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06 Oct 2010, 07:11
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C
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Difficulty:
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70% (01:07) correct
30%
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wrong
based on 757
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06 Oct 2010, 07:21
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agnok
Is |x – 3| < 7 ?
(1) x > 0
(2) x < 10
Please explain
(1) x > 0
(2) x < 10
Please explain
Is \(|x-3|<7\)?
Let's see for which range(s) of \(x\) this inequality holds true. One check point \(x=3\) (check point the value of \(x\) when the expression in || equals to zero), so we should check two ranges:
When \(x<3\) --> \(|x-3|\) becomes \(-x+3\) --> \(-x+3<7\) --> \(-4<x\) --> \(-4<x<3\);
When \(x\geq{3}\) --> \(|x-3|\) becomes \(x-3\) --> \(x-3<7\) --> \(x<10\) --> \(3\leq{x}<10\);
So inequality \(|x-3|<7\) holds true for \(-4<x<10\).
(1) \(x>0\). Not sufficient.
(2) \(x<10\). Not sufficient.
(1)+(2) \(0<x<10\) --> we know that in this range inequality \(|x-3|<7\) holds true (this range is the subset of the range we've got in the beginning). Sufficient.
Answer: C.
29 Oct 2010, 06:02
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You would have read in theory of mods that mod is nothing but the distance from x = 0 on the number line.
This means that if |x| = 4, x is a point at a distance of 4 units from 0. So x could be 4 or -4 (which suits our understanding of mods)
Now, |x – 3| is the distance from the point 3 on the number line. So if I say
|x – 3| = 7, I am looking for points which are at a distance of 7 from point 3. These points will be 10 and -4. These are the solutions of x in this equation.
Coming to our question, |x – 3| < 7 means we are looking for points whose distance from 3 is less than 7. There will be many such points e.g. 4, 5, 6, -2, -1 etc that satisfy our inequality as is apparent from the diagram below:
Ques.jpg [ 5.28 KiB | Viewed 13676 times ]
Points beyond 10 on the right side of the number line and points beyond -4 on the left side of the number line will have a distance of more than 7 and hence, do not satisfy our inequality.
Statement I: x > 0
There are points greater than 0 that satisfy our inequality (e.g. 1, 5, 7 etc) and there are those that do not satisfy our inequality (e.g. 11, 12, 18 etc). Hence this statement is not sufficient.
Statement II: x < 10
Again, using the same logic as above, this statement is not sufficient.
When we combine the two statements, we see that all the points satisfying 0 < x < 10, satisfy our inequality. Hence we can say 'Yes, |x – 3| is less than 7' and we get (C) as our answer.
This means that if |x| = 4, x is a point at a distance of 4 units from 0. So x could be 4 or -4 (which suits our understanding of mods)
Now, |x – 3| is the distance from the point 3 on the number line. So if I say
|x – 3| = 7, I am looking for points which are at a distance of 7 from point 3. These points will be 10 and -4. These are the solutions of x in this equation.
Coming to our question, |x – 3| < 7 means we are looking for points whose distance from 3 is less than 7. There will be many such points e.g. 4, 5, 6, -2, -1 etc that satisfy our inequality as is apparent from the diagram below:
Attachment:
Ques.jpg [ 5.28 KiB | Viewed 13676 times ]
Points beyond 10 on the right side of the number line and points beyond -4 on the left side of the number line will have a distance of more than 7 and hence, do not satisfy our inequality.
Statement I: x > 0
There are points greater than 0 that satisfy our inequality (e.g. 1, 5, 7 etc) and there are those that do not satisfy our inequality (e.g. 11, 12, 18 etc). Hence this statement is not sufficient.
Statement II: x < 10
Again, using the same logic as above, this statement is not sufficient.
When we combine the two statements, we see that all the points satisfying 0 < x < 10, satisfy our inequality. Hence we can say 'Yes, |x – 3| is less than 7' and we get (C) as our answer.
