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some questions on general math principles

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some questions on general math principles [#permalink] New post 24 Feb 2012, 18:11
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I'm taking my test in a few days and ran across a couple of things over the course of my studies that are nagging me a bit...

1) Is 0 not included in a set of the first n integers/ set of consecutive integers? eg are the first 6 integers 1-6 or 0-5? i am rethinking this because the formula for the sum of even ints n(n+1) ignores 0 but 0 is even.

2) When can you divide both sides of an equation by a variable? with inequalities i know the answer depends on whether you know the sign. However, with an equals sign in between it seems like sometimes you can but other times it breaks a quadratic... You can always factor out variable if they are on the same side, right? Example: 3w^2=6w clearly cant factor out. But in system but in system a/b=16 and a/b^2=8, you get b^2a/ba = 2. Another example: y((3x+5)/2) = y & y does not = 0. Is the trick simply knowing that the variable isnt 0? (so in the a/b example, you can becuase they have to both be non zero otherwise they'd be undefined)

3) Only undoing a square results in an absolute value scenario eg X^2 = 9 gives +/- 3, but sqr9 = 3 only is positive because you cannot squar a neg (changes if an odd root)

4) Do i need to test validity of answers to all quadratic equations (paying attention to 0 in the denominator of course) or only in quadratic inequalities?

5) if quadratic numerator has a variable on the outside x(blah blah blah) and the denominator is x, can i factor out since i know they arent 0?

For whatever reason these are the few nagging things that I never found closure on. Any help is much appreciated!
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Re: some questions on general math principles [#permalink] New post 25 Feb 2012, 02:39
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alphastrike wrote:
1) Is 0 not included in a set of the first n integers/ set of consecutive integers? eg are the first 6 integers 1-6 or 0-5? i am rethinking this because the formula for the sum of even ints n(n+1) ignores 0 but 0 is even.

0 is certainly an even integer, but is neither positive nor negative. Formulas you are referring to are:

Sum of n first positive integers: 1+2+...+n=\frac{1+n}{2}*n

Sum of n first positive odd numbers: a_1+a_2+...+a_n=1+3+...+a_n=n^2, where a_n is the last, n_{th} term and given by: a_n=2n-1. Given n=5 first positive odd integers, then their sum equals to 1+3+5+7+9=5^2=25.

Sum of n first positive even numbers: a_1+a_2+...+a_n=2+4+...+a_n=n(n+1), where a_n is the last, n_{th} term and given by: a_n=2n. Given n=4 first positive even integers, then their sum equals to 2+4+6+8=4(4+1)=20.

alphastrike wrote:
2) When can you divide both sides of an equation by a variable? with inequalities i know the answer depends on whether you know the sign. However, with an equals sign in between it seems like sometimes you can but other times it breaks a quadratic... You can always factor out variable if they are on the same side, right? Example: 3w^2=6w clearly cant factor out. But in system but in system a/b=16 and a/b^2=8, you get b^2a/ba = 2. Another example: y((3x+5)/2) = y & y does not = 0. Is the trick simply knowing that the variable isnt 0? (so in the a/b example, you can becuase they have to both be non zero otherwise they'd be undefined)


Never multiply or divide inequality by a variable (or by an expression with variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality.

As for equations such as 3w^2=6w: if you reduce by w then you are assuming with no ground for it that w doesn't equal to zero and thus excluding one legitimate solution (though you can safely reduce it by 3). Correct approach would be: w^2=2w --> w^2-w=0 --> w(w-1)=0 --> w=0 or w=1;

Another example: \frac{a}{b}=16 and \frac{a}{b^2}=8. On the GMAT you will definitely be told that b does not equal to zero since it's in denominator but there won;t be any restrictions on a. Solving: \frac{a}{b^2}=8 --> \frac{a}{b}*\frac{1}{b}=8 --> 16*\frac{1}{b}=8 --> b=2 --> a=32;

Next, if given that \frac{y(3x+5)}{2}=y and y\neq{0}, then you can safely reduce by y and write: \frac{3x+5}{2}= 1 --> x=-1

alphastrike wrote:
3) Only undoing a square results in an absolute value scenario eg X^2 = 9 gives +/- 3, but sqr9 = 3 only is positive because you cannot squar a neg (changes if an odd root)

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt[4]{x}, then the only accepted answer is the positive root.

That is, \sqrt{25}=5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, \sqrt{25}=+5 and -\sqrt{25}=-5. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \sqrt[3]{125} =5 and \sqrt[3]{-64} =-4.

3. \sqrt{x^2}=|x|.

The point here is that as square root function can not give negative result then \sqrt{some \ expression}\geq{0}.

So \sqrt{x^2}\geq{0}. But what does \sqrt{x^2} equal to?

Let's consider following examples:
If x=5 --> \sqrt{x^2}=\sqrt{25}=5=x=positive;
If x=-5 --> \sqrt{x^2}=\sqrt{25}=5=-x=positive.

So we got that:
\sqrt{x^2}=x, if x\geq{0};
\sqrt{x^2}=-x, if x<0.

What function does exactly the same thing? The absolute value function: |x|=x, if x\geq{0} and |x|=-x, if x<0. That is why \sqrt{x^2}=|x|.

alphastrike wrote:
4) Do i need to test validity of answers to all quadratic equations (paying attention to 0 in the denominator of course) or only in quadratic inequalities?


I'm not sure that understand the question: if you solve quadratic equation correctly you most certainly don't need to check. As for quadratic inequalities: in most of the cases you'll get one or two ranges and again if you solved correctly there is no need of checking.

Solving and Factoring Quadratics:
http://www.purplemath.com/modules/solvquad.htm
http://www.purplemath.com/modules/factquad.htm

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

alphastrike wrote:
5) if quadratic numerator has a variable on the outside x(blah blah blah) and the denominator is x, can i factor out since i know they arent 0?


Are you talking about something like \frac{x^2*(y+1)}{x}? If yes, then again on the GMAT you'll be told that x\neq{0} (since its in denominator and division by zero is not allowed) and you can reduce by x and write x*(y+1).

Hope it helps.
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Re: some questions on general math principles [#permalink] New post 25 Feb 2012, 21:58
thank you sir. that cleared everything up for me. i'm good to go now!
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Re: some questions on general math principles   [#permalink] 25 Feb 2012, 21:58
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