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Bunuel
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Bunuel
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Hello,

Where can I get some theory and a lot of practice questions on absolute numbers.

Thank you.
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Bunuel
Official Solution:


(1) \((x - 1)^2 \le 1\). Since both sides of the inequality are non-negative then we can take square root from both parts: \(|x-1| \le 1\), so \(|x-1|\) can be less than 1 (answer YES), as well as equal to 1, for \(x=2\) or \(x=0\) (answer NO). Not sufficient. Notice that \(|x-1| \le 1\), means \(0 \le x \le 2\).

(2) \(x^2 - 1 \gt 0\). Rearrange: \(x^2 \gt 1\). Again, since both sides of the inequality are non-negative then we can take square root from both parts: \(|x| \gt 1\). then the answer is YES but if \(x=2\) then the answer is NO. Not sufficient.

(1)+(2) \(x=1.5\) and \(x=2\) satisfy both statements and give different answer to the question. Not sufficient.


Answer: E

Hi!

Can somebody please explain me how did we get this highlighted part? I did this question via Critical Key Point method and got it right. Trying to understand how
x^2>1 ---> |x| > 1

Thankyou!
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578vishnu
Bunuel
Official Solution:


(1) \((x - 1)^2 \le 1\). Since both sides of the inequality are non-negative then we can take square root from both parts: \(|x-1| \le 1\), so \(|x-1|\) can be less than 1 (answer YES), as well as equal to 1, for \(x=2\) or \(x=0\) (answer NO). Not sufficient. Notice that \(|x-1| \le 1\), means \(0 \le x \le 2\).

(2) \(x^2 - 1 \gt 0\). Rearrange: \(x^2 \gt 1\). Again, since both sides of the inequality are non-negative then we can take square root from both parts: \(|x| \gt 1\). then the answer is YES but if \(x=2\) then the answer is NO. Not sufficient.

(1)+(2) \(x=1.5\) and \(x=2\) satisfy both statements and give different answer to the question. Not sufficient.


Answer: E

Hi!

Can somebody please explain me how did we get this highlighted part? I did this question via Critical Key Point method and got it right. Trying to understand how
x^2>1 ---> |x| > 1

Thankyou!

Recall that \(\sqrt{x^2}=|x|\).

Why?

The point here is that square root function cannot give negative result: which means that \(\sqrt{some \ expression}\geq{0}\).

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! That is why \(\sqrt{x^2}=|x|\)
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Hi Bunuel,

Can you help me with breaking the questions stem?

|x−1|<1 means 0 < x < 2, Is that correct?

then from statement 1, we get 0 <= x <= 2
and statement 2, x > 1 or x < -1,

statement 1 and statement 2 individually are not possible, but if we combine them we do get 1.5 which matches the original stem? Can we not conclude that by combining both the statements we get a solution?

Regards,
RK
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Rajaking
Hi Bunuel,

Can you help me with breaking the questions stem?

|x−1|<1 means 0 < x < 2, Is that correct?

then from statement 1, we get 0 <= x <= 2
and statement 2, x > 1 or x < -1,

statement 1 and statement 2 individually are not possible, but if we combine them we do get 1.5 which matches the original stem? Can we not conclude that by combining both the statements we get a solution?

Regards,
RK

Is |x − 1| < 1 ?
Is -1 < x − 1 < 1 ?
Is 0 < x < 2 ?

(1) (x - 1)^2 <= 1 gives: 0<=x<=2


(2) x^2 - 1 > 0 gives: x < -1 or x > 1

So, when combining the range is 1 < x <=2.

If 1 < x < 2, then the answer to the question whether 0 < x < 2 is YES.
If x = 2, then the answer to the question whether 0 < x < 2 is NO.

Two different answers, hence E.

Hope it's clear.
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this has to be one of my favorite questions, thank you Bunuel.
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Bunuel
Is \(|x - 1| < 1\) ?



(1) \((x - 1)^2 \leq 1\)

(2) \(x^2 - 1 > 0\)

Simplifying the given question : is |x-1|<1
it basically asks us if 0<x<2

Rough workings:
|x-1|<1
So a) x-1<1
x<2

or
-x+1<1
x>0

Back to question:

1) \((x - 1)^2 \leq 1\)
Simplifying we get x<=0 or x<=2
So clearly not sufficient

2) \(x^2 - 1 > 0\)
simplifying we get x>1 or x>-1

So we can see individually or together, we dont get the answer we're looking for.
Hence E (none)
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