which we don't want. Okay, so this is going to be not included in our solution. Alright now for the sector between two and let's just do this above. So we have a bit more room. Okay, so between two and three so you can pick a point say 2. X -3, it's going to be less than zero X plus two. It's going to be bigger than zero. So our numerator is going to be less than zero X minus four. It's going to be less than zero X minus two. It's going to be bigger than zero. So our denominator is also less than zero. We have a negative times a positive. That gives us a negative. So now our numerator is negative. Our denominator is negative. When we do a negative divided by a negative, we're going to get a positive. So G F X here is going to be bigger than zero. Okay, so this portion is actually going to be included in our solution. We'll draw in blue here, we're going to include this. Okay, now what happens at X equals two? If X is equal to two. Okay, we have a zero in the denominator. The function is not defined. So that is not a solution. Okay, what about that? X equals three. If x is equal to three. The numerator is zero. The left hand side of zero. And again same as before. Zero is not greater than zero. Okay. We don't have the equality in our inequality so we don't include three. Okay, so we leave those as open circles. All right. We only have two sectors left. We have between three and four and then we have four onwards. So between three and four. Okay, so choose the points. A 3.5 X minus three. If x is 3.5 that's going to be positive. X-us two. Also positive. Okay, so the numerator is positive. X -4. Well that's gonna be negative. Okay, 3.5 -4 is less than zero. X minus two. Tiny bit more room. It's going to be bigger than zero. And so the denominator negative times a positive is going to be less than zero. So we have a positive number divided by a negative number. This is going to give us G f X less than zero, which is not what we want. So this is not included in our solution And then we have one last sector. Okay, one X is bigger than four. Alright, effects is bigger than four. Pick a number, say five X -3 is positive. X-plus two is also going to be positive. So this tells us that our numerator is positive X minus four K. If x is five, this is going to be bigger than zero, positive X -2 is also going to be positive bigger than zero. Which means that our denominator is bigger than zero positive times a positive gives us a positive. And so our function G of X A positive divided by a positive is going to be bigger than zero. So this is included. Okay, that does satisfy the inequality. We want this function G to be bigger than zero. So this is satisfies the inequality So we're gonna include everything to the right of four. Alright, and again we leave this as an open circle because if X is equal to four and then our function does not exist. Alright. So there we have it. We looked at all those sectors and this is the graph of our solution. Okay, so we have everything included to the left of -2. We have between two and three included. We have everything to the right of four included. And if we look at our possible answer traces, we see that this matches up with the number line for B. Okay. So we found that the interval from negative infinity to negative into two. Non inclusive. The interval from 2-3 not inclusive and the interval from four to infinity not inclusive is our solution. So we have to be here. 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. (x^2 -x - 2)/(x^2 - 4x + 3) > 0

Key Concepts

Inequalities

Rational Functions

Graphing Solution Sets

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