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# If x < 12, then it must be true that

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Re: If x < 12, then it must be true that [#permalink]
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janani28 wrote:
Hi Bunuel,

x<12

From E,we get x<13 ,what if x is 12 .This contradicts the question x<12 right ?

It's the other way around. We are told that x is less than 12 (it's given as a fact) and asked which of the options is true. Since x is less than 12, then it must be less than 13 too.
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Re: If x < 12, then it must be true that [#permalink]
jlgdr wrote:
If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

E wil always hold true.
But so will B???

if x=10, -12 <14 true.
if x=0, -2<14 true
if x=5, -7<14 true

Can someone pls explain where I am going wrong.
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Re: If x < 12, then it must be true that [#permalink]
jlgdr wrote:
If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

E wil always hold true.
But so will B???

if x=10, -12 <14 true.
if x=0, -2<14 true
if x=5, -7<14 true

Can someone pls explain where I am going wrong.

B gives: x > -16. If x is less than -16, B won't be right. For example, if x = -100 < 12, then B is not true.
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Re: If x < 12, then it must be true that [#permalink]
Bunuel wrote:
jlgdr wrote:
If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

A. -x < -12 --> x>12. Not true.

B. -x - 2 < 14 --> x>-16. Not true, consider x=-20.

C. -x + 2 < -10 --> x>12. Not true.

D. x + 2 < 10 --> x<8. Not true, consider x=10.

E. x - 2 < 11 --> x<13. Since given that x<12, then x is for sure less than 13. Thus this statement must be true.

x<12
A. x<12
-x>-12

B. x<12
-x>-12 , -x -2 > -14

C. -x + 2 > -10

D. x + 2 < 14

E. x < 12 , So, x must be <13. This statement must be true.

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Re: If x < 12, then it must be true that [#permalink]
jlgdr wrote:
If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

Let’s analyze each choice.

A) -x < -12

If we multiply both sides of x < 12 by -1, we will have -x > -12 (notice that we have to switch the inequality sign since we’ve multiplied by a negative number). So A is not correct.

B) -x - 2 < 14

Starting with x < 12, we have:

x < 12

-x > -12

-x - 2 > -14

We see that B is not correct either.

C) -x + 2 < -10

x < 12

-x > -12

-x + 2 > -10

We see that C is not correct either.

D) x + 2 < 10

x < 12

x + 2 < 14

We see that D is not correct either. So the correct answer must be E. But let’s verify it anyway.

E) x - 2 < 11

x < 12

x - 2 < 10

If x - 2 < 10, then x - 2 is less than 11 also. So E is correct.

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Re: If x < 12, then it must be true that [#permalink]
If x < 12, then it must be true that

(a) -x < -12

x > 12; not must be true.

(b) -x - 2 < 14

x > 16; not must be true.

(c) -x + 2 < -10

x > 12; not must be true.

(d) x + 2 < 10

x < 8; not must be true.

(e) x - 2 < 11

x < 13. This choice must be true. If x < 12, then x < 13.
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Re: If x < 12, then it must be true that [#permalink]
To remove the negative sign, when we multiply the inequality by (-1) on both sides, the sign of inequality gets flip. So, A and C are eliminated as those options say x > ??

Option B says x > -16 not necessarily TRUE if x = -18 which is less than 12.

Option D says x < 8 not necessarily TRUE if x = 11 which is less than 12.

Option E says x < 13 is TRUE because if x is less than 12 it has to be less than 13.

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Re: If x < 12, then it must be true that [#permalink]
jlgdr wrote:
If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

Given x < 12
The only thing it tells me for sure is that x is less than 12 in every case so it is less than 13 and less than 14 and less than 15 etc in every case too. So I am looking for something like x < 13 or x < 14 etc

I ignore options where x has a negative sign since making it positive will flip the inequality.
Option (D) gives x < 8 and option (E) gives x < 13.

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If x < 12, then it must be true that [#permalink]
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jlgdr wrote:
If x < 12, then it must be true that

(A) -x < -12
(B) -x - 2 < 14
(C) -x + 2 < -10
(D) x + 2 < 10
(E) x - 2 < 11

In this case, we can easily test values that satisfy the given inequality.
From here, I'll give myself 15-20 seconds to identify a faster approach.....
We can also solve the question by analyzing each answer choice to see if it adheres to the given inequality algebraically, but since I'm less likely to make mistakes testing values, I'll go that route

When looking for a value to test, I typically start with 0, since it's such an easy value to work with.

Since it turns out that x = 0 is solution to the inequality x < 12, let's plug x = 0 into each answer choice to see if it's a solution:
(A) -0 < -12, which simplifies to 0 < -12. Not true. Eliminate A.
(B) -0 - 2 < 14, which simplifies to -2 < 14. True. KEEP.
(C) -0 + 2 < -10, which simplifies to 2 < -10. Not true. Eliminate C.
(D) 0 + 2 < 10. True. KEEP.
(E) 0 - 2 < 11. True. KEEP.

We're down to choices B, D and E

Now let's test an "extreme" possible value that satisfies the inequality x < 12
We can choose a value close to 12 (like x = 11.5) or a value far away (like x = -20)

Let's test x = 11.5, by plugging it into the remaining three answer choices:
(B) -11.5 - 2 < 14, which simplifies to -13.5 < 14. True. KEEP.
(D) 11.5 + 2 < 10. Doesn't work. Eliminate D
(E) 11.5 - 2 < 11. True. KEEP.

We're down to choices B and E

Now let's test another extreme value, x = -20:
(B) -(-20) - 2 < 14, which simplifies to 18 < 14. Doesn't work. Eliminate B
(E) -20 - 2 < 11. True. KEEP.

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Re: If x < 12, then it must be true that [#permalink]
If x < 12, then

(a) -x < -12. =>. x >12 (not true)
(b) -x - 2 < 14. => -x <16 => x>-16 (not true)
(c) -x + 2 < -10 => -x < -12 => x>12 (not true)
(d) x + 2 < 10 => x< 8 (not true)
(e) x - 2 < 11 => x<13 (correct answer)
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Re: If x < 12, then it must be true that [#permalink]
BrentGMATPrepNow wrote:
jlgdr wrote:
If x < 12, then it must be true that

(A) -x < -12
(B) -x - 2 < 14
(C) -x + 2 < -10
(D) x + 2 < 10
(E) x - 2 < 11

In this case, we can easily test values that satisfy the given inequality.
From here, I'll give myself 15-20 seconds to identify a faster approach.....
We can also solve the question by analyzing each answer choice to see if it adheres to the given inequality algebraically, but since I'm less likely to make mistakes testing values, I'll go that route

When looking for a value to test, I typically start with 0, since it's such an easy value to work with.

Since it turns out that x = 0 is solution to the inequality x < 12, let's plug x = 0 into each answer choice to see if it's a solution:
(A) -0 < -12, which simplifies to 0 < -12. Not true. Eliminate A.
(B) -0 - 2 < 14, which simplifies to -2 < 14. True. KEEP.
(C) -0 + 2 < -10, which simplifies to 2 < -10. Not true. Eliminate C.
(D) 0 + 2 < 10. True. KEEP.
(E) 0 - 2 < 11. True. KEEP.

We're down to choices B, D and E

Now let's test an "extreme" possible value that satisfies the inequality x < 12
We can choose a value close to 12 (like x = 11.5) or a value far away (like x = -20)

Let's test x = 11.5, by plugging it into the remaining three answer choices:
(B) -11.5 - 2 < 14, which simplifies to -13.5 < 14. True. KEEP.
(D) 11.5 + 2 < 10. Doesn't work. Eliminate D
(E) 11.5 - 2 < 11. True. KEEP.

We're down to choices B and E

Now let's test another extreme value, x = -20:
(B) -(-20) - 2 < 14, which simplifies to 18 < 14. Doesn't work. Eliminate B
(E) -20 - 2 < 11. True. KEEP.

Great explanation BrentGMATPrepNow. In regard to D, if solving as x + 2 < 10 --> x<8. This mean the greatest of x will only be 7. Not sure why is this not true?
E, x<13 which mean x could be 12, isn't this go against given x <12 ? Thanks Brent
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If x < 12, then it must be true that [#permalink]
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Kimberly77 wrote:
BrentGMATPrepNow wrote:
jlgdr wrote:
If x < 12, then it must be true that

(A) -x < -12
(B) -x - 2 < 14
(C) -x + 2 < -10
(D) x + 2 < 10
(E) x - 2 < 11

In this case, we can easily test values that satisfy the given inequality.
From here, I'll give myself 15-20 seconds to identify a faster approach.....
We can also solve the question by analyzing each answer choice to see if it adheres to the given inequality algebraically, but since I'm less likely to make mistakes testing values, I'll go that route

When looking for a value to test, I typically start with 0, since it's such an easy value to work with.

Since it turns out that x = 0 is solution to the inequality x < 12, let's plug x = 0 into each answer choice to see if it's a solution:
(A) -0 < -12, which simplifies to 0 < -12. Not true. Eliminate A.
(B) -0 - 2 < 14, which simplifies to -2 < 14. True. KEEP.
(C) -0 + 2 < -10, which simplifies to 2 < -10. Not true. Eliminate C.
(D) 0 + 2 < 10. True. KEEP.
(E) 0 - 2 < 11. True. KEEP.

We're down to choices B, D and E

Now let's test an "extreme" possible value that satisfies the inequality x < 12
We can choose a value close to 12 (like x = 11.5) or a value far away (like x = -20)

Let's test x = 11.5, by plugging it into the remaining three answer choices:
(B) -11.5 - 2 < 14, which simplifies to -13.5 < 14. True. KEEP.
(D) 11.5 + 2 < 10. Doesn't work. Eliminate D
(E) 11.5 - 2 < 11. True. KEEP.

We're down to choices B and E

Now let's test another extreme value, x = -20:
(B) -(-20) - 2 < 14, which simplifies to 18 < 14. Doesn't work. Eliminate B
(E) -20 - 2 < 11. True. KEEP.

Great explanation BrentGMATPrepNow. In regard to D, if solving as x + 2 < 10 --> x<8. This mean the greatest of x will only be 7. Not sure why is this not true?
E, x<13 which mean x could be 12, isn't this go against given x <12 ? Thanks Brent

You are reading the question backwards.
We are told that x < 12. This part is guaranteed.
This means x COULD equal values such as 8, 11.5, 7, 0, -500, 11.999999, 5, 4.1, etc.

Given this wide range of possible x-values, we're looking for an answer choice that is must be true for all possible values of x (i.e., 8, 11.5, 7, 0, -500, 11.999999, 5, 4.1, etc.)

(D) x + 2 < 10
When we simplify D, we get x < 8.
Given all of the possible values of x (i.e., 8, 11.5, 7, 0, -500, 11.999999, 5, 4.1, etc.), must it be true that x < 8?
No.
Notice that, among the possible values of x, we have numbers like 8, 11.5 and 11.999999, all of which are larger than x.
So we certainly can't conclude that x MUST be less than 8.

(E) x - 2 < 11
When we simplify E, we get x < 13.
Given all of the possible values of x (i.e., 8, 11.5, 7, 0, -500, 11.999999, 5, 4.1, etc.), must it be true that x < 13?
Yes.
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Re: If x < 12, then it must be true that [#permalink]
Brilliant explanation always. Thanks BrentGMATPrepNow
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