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Must be True

by greenwich » Fri Jan 17, 2014 5:17 pm
f 4<[(7-x)/3], which of the following must be true?
I. 5<x
II. |x+3|>2
III. -(x+5) is positive

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II and III

For II, x+3>2, x>-1 or -x-3>2, x<-5, can we say II is true even if only one of the two solutions satisfies the MUST be true question?

Brent, for the 3 steps to solve the Absolute Value:

1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Does #3 Plug in the solutions to check for extraneous roots also applies to Absolute Value with Inequality such as II in the above question (|x+3|>2)?
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by Brent@GMATPrepNow » Fri Jan 17, 2014 5:26 pm
greenwich wrote: Brent, for the 3 steps to solve the Absolute Value:

1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Does #3 Plug in the solutions to check for extraneous roots also applies to Absolute Value with Inequality such as II in the above question (|x+3|>2)?
Hi greenwich,

I believe you're referring to the advice I gave in this post: https://www.beatthegmat.com/is-x-0-t273441.html

That advice deals with equations with absolute value, not inequalities.

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by greenwich » Fri Jan 17, 2014 5:39 pm
Brent@GMATPrepNow wrote:
greenwich wrote: Brent, for the 3 steps to solve the Absolute Value:

1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Does #3 Plug in the solutions to check for extraneous roots also applies to Absolute Value with Inequality such as II in the above question (|x+3|>2)?
Hi greenwich,

I believe you're referring to the advice I gave in this post: https://www.beatthegmat.com/is-x-0-t273441.html

That advice deals with equations with absolute value, not inequalities.

Cheers,
Brent
Hi Brent,

For the question above, II is with inequalities. Are steps 1 & 2 sufficient to solve absolute value with inequalities? --->x+3>2, x>-1 or -x-3>2, x<-5

Also, can we say II is true even if only one of the two solutions satisfies the MUST be true question?

Thanks again...
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by Brent@GMATPrepNow » Fri Jan 17, 2014 5:54 pm
greenwich wrote:If 4<[(7-x)/3], which of the following must be true?
I. 5<x
II. |x+3|>2
III. -(x+5) is positive

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II and III
First, let's deal with the given inequality.
4 < (7-x)/3
Multiply both sides by 3 to get: 12 < 7 - x
Add x to both sides: x + 12 < 7
Subtract 12 from both sides to get: x < -5

So, if x < -5, which of the following statements MUST be true?

Aside: When dealing with "MUST be true" questions, we can eliminate a statement if we can find an instance where it is not true.

I. 5 < x (MUST this be true?)
No!
If x < -5, then it could be the case that x = -7, and -7 is NOT greater than 5
So, statement I need NOT be true.

II. |x+3| > 2 (MUST this be true?)
The answer is Yes. Here's why:
IMPORTANT CONCEPT: |x - k| represents the DISTANCE between x and k on the number line.
So, for example, we can think of |4 - 7| as the distance between 4 and 7 on the number line.
Notice that |4 - 7| = |-3| = 3, and 3 is indeed the distance between 4 and 7 on the number line.

Now let's examine |x+3|
We can rewrite this as |x - (-3)|
This represents the DISTANCE between x and -3 on the number line.
So, the inequality |x-(-3)| > 2 is stating that the DISTANCE between x and -3 on the number line is GREATER THAN 2
Well, since we're told that x < -5, we can be certain that the DISTANCE between x and -3 on the number line is definitely GREATER THAN 2
[If you're not convinced, sketch a number line, and place a big dot at -3. Then choose ANY value for x such that x < -5. You'll see that the distance between x and -3 is greater than 2]
So, statement II MUST be true.

III. -(x+5) is positive
This is the same as saying -(x+5) > 0 (MUST this be true?)
The answer is Yes. Here's why:
We're told that x < -5
If we add 5 to both sides we get x+5 < 0
Now, if we multiply both sides by -1, we get -(x+5) > 0
[aside: notice that, since I multiplied both sides by a negative value, I reversed the direction of the inequality]
As we can see, statement III MUST be true.

Answer: D

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Brent
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by Brent@GMATPrepNow » Fri Jan 17, 2014 6:01 pm
greenwich wrote:
Brent@GMATPrepNow wrote:
greenwich wrote: Brent, for the 3 steps to solve the Absolute Value:

1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Does #3 Plug in the solutions to check for extraneous roots also applies to Absolute Value with Inequality such as II in the above question (|x+3|>2)?
Hi greenwich,

I believe you're referring to the advice I gave in this post: https://www.beatthegmat.com/is-x-0-t273441.html

That advice deals with equations with absolute value, not inequalities.

Cheers,
Brent
Hi Brent,

For the question above, II is with inequalities. Are steps 1 & 2 sufficient to solve absolute value with inequalities? --->x+3>2, x>-1 or -x-3>2, x<-5

Also, can we say II is true even if only one of the two solutions satisfies the MUST be true question?

Thanks again...
Good question.
Let's deal with statement II in a way that's different from my first solution.
If |x+3| > 2, then EITHER x+3 > 2 OR x+3 < -2
If x+3 > 2, then x > -1
If x+3 < -2, then x < -5
So, the inequality |x+3| > 2 leads to two different cases.
case a: x > -1
case b: x < -5
So, statement II is saying that EITHER case a is true OR case b is true
Since it's given that x < -5, we can be certain that case b is true.
So, statement II MUST BE TRUE

Cheers,
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by greenwich » Fri Jan 17, 2014 6:09 pm
Brent@GMATPrepNow wrote:
greenwich wrote:If 4<[(7-x)/3], which of the following must be true?
I. 5<x
II. |x+3|>2
III. -(x+5) is positive

A) II only
B) III only
C) I and II only
D) II and III only
E) I, II and III
First, let's deal with the given inequality.
4 < (7-x)/3
Multiply both sides by 3 to get: 12 < 7 - x
Add x to both sides: x + 12 < 7
Subtract 12 from both sides to get: x < -5

So, if x < -5, which of the following statements MUST be true?

Aside: When dealing with "MUST be true" questions, we can eliminate a statement if we can find an instance where it is not true.

I. 5 < x (MUST this be true?)
No!
If x < -5, then it could be the case that x = -7, and -7 is NOT greater than 5
So, statement I need NOT be true.

II. |x+3| > 2 (MUST this be true?)
The answer is Yes. Here's why:
IMPORTANT CONCEPT: |x - k| represents the DISTANCE between x and k on the number line.
So, for example, we can think of |4 - 7| as the distance between 4 and 7 on the number line.
Notice that |4 - 7| = |-3| = 3, and 3 is indeed the distance between 4 and 7 on the number line.

Now let's examine |x+3|
We can rewrite this as |x - (-3)|
This represents the DISTANCE between x and -3 on the number line.
So, the inequality |x-(-3)| > 2 is stating that the DISTANCE between x and -3 on the number line is GREATER THAN 2
Well, since we're told that x < -5, we can be certain that the DISTANCE between x and -3 on the number line is definitely GREATER THAN 2
[If you're not convinced, sketch a number line, and place a big dot at -3. Then choose ANY value for x such that x < -5. You'll see that the distance between x and -3 is greater than 2]
So, statement II MUST be true.

III. -(x+5) is positive
This is the same as saying -(x+5) > 0 (MUST this be true?)
The answer is Yes. Here's why:
We're told that x < -5
If we add 5 to both sides we get x+5 < 0
Now, if we multiply both sides by -1, we get -(x+5) > 0
[aside: notice that, since I multiplied both sides by a negative value, I reversed the direction of the inequality]
As we can see, statement III MUST be true.

Answer: D

Cheers,
Brent
Thanks Brent. Are there any specific steps I need to follow with absolute value with inequalities?
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by Brent@GMATPrepNow » Fri Jan 17, 2014 6:28 pm
Here's what you need to know:
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by GMATGuruNY » Sat Jan 18, 2014 4:28 am
If 4 < (7-x)/3,which of the following must be true?
  • I. 5<x
    2. |x+3|>2
    3. -(x+5) is positive.

    A. II only
    B .III only
    C. I and II only
    D. II and III only
    E. I,II and III
Simplify the expression in the question stem:
4 < (7-x)/3
12 < 7-x
x < -5.
Question stem rephrased:
If x < -5, which of the following must be true?

I: x > 5
Since x is negative, it is not possible that x>5.
Eliminate C and E, which include I.

II: |x+3| > 2.
|x-(-3)| > 2
|a-b| = the DISTANCE between a and b.
Thus, |x-(-3)| > 2 implies the following:
The distance between x and -3 is greater than 2.
Since x<-5, it must be true that x is more than 2 places from -3.
Eliminate B, which does not include II.

III: -(x+5)>0
x+5 < 0
x < -5.
Eliminate A, which does not include III.

The correct answer is D.
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by shahfahad » Fri Nov 13, 2015 8:14 am
Hi Brent. I just wanted to clear something. When you said:

" So, statement II is saying that EITHER case a is true OR case b is true
Since it's given that x < -5, we can be certain that case b is true.
So, statement II MUST BE TRUE "

My question is that in a DATA SUFFICIENCY question related to absolute values, when we yield two different results and one is conflicting with the statement but the other satisfies it. We classify it as INSUFFICIENT. How can we say that STATEMENT 2 is SUFFICIENT when the above statement is the same scenario. We have 2 results, one matches the target question and other does not.

Could you please explain?

Thanks.
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by Brent@GMATPrepNow » Fri Nov 13, 2015 9:15 am
shahfahad wrote:Hi Brent. I just wanted to clear something. When you said:

" So, statement II is saying that EITHER case a is true OR case b is true
Since it's given that x < -5, we can be certain that case b is true.
So, statement II MUST BE TRUE "

My question is that in a DATA SUFFICIENCY question related to absolute values, when we yield two different results and one is conflicting with the statement but the other satisfies it. We classify it as INSUFFICIENT. How can we say that STATEMENT 2 is SUFFICIENT when the above statement is the same scenario. We have 2 results, one matches the target question and other does not.

Could you please explain?

Thanks.
The original question here is not a DS question, so your question doesn't really apply.
In the original question, we are asked to determine which statements MUST be true.


Your question includes the following: My question is that in a DATA SUFFICIENCY question related to absolute values, when we yield two different results and one is conflicting with the statement but the other satisfies it.
In this case, if there are 2 results, but one is conflicting with the statement, then we can be certain that the statement is sufficient.

Consider this example:
If |x| = 3, what is the value of x
(1) x is positive
If |x| = 3, we can be certain that x = 3 OR x = -3
Statement 1 tells us that x is positive
So, at this point, we can be certain that x = 3
So, statement 1 is sufficient.

Does that help?

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by shahfahad » Fri Nov 13, 2015 9:36 am
1) By conflicting i mean that first value is x < -5 and second value is x > - 1. How can it be possible that X is greater than -1 but less than -5. So my questions is that IF X COULD be x > -1 however the value COULD be less than -5 in that case the statement is SUFFICIENT. So:

- You can say that X "COULD" be less than -5. My question is you are saying X "MUST" be less than -5. So my problem is with "MUST". It could be > -1 also. So how can u say IT MUST BE TRUE.

- Also, what would be the answer if this was a DS question. Statement INSUFFICIENT?
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by Brent@GMATPrepNow » Fri Nov 13, 2015 11:24 am
shahfahad wrote:1) By conflicting i mean that first value is x < -5 and second value is x > - 1. How can it be possible that X is greater than -1 but less than -5. So my questions is that IF X COULD be x > -1 however the value COULD be less than -5 in that case the statement is SUFFICIENT.
First, it's important to note that if |x+3| > 2 then EITHER x < -5 OR x > -1. We are not sating that x < -5 AND x > -1

Second, this is not a (data) sufficiency question. We're not trying to determine whether or not x < -5. We are told that x < -5. We are told this in a roundabout with the given information that says 4 < [(7-x)/3]

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by shahfahad » Fri Nov 13, 2015 11:28 am
OK thanks
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by Matt@VeritasPrep » Fri Nov 13, 2015 2:41 pm
Brent@GMATPrepNow wrote:
First, it's important to note that if |x+3| > 2 then EITHER x < -5 OR x > -1. We are not sating that x < -5 AND x > -1 [/quote]

One nice to way to remember this, Fahad: |x + 3| > 2 means that x is further than 2 units from -3 on the number line. So it could be more than 2 units away to the LEFT (= less than -5) or it could be more than 2 units away to the RIGHT (= greater than -1).
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by shahfahad » Sat Nov 14, 2015 3:40 am
Brent@GMATPrepNow: Ok i get it. The target question has already told us that x < -5 so we dont need to TEST it whether it is true or not. That was the mistake i was making. Right.

Matt@VeritasPrep: Do we always consider these kind of results in an "OR" manner?
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