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# If y is an integer, how many values of y are possible such that 7<(y-1

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Joined: 23 Feb 2015
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If y is an integer, how many values of y are possible such that 7<(y-1  [#permalink]

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19 Oct 2019, 09:05
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Difficulty:

65% (hard)

Question Stats:

53% (01:39) correct 47% (01:45) wrong based on 36 sessions

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If y is an integer, how many values of y are possible such that $$7<(y-1)(y-1)<49$$?
A) 10
B) 8
C) 6
D) 5
E) 3

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Joined: 02 Aug 2009
Posts: 8167
If y is an integer, how many values of y are possible such that 7<(y-1  [#permalink]

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19 Oct 2019, 19:41
If y is an integer, how many values of y are possible such that $$7<(y-1)(y-1)<49$$?
A) 10
B) 8
C) 6
D) 5
E) 3

$$7<(y-1)(y-1)<49$$.....$$7<(y-1)^2<7^2$$....

(I) $$(y-1)^2>7......y-1>|\sqrt{7}|$$,
As y is an integer, y-1 will always be an integer, so we can take the next integer value greater than $$\sqrt{7}$$, which will be $$\sqrt{9}$$
so $$y-1\geq|9|.....y-1\geq{3}......-3\geq{y-1}......y-1\geq{3}$$, so $$y\leq{-2}$$ and $$y\geq{4}$$

(II) $$(y-1)^2<49......y-1<|\sqrt{49}|$$,
$$-7<y-1<7.....-6<y<8$$

Combining two ranges we get
$$-7<y\leq{-2}....y=-5,-4,-3,-2$$---FOUR negative values
$$4\leq{y}<8.......y=4,5,6,7$$-----FOUR positive values

Total = 4+4=8

B
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Re: If y is an integer, how many values of y are possible such that 7<(y-1  [#permalink]

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19 Oct 2019, 21:36
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If y is an integer, how many values of y are possible such that $$7<(y-1)(y-1)<49$$?
A) 10
B) 8
C) 6
D) 5
E) 3

Asked: If y is an integer, how many values of y are possible such that $$7<(y-1)(y-1)<49$$?

$$2<\sqrt{7}<|y-1|<\sqrt{49}=7$$
2<|y-1|<7

|y-1|>2
y>3 or y<-1

|y-1|<7
-6<y<8

Combining
3<y<8; y = {4,5,6,7}
-6<y<-1; y = {-5,-4,-3,-2}
y = {-5,-4,-3,-2, 4,5,6,7}

IMO B
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Re: If y is an integer, how many values of y are possible such that 7<(y-1   [#permalink] 19 Oct 2019, 21:36
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