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Absolute Value Inequalities - Problem 2
This is an absolute value inequality problem where you have to remember not only your absolute value techniques but also your inequality techniques and the graphing. Here is what I mean for absolute value that means you are going to be solving two different problems the negative distance from zero and the positive distance. For inequalities you have to keep in mind that any time you multiply or divide by a negative number this inequality sign is going to change direction. And then for graphing you have to remember the open circle closed circle business. Let's go through and see how it works.
First thing I want to do is isolate the absolute value sign and the way I'm going to do that is to get rid of the 4 by subtracting 4 from both sides. And then please, please, please my people, be very careful about this negative sign, this one right there. That doesn't just go away, like what that is means is -1 times the absolute value inequality. I can't just like forget about it, it still stays down there even though that 4 is gone I still have that -1.
So to get rid of that I'm going to be dividing both sides by -1. Those will cancel out and I'll have absolute value of 2x take away 6, now that's going to become -4 and since I multiplied or divided by -1 that absolute value sign is going to change direction, it's going to look like that now. I drew a little swirly to show you guys what I meant. Since I divided by a negative value my inequality that used to be less than or equal to is now greater than an or equal to.
That was an absolute inequality trick, now I have to use an absolute value trick. 2x take away 6 is going to bigger than or equal to -4. I have to solve that inequality and then I also need to solve 2x take away 6 is going to be, change the sign and +4, less than or equal to +4. Again I switched the direction becomes although that used to be a -4, I'm making it positive value now, it's like dividing by a negative number.
Once you have that all set up these are pretty straight forward problems. 2x is bigger than or equal to 2, x is bigger or equal to 1, that's going to be one of my solution areas. The other one is going to come from when I add 6 to both sides so 2x is less than or equal to 10, x is less than or equal to 5.
Those are my two solutions algebraically but the problem also asks us to look at it graphically which is good for people who are visual like me it really helps me to see a picture of things so this is actually good for me. Okay one, one two, one, two, three, four, five okay. I want to mark all numbers that are bigger than or equal to one with a closed circle, closed circle because it's greater than or equal to.
So there it is, those are all numbers that are bigger than or equal to 1. I also want to mark all numbers that are less than or equal to 5. So it looks kind of funny I have a closed circle on 5 and it is going forever in that direction.
So really when you look at the graph you can tell that any number is going to make this problem true, any value it can be way out of here negative it could be negative 100 and gazillion ,if that's a real number, or it could be any number out in this direction, it could be 10 and a half. Any value I plug in there for x is going to be a solution and to honest I might not have seen that just looking at this algebraic answers it helps me to see the graph.
I can see that all numbers are marked. You probably wouldn't have known that looking at the original problem that's why it's really important to show all your work. If these swirlys help you, draw the swirlys to show is to how when the inequality sign changes. The negatives will throw you off but you guys I promise you can do these problems if you just stay focused and remember the stuff you learned about absolute values inequalities and graphing.