Solving absolute value inequalities
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Solving absolute value inequalities 2
Solve for p. We have the absolute value of p minus 12 plus 4 is less than 14. So, let's just do this one step at a time. The first thing we wanna do is we really just wanna isolate the part that has the absolute value. So, to do that we can just get rid of this positive 4 here. We can do that by subtracting 4 from both sides of the inequality. And so the left hand side the positive 4 and the negative 4 cancel out we're just left with the absolute value of p minus 12. And on the right hand side it, we get 14 minus 4 is 10 and we still have the less than sign. So we have the absolute value of p minus 12 is less than 10. So let's just think about it a little bit. If I were to tell you, if I were to tell you that the absolute value of x is less than 10 what does that mean? That means that the distance from x to 0 has to be less than 10. So if I were to draw a number line, if I were to draw a number line and put 0 here we can only go up 10 away and even that's too far. It has to be less than 10. So if this is positive 10 it would have to be less than positive 10 cuz 10 is exactly 10 away. We have to be less than 10 away from 0. And then we could go all the way to the left until negative 10 and even that we wouldn't be able to include because its absolute value is 10. It's not less than 10 but negative 9 negative 9.999 we could include all of those things. The absolute value of any of those things is gonna be less than 10. So another way to write this, this absolute value inequality is that x could be, x could be greater than negative 10, x could be greater than negative 10 and x needs to be less than 10 or we can write this as x is between. So negative 10 is kinda the bottom boundary and we're not going to include it. X is going to be greater than that and is less than 10. So this is another way to write the absolute value of x is less than 10. Is essentially saying that x has to be between negative 10 and 10 and it can't be either negative 10 or 10. We're not, there's no equals sign here. So, the same exact logic. Instead of an x we have a p minus 12 here. So, we can write the absolute value of p minus 12 is less than 10 is saying that negative 10 is less than p minus 12 which is less than 10. And we can just solve this compound inequality all at once by isolating the p in the middle. The best way to isolate the p in the middle we wanna get rid of this negative 12. So let's add 12 to all three, all three sections of this compound inequality. And so we get negative 10 plus 12 is positive 2. And then is less than p minus 12 plus 12 is just p. And then is less than 10 plus 12 is 22. So p is greater than 2 and less than 22. So if we were to plot it on a number line our solution set looks like this. So this might be 2 over here. This might be 22 over here maybe 0 sitting right over here. P is greater than 2. P is greater than 2. It's not greater than or equal to so we have to not fill this in. It has to be an open circle it's only greater than and it's less than 22. It's not less than or equal to so we're not gonna fill that circle in. And it's everything, it's everything in between. And we can verify it for ourselves. Let's try a value that might work out well. Well, 12 is in between these two numbers. It's in our magenta region right here. So let's try p is equal to 12. So if you have 12 minus 12. So it's the absolute value of 12 minus 12 plus 4 which should be less than 14. So this is 0 plus 4 which needs to be less than 14 and 4 is definitely less than 14. So 12 worked 0 shouldn't work. Lets try 0. Zero minus 12 so it's the absolute value of 0. I wanna do this in a different color. It is the absolute value of 0 minus 12 plus 4. This should not be less than 14. This should not work. So we get the absolute value of negative 12. Plus 4 should not be less than 14. Then we should end up with a contradiction here. So we have 12 plus 4 less than 14 we end up with 16 is less than 14 which is not true. So 0 does not work so at least we're feeling pretty good. We took something outside of our solution set didn't work something inside of our solution set it did work.