Quant-Technical Flashcards
|3x – 7| = 5
3x – 7 = 5 or 3x – 7 = -5
3x = 12 or 3x = 2
x = 4 or x = 2/3
-1 < x < 9 as an absolute value inequality
The first step is to find the midpoint of the region: 4 is exactly halfway between -1 and 9.
Now the distances: 9 is a distance of 5 from 4, and so is -1.
So the distance from 4 (viz. |x – 4|) can’t equal 5, but it can be anything up to 5. Thus
|x – 4|< 5 is the absolute value inequality representation of the region -1 < x < 9
If x + |x| + y = 7 and x + |y| – y = 6 , then x + y =
Determine whether x is positive or negative, and whether y is positive or negative. This will help us simplify the initial equations.
To test whether y is positive or negative, we’ll first assume that y is positive. So we’ll plug in a positive number for y. We’ll use “pos” to represent that positive number:
x + |y| – y = 6
x + |pos| – (pos) = 6
We know that if we take the absolute value of a positive number, and subtract that same positive number from that, we get 0. So the terms |pos| and – (pos) cancel out, and we are left with:
x = 6
Now, to test whether this makes sense, let’s plug x=6 into our first equation:
x + |x|+ y = 7
6 + |6|+ y = 7
12 + y = 7
y = –5
So our result shows us that y is negative. But this is a contradiction, because before we assumed that y was positive. So this means that y cannot be positive; we can conclude that Y MUST BE NEGATIVE
Now let’s examine x. Let’s assume that x is a negative number, and see what happens when we plug this into our first equation. We’ll use “neg” to represent a negative number:
x + |x| + y = 7
(neg) + |neg| + y = 7
Here, the terms (neg) and |neg| will cancel out. This is because we are adding a negative number (neg) to the absolute value of that negative number, which is actually positive (because of the absolute value). The result of this is zero. So:
(neg) + |neg| + y = 7
y = 7
However, we earlier determined that y must be a negative number, so here we have a contradiction (since we found that y = 7, which is positive). So therefore, x cannot be negative. Hence we can conclude that X IS POSITIVE.
Now that we know this, we can simplify our equations.
Since x is positive, we know that |x| = x. (Keep in mind that if we take the absolute value of a positive number, the number stays the same). So the equation x + |x| + y = 7 can be simplified to x + x + y = 7, which is 2x + y = 7.
And since we know that y is negative, the equation x + |y| – y = 6 can be simplified to x – 2y = 6. (EXPLAINED BELOW)
Now we have a system of equations:
2x + y = 7
x – 2y = 6
We can plug this value of x into the first equation:
2x + y = 7
2(6 + 2y) + y = 7
12 + 4y + y = 7
12 + 5y = 7
5y = –5
y = –1
Now we can plug this value of y into x = 6 + 2y, to get x = 6 – 2, and x =4.
Therefore x + y = 4 - 1 = 3
**FAQ: why x + |y| – y = 6 doesn’t become x + 2y = 6?
We have a variable y that must represent a negative number:
y = neg
Let’s now replace “y” with “neg” in our equation:
x + |y| – y = 6
x + |neg| – neg = 6
The absolute value of that negative number must result in the positive version of that number. Likewise, subtracting that negative number must result in the positive version of that number. So we end up with the following where “pos” represents the positive version of that number:
x + pos + pos = 6
x + 2 × pos = 6
but y cannot be pos so we can’t sub in y to this above quation, but to make it into a negative multiply by -1 so it equals x + 2 x neg = 6 -> x -2neg = 6 -> x-2y=6
https://gmat.magoosh.com/questions/131/a/413137129#text_explanation
What does it mean to “Express y in terms of x”?
The command “Express y in terms of x” means: get y alone, all by itself, on one side of the equation, and the other side should be some algebraic expression involving x.
Can you divide 5x=x^2 by x?
No. You cannot divide by a variable if the variable could equal zero. Instead, you have to group on one side and factor out the x:
5x= x2
0= x2-5x=x(x-5)
x=0 or x=5
Is this true or false?
root(P+Q) = root(P) + root(Q)
False, we cannot separate radicals by addition
True of False
Given that P >0 and Q>0
root(P x Q) = root(P) x root(Q)
True. We can separate radicals by multiplication.
True or False
If x^3 > 0, then x>0
True.
The cube of a number has the same sign (negative to negative, and positive to positive), so if the cube of x is positive, then x has to be positive.
The ratio 3 to 5 can be written in what two other ways?
3:5 or 3/5
True of False
1/(A/B) = (1/A) / (1/B)
True. Both sides of the equation simplify to just B/A.
a 50% increase, followed by a 50% decrease, is equivalent to what single percent change overall?
25% decrease
ex: start at 100
100+50% increase = 150
150-50% decrease=75
What is the multiplier for a 30% decrease?
Change 30% to a decimal, 0.30, and subtract this from 1.
1 – 0.30 = 0.70
The decimal 0.70 is the multiplier for a 30% decrease. We can decrease anything by 30% when we multiply it by 0.70.
If 120 Gadgets cost $44, then, at the same price, 180 gadgets would cost what?
180/120 = x/44
(44*180)/120=x
simplify
(44*18)/12=x
(44*3)/2=x
(22*3)=x
66=x
ANOTHER WAY OF DOING IT
180 is 50% greater than 120, so the cost should be 50% greater than $44.
half of $44 is $22, and 44 + 22 = $66.
What is the formula for percent change (percent increase or percent decrease)?
(new-original)/original x100%
=change/original x100%
Whats the difference between a ratio and a proportion?
A ratio is a single fraction. A proportion is an equation of the form: fraction equal fraction.
If |x| < 20 and |x – 8| > |x + 4|, which of the following expresses the allowable range for x?
(A) –12 < x < 12
(B) –20 < x < 2
(C) –20 < x < –12 and 12 < x < 20
(D) –20 < x < –8 and 4 < x < 20
(E) –20 < x < –4 and 8 < x < 20
https://magoosh.com/gmat/absolute-value-inequalities/
|x – 8| > |x + 4|. All this says is that we are looking for points such that the distance to x = 8 is greater than the distance from x = –4; in other words, we want all the points that are closer to x = –4 and farther from x = 8.
The midpoint between x = –4 and x = 8 is the point x = 2. This point is not included because it’s equidistant from both points, but everything to the left of this point on the number line is closer to x = –4 than it is to x = 8.
Combine that with the first inequality, |x| < 20, which in the negative realm means that x must be greater than –20. Thus, the allowed region is –20 < x < 2.
https://magoosh.com/gmat/absolute-value-inequalities/
B
What is a prime number? Is 1 a prime number?
a whole number divisible by only 1 and itself.
1 is not a prime since it only has one divisor
The axis of symmetry of y=ax^2 + bx + c is?
x = -b / 2a
what are the prime numbers less than 100?
What is the easy way of memorizing this?
2, 3,5,7,
11, 13, 17, 19,
23, 29,
31, 37,
41, 43, 47
53, 59
61, 67
71, 73, 79
83 89
97
HOW TO MEMORIZE
1. Write down the first three rows as done above
- Continue pattern with the 23,29 by adding 30 twice, thus 53,59 83,89
- Continue pattern with 11 and 17 by writing 31,37 61,67 and 97 (NEED TO REMEMBER NO 91)
- Continue pattern with 11, 13,17 by writing 41,43,47 and 71,73 and then write 79 instead of 77
What is the prime factorization of 75 (using exponential notation)?
Step 1. Find the multiplication factor that uses the lowest prime number that goes into 75
3 * 25 = 75
Step 2. Break down that equation further by doing the same as step 1 on the non-prime number until can’t anymore
3 * 25 = 75 3 * (5*5) = 75
Step 3. Convert to exponential notation
ANSWER = 3 x (5^2)
