Guide 2, Algebra, 6th Edition, Inequalities, Page 131

[LiliiaG24]

I cannot figure out why do we have to add second inequality twice? Here is the problem:

Is a+2b<c+2d?

(1) a<c
(2) d>b

In order to test the two statement together, add the inequalities together to see whether they match the question. First line up the inequalities so that they are all facing the same direction. Then take the sum of two inequalities to prove the result. You will need to add second inequality twice. I cant understand why twice. Thanks.

[RonPurewal]

the question is about a + 2b and c + 2d. in order to get those "2"s, you add the second inequality twice.

...or you can multiply it by 2 first, and then add it once (= same result overall).

does that make sense?

[LiliiaG24]

Thank you Ron, yes, it does make sense. While looking for the solution I completely forgot about the question.

[RonPurewal]

excellent.

[BrianW172]

I just want to ensure I fully understand... Is it really necessary to add the second equation twice?

Once we add statements I and II and arrive at the equation A + B < C + D isn't that enough to validate that A + 2B < C + 2D? Since terms B and D are being multiplied by the same number (2) can't we think of 2 as a constant? If that line of thinking is correct, couldn't you multiply B and D by anything and the statement would remain true?

Thanks for the help!!

[tim]

No. First, if you have to ask whether you can do something in a math problem, YOU CAN'T! Only use steps that you KNOW work. That's why it's important to learn as many correct mathematical facts and processes as you can.

If you're not convinced that your assertion doesn't work, consider this: Without knowing that B<D (and adding that as many times as you need), you could end up with something like

1 + 7 < 6 + 3

BUT it is NOT true that

1 + 2*7 < 6 + 2*3

[RonPurewal]

well, i'm not really capable of memorizing "rules", but my approach is basically just 2 principles:
1/ understand WHY and HOW things work
2/ when in doubt, TEST things with EASY NUMBERS

as far as #1 -- i have a very bad memory. there is ABSOLUTELY NO WAY i could just "memorize rules" without understanding WHY THEY ARE TRUE.
...so, if you are just trying to commit random meaningless "rules" to memory, this is probably your issue right here.
even if your memory is better than mine (which it almost certainly is), eventually you'll reach a point where you simply won't be able to do this anymore, and/or confusion will begin to set in.
on the other hand, if you understand WHY things work the way they do, this won't be an issue.

in terms of #2, just try simple numbers.
for instance,
1 + 5 < 4 + 3 ...this is true
but
1 + 2(5) < 4 + 2(3) ...nope, not true.

[BrianW172]

Thanks for the help, guys! Makes sense. Now I realize I should have just tested numbers and answered the question myself!

[RonPurewal]

you're welcome.
if you have any other questions, feel free to ask.