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- Quant - Algebra

Kunal K.

• 39

cards
Test-taking trick

If ≠ 2, which of the following is equal to [b * (a^2 - 4)] / [ab - 2b]?

(a) ab (b) a (c) a+2 (d) a^2 (e) 2b

What are two routes/ways for solving this problem?

+ Route 1: simplify the expression in the question stem until it looks like one of the answer choices

+ Route 2: plug in numbers

--> Notice how you can use Route 2 if you are having problems with Route 1 (i.e. the expression is hard to simplify, as could happen with e.g. absolute values)

--> Note that there are some pairs of numbers that lead you to just 1 answer, and there are some pairs that lead to 2 Just try diff #'s if I end up with 2 answers.

Algebra tips

+ Sometimes I can get away with just solving for the combined equation, rather than each individual variable that goes into the combined equation.

e.g. What is x/y?

(1) (x+y)/y = 3 (2) y=4

--> Here, I can manipulate statement 1 to calculate the value of x/y. I don't need to solve for x, and then solve for y. The better strategy is to solve for x/y!! Be direct!

+ Sometimes I can rewrite the original DS question in order to come up with a new rearrangement that is FAR easier to test.

e.g. If xy ≠ 0 and sqrt (xy/3) = x, what is y?

(1) x/y = 1/3 (2) x = 3

--> Best way to work this problem is to first do the math on the original question, rather than plugging in statements 1 and 2.

Then, the equation turns into y = 3x. Thus, the rephrased question is "what is x?", so the answer is B

Random Exponents and square roots

+ Remember that sqrt (x^2) = abs (x)

+ Even exponents "hide" the sign of the base.

e.g. If x^2 = 25, then abs(x) = 5, and x = +5 or -5

e.g. a^2 - 5 = 12, then a^2 = 17, then a = +sqrt(17) or -sqrt(17)

+ Even exponents hide the sign of the base

+ Odd exponents keep the sign of the base. Also, equations that involve only odd exponents (like a cube root) only has 1 solution

Exponents - base of 0 or 1

0 raised to any power equals 0

1 raised to any power equals 1

If x = x^2, then you know that x is either 0 or 1!

Exponents - basic rules

+ When raising fractions to a power, you can distribute the exponent to both the numerator and denominator

e.g. (3/4)^2 = 3^2 / 4^2

+ Similarly, exponents can be distributed to a product

e.g. (3x)^4 = 3^4 * x^4

+ When two exponential terms have the same base, you can add/subtract the exponents upon multiplication/division, respectively

+ Anything raised to the 0 power equals 1! The only exception is 0^0, which is undefined

+ When you raise an exponential term to an exponent, multiply the exponents! e.g. (z^2)^3 = z^6

Exponents - negative bases

When dealing with negative bases, pay attention to PEMDAS.

--> If the negative sign is inside parentheses, the exponent does not distribute.

-2^4 ≠ (-2)^4

Any negative base will follow the same pattern as -1. That is, negative bases raised to an odd exponent will be negative, and any negative bases raised to an even exponent will be positive.

Normally you can't do much with two exponential terms that are added/subtracted to each other (we haven't yet seen an example of this in previous flash cards!).

BUT, if the exponential terms have the SAME BASE, then you can factor out a common term.

Equations with odd exponents only have 1 solution...NOT 2 as you get when, e.g., you square something

You must be careful when dealing with equations that have the same base (or could have the same base) on both sides of the equation.

Tricky exponents

What is (1/8) ^ (-4/3)?

You can simplify roots by either combining or separating them in multiplication and division.

sqrt(25×16) = sqrt(25) × sqrt(16) = 5×4 = 20

sqrt(50) × sqrt(18) = sqrt(50×18) = sqrt(900) = 30

sqrt(144÷16) = sqrt(144) ÷ sqrt(16) = 12÷4 = 3

sqrt(72) ÷ sqrt(8) = sqrt (72÷8) = sqrt(9) = 3

But, you CAN'T combine or separate roots in addition and subtraction.

Memorize square and square roots

The GMAT will try to trick you by disguising quadratics - that is, they'll put it in a form that doesn't look like ax^2 + bx + c = 0.

Example #1: Solve for x, given that 36/x = x-5

Example #2: Another example: 3x^2 = 6x.

** How do you solve this example??

Some quadratics are easier to solve when you do NOT set one side equal to 0 -- that is, when the other side of the quadratic is a perfect square. To solve, simply take the square root of both sides of the equation.

e.g. if (x+3)^2 = 25, what is x?

Remember when dealing with quadratics and finding solutions that you CAN'T allow any division by 0. Thus, no solutions can yield a denominator of 0.

Never forget to completely set one side of a quadratic equation equal to 0. Otherwise, I'll miss solutions

How would I solve (x^2 + 6x + 9) / (x+3) = 7?

What's the methodology (don't need the answer)?

+ Adding constants to both sides of an inequality --> this is allowed

+ Adding variables to both sides --> this is allowed

+ Multiplying/dividing by a positive number on both sides of an inequality --> this is allowed

+ Multiply/dividing by a negative number on both sides of an inequality --> This is NOT ALLOWED! If you do so, you MUST FLIP the inequality sign.

You can combine 2 inequalities by adding them together. In order to add them together, you have to ensure that the inequalities are facing the same side.

Question: Is a+2b < c+2d?

(1) a < c

(2) d > b

+ You can ADD inequalities together - in fact this is a powerful method on the GMAT

+ HOWEVER< you can't SUBTRACT or DIVIDE two inequalities.

+ FURTHERMORE, you can only MULTIPLY inequalities together under certain conditions

Inequalities - adding them up (3/3)

Is mn < 10?

(1) m < 2

(2) n < 5

Question:

If r > 0 and s > 0, is r/s < s/r ?

(1) r/3s = 1/4

(2) s = r+4

Fractions

You must know what happens when you raise a fraction to a power.

The results depends on the SIZE and the SIGN of the fraction.

*I will need to regenerate the following scenarios on the GMAT

Any positive proper fraction raised to a power greater than 1 will result in a number smaller than the original fraction.

Any positive proper fraction raised to a power between 0 and 1 will result in a number larger than the original fraction.

Whenever you see 2 variables being multiplied together and you need to know if one of the variables is positive/negative, then simply make a table and do positive/negative analysis!

That's the only way I can prevent myself from getting confused and selecting the wrong answer!

You can either take an algebraic solve-the-formula approach, or you can take a positive/negative analysis approach.

The solutions for a quadratic formula ax^2 + bx + c = 0 are:

x = [-b ± sqrt(b^2 - 4ac)] / 2a

If you have a sqrt in the denominator, it's easy to simplify (just multiply it out)...but if you have an expression involving a sqrt in the denominator, it becomes tricky.

e.g. 4 / [3-sqrt(2)]

Remember that the "domain" is the possible inputs to a formula. The "range" is the possible outputs.

y = kx

y/x = k

Inverse proportions

y = k * (1/x)

xy = k

I need to know BOTH of these equations so I can work these problems on the GMAT in under 2 minutes!!!

You can recognize an exponential growth question when it says something like "the ratio of values in any 2 consecutive years is constant", which implies exponential growth.

Also when it says something like "by what factor...", you can infer that it's an exponential growth question since you are dealing with a multiplier of some original quantity.

If something increases 8-fold in an hour, then how much does it increase in 10 minutes?

What are the two ways I can solve this question???

If you have a problem where you have to find the maximum (or minimum) possible value, then create a table that tests the extreme scenarios for x and y (that is, the situations where x and y are minimimzed/maximized)

Inequalities become particularly tricky when you're dealing with one of two scenarios:

(a) taking the reciprocal of an inequality

(b) squaring an inequality

Taking reciprocals of inequalities is similar to multiplying/dividing by negative numbers.

You need to consider the positive and negative cases of the variables involved.

(3) If x is negative and y is positive, then 1/x < 1/y --> DO NOT flip the inequality

e.g. if -6 < 7, then -1/6 < 1/7.

(4) If you don't know the signs of x and y, you can't take the reciprocal

As with reciprocals, you can't square both sides of an inequality unless you know the signs of both sides of the inequality.

However, the rules for squaring inequalities is different than those for reciprocating inequalities.

(2) If both sides are known to be positive, then don't flip the inequality sign when you square.

e.g. if x > 3, then you can say x^2 > 9 no problem -- after all, you know that both sides are positive.

BUT if x < 3, then you can't say x^2 < 9 (i.e. you can't square both sides) since you don't know if x is positive (in which case x^2 < 9) or if x is negative (in which case x^2 can be greater or less than 9)

When working inequality problems, I must be sure to write everything down. These problems become tricky when you multiply across and have to flip signs, deal with positive/negatives, etc.

e.g. 4/x < -1/3

How do you solve this??

You can't divide a variable out unless you're sure it's not 0.

i.e. it’s an illegal operation if the variable can possibly equal zero. Conversely, if you know FOR SURE that y is not zero, then you can divide by y.

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