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Solving and Graphing Compound Inequalities - Problem 2
When solving a compound inequality, which has more than one inequality sign, you can either treat it as two separate problems or keep it as a compound inequality and solve by making sure to use inverse operations on all sides. If you subtract 3 from one side, you must subtract it from every side (left, middle, right). If you divide by 2 on one side, you must divide by 2 on every side. Remember that if you divide or multiply all sides by a negative number, you must switch all the inequality signs to the opposite direction. If the compound inequality is an "and" inequality, only the area between the two circles should be filled in. If it is an "or" compound inequality, the arrows should point outwards.
This is a compound inequality that's written with two inequality signs. It's one of those 'and' problems. So there's two different ways to solve this. One way your teacher might have shown you is to separate this in two different problems. I'll show you how to start it but I'm not going to do it this way, it's just my personal preference.
You could write it as two different problems like this, solve that problem and then solve the other problem also. You have to do that guy and that guy. That's one way to solve this problem. I have a little short cut though that I like and I'll show you guys how to do the shortcut.
What I like to do is just treat this as if it had two equal signs. Like for example, in order to get this x by itself the first thing I would do would be to subtract 3. So I'm just going to subtract 3 here, here and here. It's kind of tricky because it's like a triple equation almost, a triple inequality. -8 is less than or equal to 2x which is less than 8, so that's the first step.
Next thing I want to do to get x all by itself is divide it by 2. So I'm not just going to divide by two here, I'm also going to divide by 2 and 2 there. Keep in mind that if that were a -2, I'd have to change both of these inequalities but it's =2.
-8 divided by two 2 is -4, x is less than 4, that's my final solution and I like it, this method because you only have to solve one problem but you do have deal with like the two inequalities, it's kind of tricky.
When you're asked to graph it, be really careful with the open circle, closed circle business. I want a closed circle at -4, how did that happen that's -2 try it again. -4 would be one, two, three four about there. Okay closed circle at -4 then I want an open circle at +4. One, two, three four and then my x values are everything in between there. Let's make that a little more clear. Closed circle at -4, open circle at +4 and my solutions are any x value that falls in the middle there.
So when you see a compound inequality that's written like this with two signs you have an option. You could just write it as two different problems and solve the two different problems with the word 'and' in between them or you can do this kind of short cut way where you just treat it as two different equal signs.