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The point here is that square root function cannot give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Hi Bunuel,
I have got lil confused here.. if x^2=25 => x=sqrt(25) => x=5 right ? then why x^2=25 => +5,-5 if the above can be written as x^2=25 => x=sqrt(25) => x=5 ? Please help me understand this !!
Thanks a ton in advance
\(x^2=25\) has two solutions: \(x=\sqrt{25}=5\) and \(x=-\sqrt{25}=-5\). But, \(\sqrt{25}=5\), because square root function cannot give negative result.
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Hi Bunuel,
In this step: \(|\frac{15}{4}|-|-\frac{3}{5}|=\frac{15}{4}-\frac{3}{5}\)
How did you take \(|-\frac{3}{5}| as \frac{3}{5}\) because \(\sqrt{(y-1)^2}\)= |y-1| and now |y-1| can have two values (y-1), if y-1>0 => y>1 or -(y-1) if y-1<0 => y<1 .... and here y is given as 2/5 (y<1), that means value of \(\sqrt{(y-1)^2}\) = |y-1| = -(y-1) that means \(|\frac{15}{4}|-|-\frac{3}{5}|=\frac{15}{4}-(-\frac{3}{5}) = \frac{15}{4}+\frac{3}{5} = \frac{87}{20}\)
please, let me know if I am doing anything wrong here.
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Hi Bunuel,
In this step: \(|\frac{15}{4}|-|-\frac{3}{5}|=\frac{15}{4}-\frac{3}{5}\)
How did you take \(|-\frac{3}{5}| as \frac{3}{5}\) because \(\sqrt{(y-1)^2}\)= |y-1| and now |y-1| can have two values (y-1), if y-1>0 => y>1 or -(y-1) if y-1<0 => y<1 .... and here y is given as 2/5 (y<1), that means value of \(\sqrt{(y-1)^2}\) = |y-1| = -(y-1) that means \(|\frac{15}{4}|-|-\frac{3}{5}|=\frac{15}{4}-(-\frac{3}{5}) = \frac{15}{4}+\frac{3}{5} = \frac{87}{20}\)
please, let me know if I am doing anything wrong here.
thanks
-K
Without any connection to where it came from, \(|-\frac{3}{5}|=\frac{3}{5}\). Absolute value expresses distance and can never be negative. \(|y-1|\) can never have two values. \(|8|=8\), while \(|-8|=-(-8)=8\). So, \(|y-1|=|2/5-1|=1-2/5=3/5\) or \(|2/5-1|=|-3/5|=3/5.\) _________________
PhD in Applied Mathematics Love GMAT Quant questions and running.
If \(x<0\), then \(|x|=-x\). \(|-8|=8=-(-8)\). If \(x>0\), then \(|x|= x. \,\,|0|=0.\) \(|x|\) means the distance on the number line between \(x\) and \(0.\) Distance between \(5\) and \(0\) is the same as the distance between \(-5\) and \(0.\) So, when the number is positive, absolute value of it is the number itself. When the number is negative, the absolute value of the number is that number without the minus sign. There is no mathematical operation of "drop the sign of the negative number." But if we multiply a negative number by \(-1\), we get that number without its negative sign. _________________
PhD in Applied Mathematics Love GMAT Quant questions and running.
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Hi Bunuel,
I have got lil confused here.. if x^2=25 => x=sqrt(25) => x=5 right ? then why x^2=25 => +5,-5 if the above can be written as x^2=25 => x=sqrt(25) => x=5 ? Please help me understand this !!
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Bunuel, another dubious question. . If I substitue the value of x and y here, then I dont have to take the mod thing because root of square of a number is always positive and the value results in 87/20 \(\sqrt{x^2 +6x +9}-\sqrt{y^2 -2y +1}=\sqrt{(x+3)^2}-\sqrt{(y-1)^2}\) _________________
hope is a good thing, maybe the best of things. And no good thing ever dies.
The point here is that square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
About \(\sqrt{x^2}=|x|\): from above we have that \(\sqrt{x^2}\geq{0}\). 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 \(\sqrt{x^2}=|x|\).
Hope it's clear.
Bunuel, another dubious question. . If I substitue the value of x and y here, then I dont have to take the mod thing because root of square of a number is always positive and the value results in 87/20 \(\sqrt{x^2 +6x +9}-\sqrt{y^2 -2y +1}=\sqrt{(x+3)^2}-\sqrt{(y-1)^2}\)
There is nothing wrong with this questions as well! The answer is 63/20. Check your math. _________________
Square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
Square root function can not give negative result --> \(\sqrt{some \ expression}\geq{0}\), for example \(\sqrt{x^2}\geq{0}\) --> \(\sqrt{25}=5\) (not +5 and -5). In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5, because both 5^2 and (-5)^2 equal to 25.
Re: If x=3/4 and y=2/5, what is the value of [#permalink]
10 Jul 2014, 08:01
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Re: If x=3/4 and y=2/5, what is the value of [#permalink]
12 Sep 2015, 09:29
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________
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