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# If x2 - 100 < 300, how many integers x satisfy this condition?

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Intern
Joined: 26 Apr 2018
Posts: 38
If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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01 Oct 2018, 07:18
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If x^2 - 100 < 300, how many integers x satisfy this condition?

a) 42

b) 39

c) 38

e) 37

d) 19

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Joined: 02 Aug 2009
Posts: 8593
Re: If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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01 Oct 2018, 08:23
bettatantalo wrote:
If x^2 - 100 < 300, how many integers x satisfy this condition?

a) 42

b) 39

c) 38

e) 37

d) 19

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$$x^2-100<300.....x^2<400....x<|20|.........-20<x<20$$
So x can take any value from -19 to 19..
Total 19 positive integers + 19 negative integers +0=19+19+1=39

B
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Re: If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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01 Oct 2018, 08:24
Top Contributor
bettatantalo wrote:
If x² - 100 < 300, how many integers x satisfy this condition?

a) 42
b) 39
c) 38
e) 37
d) 19

Take: x² - 100 < 300
Add 100 to both sides to get: x² < 400

We know that 20² EQUALS 400, so x must be less than 20
For example, 19² < 400, so one possible solution is x = 19
Likewise, 18² < 400, so another possible solution is x = 18
Keep going to see that...
Another possible solution is x = 17
Another possible solution is x = 16
.
.
.
Another possible solution is x = 1
Another possible solution is x = 0

ARE WE DONE?
NO!

Negative values also work.
For example, (-1)² = 1, and 1 < 400. So another possible solution is x = -1
And (-2)² = 4, and 4 < 400. So another possible solution is x = -2
And (-3)² = 9, and 9 < 400. So another possible solution is x = -3
.
.
.
And (-19)² = 361, and 361 < 400. So another possible solution is x = -19
HOWEVER, (-20)² = 400, and 400 is NOT less than 400.

So, the integer solutions are: -19, -18, -17, . . . . 17, 18, 19
There are 39 such solutions.

Cheers,
Brent
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If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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01 Oct 2018, 10:01
bettatantalo wrote:
If x^2 - 100 < 300, how many integers x satisfy this condition?

a) 42

b) 39

c) 38

e) 37

d) 19

source gmat tutor

$$x^2 - 100 < 300$$ (add 100 to both sides)

$$x^2 - 100+100 < 300+100$$

$$x^2 < 400$$ square both sides

$$x<20$$

So x can be -19 or +19 , because $$\sqrt{x^2}=|x|$$

$$-19<x<19$$

$$19+19 +0 = 39$$ (dont forget to count zero )

IMPORTANT PROPERTIES

1. $$|x|≥0$$

2. $$\sqrt{x^2}=|x|$$

3. $$|0|=0$$

4. $$|−x|=|x|$$

5. $$|x−y|=|y−x|$$ . $$|x - y|$$ represents the distance between $$x$$ and $$y$$, so naturally it equals to $$|y - x|$$, which is the distance between $$y$$ and $$x$$.

6. $$|x|+|y|≥|x+y|$$ Note that "=" sign holds for $$xy≥0$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$

7. |x|−|y|≤|x−y| Note that "=" sign holds for $$xy>0$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|≥|y|$$ (simultaneously).
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Re: If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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01 Oct 2018, 14:14
bettatantalo wrote:
If x^2 - 100 < 300, how many integers x satisfy this condition?

a) 42

b) 39

c) 38

e) 37

d) 19

source gmat tutor

$$x^2<300 +100$$

$$x^2 < 400.$$

Notes: 20^2 = 400. It means Highest value of x must be 19 and lowest value must be -19.

-19<x<19.

As we are dealing with $$x^2$$ , negative value doesn't affect our calculation.

No. of x. : 19 - (-19) +1 = 39.

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Re: If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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03 Oct 2018, 16:34
bettatantalo wrote:
If x^2 - 100 < 300, how many integers x satisfy this condition?

a) 42

b) 39

c) 38

e) 37

d) 19

Simplifying, we have:

x^2 < 400

√(x^2) < √400

|x| < 20

Since all the integers from -19 to 19, inclusive, have an absolute value less than 20, there are 39 integers.

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Re: If x2 - 100 < 300, how many integers x satisfy this condition?  [#permalink]

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02 Apr 2020, 02:41
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Re: If x2 - 100 < 300, how many integers x satisfy this condition?   [#permalink] 02 Apr 2020, 02:41