we're being asked to solve the given compound inequality and then we're gonna graft the solution set on a number line. So remember to solve these. What we're gonna do is solve both inequality separately. So let's start with the first one. To X plus three is greater than 11. So the solve for X, we're going to start by subtracting three from both sides of our inequality. So we're gonna be left with two X is greater than, well, 11 ministries. Then we're going to divide both sides by two. Well, eight divided by two is four. So we end up with a solution of X is greater than four. Okay, Now we need to solve the second inequality we have Negative three X is less than or equal to 15. So to find X, we're going to divide both sides of our inequality by negative three. Now, don't forget when you divided inequality by negative value, you have to flip your inequality sign so the less center equal to will become greater than or equal to. And then we'll divide while 15 divided by negative three is negative. Five. Okay, Perfect. Now we have our solution. X is greater than four or X is greater than equal to negative five. So what we need to do now is graft. The solution set on the number line. So we're gonna start. Let me try and make that a little straighter for us here. So we have our number line. Remember, we're gonna start by putting our key values here so foreign, negative five. But because negative five smaller that will go to the left. And remember, because it's an or statement, we're going to salt or graft them both separately. So first, let's start by Graphing X is greater than four. Well, since four is not, including the solution will have an open set or sorry, open circle it four. And because we're talking about values that are greater than for, we're going to shave everything to the right. So all of these values of vaccine, this red shaded area makes the first in the quality true. Then we need to graph X is greater than or equal to negative five. Well, because negative five is included in solution set, we're going to have a close circle and negative five. And then we're talking about all the values of X that a greater than or equal to negative five. Well, in terms of our number line, we would typically shade everything to the right. So if we shared everything to the right than negative five, take a look at what's gonna happen here. Doesn't that include for where we used to have an open circle? Well, because four does make the second inequality true, What that's going to do is kind of shade that circle in. And it's gonna be all of these values that also are included in the rates shaded area. So essentially, what you're a number line really represents is our solution. That means that any value that's greater and equal, the negative five will make both inequalities true. Now it doesn't mean that these values in between will make the first inequality true because they won't But remember what the word or means as long as it makes one of them true, it makes the whole statement true. So now this is one of our special cases for compound inequalities that involved the word Or so how would you know you have a special case? Well, take a look. You're to inequality signs. They're both greater than the or equal to doesn't matter in the fact that it's a special case, but when they're both greater than or they're both less, then you're gonna have some number line kind of like the one that we had. It's gonna win to either shade to the whole thing, to the right or the whole thing to the left. So that's just one example of special case in the next example will show you a number special case that you could potentially have.

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we're being asked to solve the given compound inequality and then we're gonna graft the solution set on a number line. So remember to solve these. What we're gonna do is solve both inequality separately. So let's start with the first one. To X plus three is greater than 11. So the solve for X, we're going to start by subtracting three from both sides of our inequality. So we're gonna be left with two X is greater than, well, 11 ministries. Then we're going to divide both sides by two. Well, eight divided by two is four. So we end up with a solution of X is greater than four. Okay, Now we need to solve the second inequality we have Negative three X is less than or equal to 15. So to find X, we're going to divide both sides of our inequality by negative three. Now, don't forget when you divided inequality by negative value, you have to flip your inequality sign so the less center equal to will become greater than or equal to. And then we'll divide while 15 divided by negative three is negative. Five. Okay, Perfect. Now we have our solution. X is greater than four or X is greater than equal to negative five. So what we need to do now is graft. The solution set on the number line. So we're gonna start. Let me try and make that a little straighter for us here. So we have our number line. Remember, we're gonna start by putting our key values here so foreign, negative five. But because negative five smaller that will go to the left. And remember, because it's an or statement, we're going to salt or graft them both separately. So first, let's start by Graphing X is greater than four. Well, since four is not, including the solution will have an open set or sorry, open circle it four. And because we're talking about values that are greater than for, we're going to shave everything to the right. So all of these values of vaccine, this red shaded area makes the first in the quality true. Then we need to graph X is greater than or equal to negative five. Well, because negative five is included in solution set, we're going to have a close circle and negative five. And then we're talking about all the values of X that a greater than or equal to negative five. Well, in terms of our number line, we would typically shade everything to the right. So if we shared everything to the right than negative five, take a look at what's gonna happen here. Doesn't that include for where we used to have an open circle? Well, because four does make the second inequality true, What that's going to do is kind of shade that circle in. And it's gonna be all of these values that also are included in the rates shaded area. So essentially, what you're a number line really represents is our solution. That means that any value that's greater and equal, the negative five will make both inequalities true. Now it doesn't mean that these values in between will make the first inequality true because they won't But remember what the word or means as long as it makes one of them true, it makes the whole statement true. So now this is one of our special cases for compound inequalities that involved the word Or so how would you know you have a special case? Well, take a look. You're to inequality signs. They're both greater than the or equal to doesn't matter in the fact that it's a special case, but when they're both greater than or they're both less, then you're gonna have some number line kind of like the one that we had. It's gonna win to either shade to the whole thing, to the right or the whole thing to the left. So that's just one example of special case in the next example will show you a number special case that you could potentially have.

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