Inequality question

[jraper13]

Hi all - in the MGMAT book for Equations, Inequalities and VICS, there is a bit about adding up inequalities... However, can someone explain why you add the second inequality twice? Is this in every time thing or just for this problem you add twice? I am really confused about that and the book doesn't explain clearly enough.

here is the example:

Is a + 2b < c + 2d?

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


And the answer:


For this problem, we can add the inequalities together to make them match the question.
First, we need to line up the inequalities so that they are all facing the same direction:
a < c
b < d

Then we can take the sum of the two inequalities to prove the result. We will need to add
the second inequality TWICE:
a < c
+ b< d
a+ b < c+ d
+ b< d
a+2b < c+ 2d
If you use both statements, you can answer the question. Therefore the answer is (C).



thanks a lot

[jnelson0612]

Sure! Look at what we are trying to answer:

Is a + 2b < c + 2d?

Statement 1 tells us that a < c.
Statement 2 tell us that d > b, or b < d.

Now, let's try to add the two together:
a < c
+b < d
a + b < c + d

So this is a step in the right direction, but what I need is a + 2b < c + 2d, because that is the original question.

How can I get there? Take a + b < c + d and add another b < d.
a + b < c + d
+ b < d
a + 2b < c + 2d

Aha! This is what I want, and this is why I add b + d twice: I need 2b and 2d.

Please let us know if we can help further! :-)

[JeremyJH]

I had the same question. The book is a little vague in its wording making the phrase "You will need to add the second inequality twice" appear to be a rule when summing inequalities rather than a circumstance of the particular problem.

[jnelson0612]

I had the same question. The book is a little vague in its wording making the phrase "You will need to add the second inequality twice" appear to be a rule when summing inequalities rather than a circumstance of the particular problem.

That's a very good point, Jeremy. I can see why there would be confusion here and that would be a good edit for a future edition.