# Math question - Inequalities

-2<x<3
9<y+x<11
find range of y-x

Someone please give the answer I just need to verify i got it correct. Preferribly with working so i know where i went wrong if i went wrong. This is a pretty simple question i just need to verify i got the method correct.
y - x = (y + x) - 2x, so you should try to minimise that.
3 < y - x < 15
Lowest x value is -1 and the highest x value is 2

So the highest y value should be 10 minus the lowest x value, which is equal to 11.

So the lowest y value should be 10 minus the highest x value, which is equal to 8.

We choose 10 because that's the only value y+x can be.

so our highest y-x value is when y is the largest value possible which is 11. so y-x is 11-(-1) which is equal to 12. the lowest y-x value is when y is the smallest value possible which is 8. so y-x is 8-2 which is equal to 6.

so range is 6-12 so 6
Okay seeing the other answer, I'm probably wrong sorry
Original post by yellowleaf23
y - x = (y + x) - 2x, so you should try to minimise that.
3 < y - x < 15

can u give a more thorough explanation?
Original post by ian101
Lowest x value is -1 and the highest x value is 2

So the highest y value should be 10 minus the lowest x value, which is equal to 11.

So the lowest y value should be 10 minus the highest x value, which is equal to 8.

We choose 10 because that's the only value y+x can be.

so our highest y-x value is when y is the largest value possible which is 11. so y-x is 11-(-1) which is equal to 12. the lowest y-x value is when y is the smallest value possible which is 8. so y-x is 8-2 which is equal to 6.

so range is 6-12 so 6

Pretty sure ur right, i got 13 to 5, but quite a few people got ur answer
Original post by ArgonYttriumAN
can u give a more thorough explanation?

As you know -2 < x < 3, you can just double to find -4 < 2x < 6.
Now think about (y + x) - 2x.
If you want to make this as large as possible, you want the (y + x) part to be as large as possible, and the 2x part to be as small as possible. So set y + x = 11, and set 2x = -4. Then (y + x) - 2x = 11 - (-4) = 15, so the largest possible value of y - x is 15.
Try following the same logic to find the smallest possible value!