Hi,
With the difficult Inequality DS questions where by you need to test different values ... does anyone have a rock hard strategy for testing the correct values.
I know you test fractions, zero, negs, positives, HOWEVER with a question like this (from MGMAT CAT):
Is X > Y ?
(1) root(x) > y
(2) X^3 > y
I would try 2 values for each statement (fractions and negs) and realize on their own BOTH are insufficient.
However when testing to see if they are sufficient together, i find it difficult to figure out (quickly) what values to test that will suit both conditions.
For example statement 1
i will try
y= 1/2, y^2= 1/4, x = 1/2 ... (x>y? no)
y = -2, Y^2= 4, x = 5 ... (x>y? yes)
NOT SUFF
statement 2
x = 2, x^3 = 8, y = 7 (x>y? no)
x= 1/2. x^3 = 1/8, y = 1/9 (x>y? yes)
NOT SUFF
(Now I know i need to test for both, however, I find it difficult (within 2 mins) to figure what values to use in order to suit both conditions, and try and figure out if there is only one possible answer)
The answer is C (both)
I will appreciate any help.
Thanks!
DS - Inequalities - Strategy help please
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Since we can't take the square root of a negative, statement 1 implies that x≥0.zueswoods wrote:Hi,
With the difficult Inequality DS questions where by you need to test different values ... does anyone have a rock hard strategy for testing the correct values.
I know you test fractions, zero, negs, positives, HOWEVER with a question like this (from MGMAT CAT):
Is X > Y ?
(1) root(x) > y
(2) X^3 > y
I would try 2 values for each statement (fractions and negs) and realize on their own BOTH are insufficient.
However when testing to see if they are sufficient together, i find it difficult to figure out (quickly) what values to test that will suit both conditions.
For example statement 1
i will try
y= 1/2, y^2= 1/4, x = 1/2 ... (x>y? no)
y = -2, Y^2= 4, x = 5 ... (x>y? yes)
NOT SUFF
statement 2
x = 2, x^3 = 8, y = 7 (x>y? no)
x= 1/2. x^3 = 1/8, y = 1/9 (x>y? yes)
NOT SUFF
(Now I know i need to test for both, however, I find it difficult (within 2 mins) to figure what values to use in order to suit both conditions, and try and figure out if there is only one possible answer)
The answer is C (both)
I will appreciate any help.
Thanks!
Thus, when we combine the two statements, if y<0, we know that y<x.
Our concern is what happens when y≥0.
One approach is to memorize the shapes of some basic graphs:
Only in the yellow region is y<√x and y<x³.
The entire yellow region is below the graph of y=x, implying that y<x throughout the entire region.
Thus, combining the two statements, we know that y<x.
SUFFICIENT.
The correct answer is C.
An alternate approach would be to use algebra to test the 3 cases: y=x, y>x, and y<x.
Case 1: y=x.
Statement 1: If y=x and y<√x, then x < √x.
Statement 2: If y=x and y<x³, then x < x³.
No value for x will work here: a number cannot be less than both its root and its cube.
Thus, y≠x.
Case 2: y>x.
Statement 1: If x<y and y<√x, then x < √x.
Statement 2: If x<y and y<x³, then x < x³.
No value for x will work here: a number cannot be less than both its root and its cube.
Thus, it is not possible that y>x.
Since it is not possible that y=x or that y>x, we know that y<x.
SUFFICIENT.
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If the question mentions √x, then x clearly cannot be negative. When we combine the two statements, then if x is 0 or 1, clearly x > y from either statement. If x is between 0 and 1, then x^3 is smaller than x, and √x is greater than x. So in this case, combining the two statements, we know that √x > x > x^3 > y. If x > 1, then x^3 > x and √x < x, so in this case, combining the two statements, we know that x^3 > x > √x > y. So in every case we find x > y, and the two statements together are sufficient.zueswoods wrote:Hi,
With the difficult Inequality DS questions where by you need to test different values ... does anyone have a rock hard strategy for testing the correct values.
I know you test fractions, zero, negs, positives, HOWEVER with a question like this (from MGMAT CAT):
Is X > Y ?
(1) root(x) > y
(2) X^3 > y
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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honestly this sum at first instance is impossible to solve in 2 mins as you will not be able to come up with the values only...
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Hi GmatguryNY
Hi again,
I wanted to make sure that I have got this technique down.
The question is:
Is x>y?
(1) x^2 > y
(2) root(x) < y
So I drew the graphs as follows
(1) y < x^2
and we see that the yellow region could be above y=x or below so its insufficient
(2) y > root(x)
still insuf.
Now when taking them both together
We still see that the yellow region is above y=x and below y=x, therefore the answer is E.
Did i work through this correctly?
Thanks so much for your time, i was really strugling on these questions
Also, when will this technique not work?
I know with absolute values I will have to work through it and solve, but other than that?
zueswoods
Hi again,
I wanted to make sure that I have got this technique down.
The question is:
Is x>y?
(1) x^2 > y
(2) root(x) < y
So I drew the graphs as follows
(1) y < x^2
and we see that the yellow region could be above y=x or below so its insufficient
(2) y > root(x)
still insuf.
Now when taking them both together
We still see that the yellow region is above y=x and below y=x, therefore the answer is E.
Did i work through this correctly?
Thanks so much for your time, i was really strugling on these questions
Also, when will this technique not work?
I know with absolute values I will have to work through it and solve, but other than that?
zueswoods
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Is X > Y ?
(1) root(x) > y
(2) X^3 > y
Explanation:
Statement 1 Alone:
If you take statement 1, you find that the inequality does not stand when 0 < x < 1. i.e for fractions between 0 and 1
Statement 2 Alone:
If you take statement 2, you find that the inequality does not stand when x > 1;
Both 1 and 2:
Both the statements meet the required conditions...
Yes in order to understand properly, you can refer to the graphical inequalities above...
Hope this helps!!!
(1) root(x) > y
(2) X^3 > y
Explanation:
Statement 1 Alone:
If you take statement 1, you find that the inequality does not stand when 0 < x < 1. i.e for fractions between 0 and 1
Statement 2 Alone:
If you take statement 2, you find that the inequality does not stand when x > 1;
Both 1 and 2:
Both the statements meet the required conditions...
Yes in order to understand properly, you can refer to the graphical inequalities above...
Hope this helps!!!
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Nice work.zueswoods wrote:Hi GmatguryNY
Hi again,
I wanted to make sure that I have got this technique down.
The question is:
Is x>y?
(1) x^2 > y
(2) root(x) < y
So I drew the graphs as follows
(1) y < x^2
and we see that the yellow region could be above y=x or below so its insufficient
(2) y > root(x)
still insuf.
Now when taking them both together
We still see that the yellow region is above y=x and below y=x, therefore the answer is E.
Did i work through this correctly?
Thanks so much for your time, i was really strugling on these questions
Also, when will this technique not work?
I know with absolute values I will have to work through it and solve, but other than that?
zueswoods
In my post above, I offered a graphical solution because y=x, y=x³ and y=√x are common graphs that can be drawn quickly.
The problem you cited also is about common graphs, so a graphical approach works well.
It might be quicker, however, simply to combine the inequalities and plug in:
Combining √x<y and y<x², we get √x < y < x².
If x=100, then 10<y<10,000.
Thus, it's possible that y<x, y=x, or y>x, indicating that the two statements combined are INSUFFICIENT.
Ideally, you'll be comfortable with a variety of approaches.
Some problems are best solved algebraically, others by plugging in values.
Graphing can be helpful, as can drawing a number line.
Many problems are best solved with a combination of techniques.
Here's another problem that I solved graphically:
https://www.beatthegmat.com/xy-plane-and ... 97228.html
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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