All the Quant, Combining Inequalities, page 236

[yo4561]

Good morning my Manhattan Prep friends!

I have a question on the question:

Is a + 2b < c + 2d
(1). a < c
(2) b < d

I understand that the solution adds b and d to get-->

a + b < c + d

But, I do not understand how it is a legal math move to add b + d again. Can you keep adding variables until you get the solution to look like what you want it to ? What is the rule I can follow here to clarify my understanding? It seems subjective, that I can perhaps then add a + c if I wanted to to get 2 a + 2b < 2c + 2d. Or, if I just multiply the whole thing by 2 after adding the first b and d, I will get 2 a + 2b < 2c + 2d, and then the answer is not correct. Thank you in advance.

[esledge]

But, I do not understand how it is a legal math move to add b + d again.

You can stack and add inequalities, but the inequality sign must be facing the same direction in both of them (and then in the result). The general idea is that:

(small < big)
+ (little < large)
------------------------------------
small + little < big + large
The sign must go this direction because two larger values must have a sum greater than the sum of two smaller values!

But if the inequality signs were facing the opposite directions:

(small < big)
+ (large > little)
------------------------------------
(small + large) < or = or > (big + little)
Either side of the inequality could be greater! The direction of the resulting sign would depend on the relative values (i.e. how big or little the various terms are).

Can you keep adding variables until you get the solution to look like what you want it to ? What is the rule I can follow here to clarify my understanding? It seems subjective, that I can perhaps then add a + c if I wanted to to get 2 a + 2b < 2c + 2d. Or, if I just multiply the whole thing by 2 after adding the first b and d, I will get 2 a + 2b < 2c + 2d, and then the answer is not correct.

The rest of your question boils down to “how do you know what to add?” Yes, you could add a < c to get 2a+2b < 2c + 2d, and that’s a mathematically true statement…but it is not a match (or answer) for the question! The short answer is: Always try to match the statement(s) to the question (or vice-versa).

The solution in the book is the proof that the statements together are sufficient to answer the question. Here’s the step-by-step:

Per Stmt (1), a < c. Leave this alone because this already matches the a … < c … part of the Combo in the question.

If b < d per Stmt(2), then it is also true that 2b < 2d. Why do this? Solely to make statement (2) match the Combo in the question better.

If you add the inequalities above, you get the exact match to the question Combo, which was the goal.