why when you have an absolute value with an inequality do you switch the sign like the esample in my book |4y+2| <10 it says that when you write the problem you need to write it as 4y+2>-10 and 4y+2<10 put together it would be -10<4y+2<10 not
4y+2<-10 and 4y+2<10 put together it would be -10>4y+2<10 this is what makes sence to me and im confused on why you switch the sign like it wants me to do.. someone please help me!!!!!
While this is not impossible, that's not really the way we would do it. If -10 is greater than some value, then 10 is also going to be greater than that value. In other words, your value would have to be less than -10 (i.e. -10.5, -12, -100). There wouldn't be much point to putting the "<10" on that because if it's less than -10, it also must be less than 10.
-10>4y+2 is the same as
4y+2<-10
So you're basically saying:
4y+2<-10<10
Well, yeah, if it's less than -10, it's also going to be less than 10. So it's useless to even bother with the "<10" part. That's why we wouldn't write it that way. (Even if it were true for this expression, which it's not.)
OK, now that we're done with why it wouldn't make a lot of sense to write it that way... how about why the original expression does not mean that...
If (4y+2) gives us a positive answer, then the absolute value part means nothing. So that would be the same as 4y+2 <10. Make sense?
What if (4y+2) gives us a negative value? Like y=-1. Then 4(-1) + 2 = -4 + 2 = -2. The absolute value of that is 2. Which is less than 10. We're still fine.
But what if y = -4. So we have 4(-4) + 2 = -16 + 2 = -14. The absolute value to that is 14. That is not less than 10. Notice that something which came out as less than -10 doesn't work when we do the absolute value of it.
Picture a number line. Think of it as a mirror with 0 in the middle. Picture -10 and 10 on either side of that mirror, as an image of itself. -14 is over to the left, past the -10. If I did the mirror image of it, it would be over to the right, past the 10. i.e. it just went bigger than 10. So that can't be in the solution. The absolute value of anything on the left will put it in the same place on the right, and the absolute value of anything below -10 on the left will be above 10 (>10) on the right. And that will not fulfill the answer to the expression.
So all negative answers to 4y + 2 must be greater than -10. If we do a mirror image of those values, when we flip them to the right, they will be under the 10. And that fulfills the expression. Hence, all negative answers to 4y+2 must be > -10. When we do the absolute value and flip them to the right side of the mirror, that will keep them <10.
Make sense?
Or....
Any answers you get for 4y+2 that comes out negative must be flipped to a positive. I can do that mathematically by multiplying both sides by -1.
-1 (4y+2) < 10 (-1)
(4y+2) > -10
Remember the sign flips when you multiply or divide a negative.
The mirror version just kind of shows why that is.
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