Is 4a^2-9b^2 <0

[dlginsberg89]

This question is from Advantage Testing's Math Test 5 Question 24:

Is 4a^2-9b^2<0?

1)2a+3b < 0

2) 2a-3b < 0
-----------------------
1)
4a^2 - 9b^2 < 0
2^2a^2-3^2b^2 < 0
2^2a^2 < 3^2b^2
2a < 2b

2) 2a-3b <0
2a < 3b

The answer is B. I must be missing a rule about square roots that made what I did in (1) wrong. Do you know what it is?

[RonPurewal]

This question is from Advantage Testing's Math Test 5 Question 24:

Is 4a^2-9b^2<0?

1)2a+3b < 0

2) 2a-3b < 0
-----------------------
1)
4a^2 - 9b^2 < 0
2^2a^2-3^2b^2 < 0
2^2a^2 < 3^2b^2
2a < 2b

2) 2a-3b <0
2a < 3b

The answer is B. I must be missing a rule about square roots that made what I did in (1) wrong. Do you know what it is?

ya, you can't "square root both sides" of an inequality unless you know the signs of both sides.
* if both sides are positive, the inequality points in the same direction as it originally did.
* if both sides are negative, the inequality "flips" and points in the opposite direction.
(these are easy enough to see if you think about specific examples:
4 < 9 and 2 < 3
4 < 9 but -2 > -3.)

in fact, you can't even "square root both sides" of an equation, much less an inequality, because you still have the same issue (you don't know whether the expressions being squared are positive or negative to begin with).

--

the point of this problem is for you to (immediately) recognize the "difference of squares" factoring pattern.
i.e., when you see 4(a^2) - 9(b^2), you should factor that into
(2a + 3b)(2a - 3b)
before you even turn your brain on and start thinking.
if you do that factoring, you should be able to see pretty easily how this problem shakes out (provided you understand the basics of the DS format).

--

this may be getting in over our heads here, but you can "square root both sides" IF you use absolute-value signs once you've done so.
for instance, if you know that a^2 < b^2, then you can say for sure that |a| < |b|.
more on that in our strategy guides if you're interested.

[kouranjelika]

Hey Ron,

You probably already saw my previous post, I'm reasearching anything and everything about this subject on here to pick up on every insight possible.

So on the question here, not sure how the student got an answer B.

I obviously made the question stem into a difference of squares right away.
Is (2a-3b)(2a+3b)<0?
each statement provides a piece to the inequality at question.
SO, don't we need both to figure out that the Answer to the question is NO, the dif of squares is not negative, it's positive. Negative Number*Negative Number = positive Number? How can the correct answer be B?

Thanks!

[RonPurewal]

The answer shouldn't be B. The answer should be C.

IF we were told that a and b are both positive, THEN the answer would be B, because in that case 2a + 3b would have to be positive.
Without such a stipulation, you're absolutely right"”we need the signs of both factors.

[kouranjelika]

Yea makes sense. Cause then the other part is def pos, hence making the whole thing negative.
Otherwise without knowing a & bs signs we need both.
Cool. Thank you!!

Btw, are you not running the study hall this week?

[RonPurewal]

Glad it's all clear.

There was no study hall this week, for various logistical reasons.
Also note that, starting with the upcoming session (April 10th 2014), the start time will be two hours later.

[kouranjelika]

Oh no! That cuts right into the entire Advanced Quant hmw session. Have you already solidified the time? Because it wouldn't be good for most Manhattan students (it's a total pain that it's the only time they offer the quant section anyway..)

Hope to catch you at least at the following one. I am in my last week of class now and test is in 3 weeks.

Thanks anyway Ron. You're grand.

[RonPurewal]

The time is definitely finalized; I've been wanting to change it for months.

If you can't make the session live, you can always watch the recording, which should be posted online within a few days.

Thanks and good luck.