If 4 < (7 - x)/3, which of the following must be true?

Solved the equation to get x < -5
As the question says must be true, it means that for any below -5 the statements have to hold true. So just assumed it to be -6 in every statement

Answer D
Time Taken 1:25
Difficulty level 550

Hi erikvm,

Let me explain the solution to you using the representation on number line. We need to evaluate if for every value in the range of the question statement i.e. x < -5, do the statements hold true.

Refer the below diagram for the range given to us in the question & the other statements:




Question Statement: The question statement gives us the range as x < -5. We need to see for every value of x < -5, are the inequality in the statements true.

Statement-I: St-I gives us the range as x > 5. We see that this range does not hold true for any value of x. Hence, it can't be true.

Statement-II: St-II gives us the range as x > -1 or x < -5. We see that for every value of x < -5 (i.e. the inequality in the question statement), we can write |x +3| > 2 holds. Hence this is a must be true statement.

Statement-III: St-III gives us the range as x < -5 which is nothing but the range in question statement. Hence this is a must be true statement.

Therefore, st-II & III are true for every value of x < -5 and hence must be true statements.

Hope its clear

Regards
Harsh

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



Sol:The given expression can be 4<(7-x)/3 (multiply by 3) we get
12<7-x or x<-5

St 1 can be ruled out

St 3 must be true So Option A,C and E can be ruled out

Consider st 2 |x+3|>2

When x+3 >/ 0 then |x+3|= x+3 and also x>/-3

or x+3> 2 or x>-1 but our condition is x>/-3 so x> -1 (because if x>-1 then x surely is greater than -3)

When x+3 <0 then |x+3|=- (x+3) and the st2 equation becomes x+3<-2 or x<-5 and thus x<-5 (because of x<-5 then x <-3 as well)

So we have x >-1 or x<-5 ------> In this range the St 2 will hold good. But we are given that x<-5 thus St 2 is true

Ans is D....

For (2):
X could be a positive number or a negative number, if x is a negative number than solving inequality would result into X < -5 which is true because it is the exact same inequality which has been given in the question.

But if x would be positive that solving inequality would result into x >-1 which would make the option (2) not true. So in other words when x is negative than statement (2) is true but when x is positive than statement (2) is not true. So statement (2) is not ALWAYS true.


Can you please tell me why im unable to get the right answer? what is flaw in my reasoning?

It should be other way around.

To elaborate more. Question uses the same logic as the examples below:

If \(x=5\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x>-10

Answer is E (x>-10), because as x=5 then it's more than -10.

Or:
If \(-1<x<10\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x<120

Again answer is E, because ANY \(x\) from \(-1<x<10\) will be less than 120 so it's always true about the number from this range to say that it's less than 120.

Or:
If \(-1<x<0\) or \(x>1\), then which of the following must be true about \(x\):
A. x>1
B. x>-1
C. |x|<1
D. |x|=1
E. |x|^2>1

As \(-1<x<0\) or \(x>1\) then ANY \(x\) from these ranges would satisfy \(x>-1\). So B is always true.

\(x\) could be for example -1/2, -3/4, or 10 but no matter what \(x\) actually is it's IN ANY CASE more than -1. So we can say about \(x\) that it's more than -1.

On the other hand for example A is not always true as it says that \(x>1\), which is not always true as \(x\) could be -1/2 and -1/2 is not more than 1.

Hope it's clear.

I still don't understand the application of this, to the question. Sure, I understand that if x = 5 its > -10.

But, here, we get two different cases. Solving the question stem we get x < -5.

In "II" we have that x is either x < -5 or x > -1. If I plug in a value where x > -1. Say x = 3. The equation doesn't hold?

We get 4 < (7-3)/3 --> 12 < 4. Not true?

How can this be a valid answer choice then

Edit: Don't know if this is how you should go about doing it, correct me if I'm wrong:

If we have \(x < -5.\)

Plugging in a value < -5, say -6 into II: \(|-6+3| > 2\). This holds true, aka sufficient

Thanks, I'm just abit confused about the whole "must be true". I know how absolute values work, but \(|x+3| > 2\) provides me with 2 different values for x, so I just assumed that this statement was not true because we have two different possibilities for X

I tried an alternate solution by plugging numbers

Given that \(x < -5\), we could pick a number that satisfies this condition, lets say \(x = -7\)

I. \(5 < x\), is \(5 < -7\), Not true, Eliminate
II. \(|x + 3| > 2\), plug in -7 in place of x, we get \(| -7 + 3 | > 2, |4| > 2\), True
III. \(- (x + 5) > 0, - ( -7 + 5) = - (-2) = 2 > 0\), True

Answer is D. II & III only

Responding to a pm:

You are given that x < -5. You obtain this because you are given that 4<(7-x)/3.
So x can take values such as -5.4, -6, -100, -54637 etc

Which of the following MUST BE TRUE?

II. |x+3|>2
Gives x < -5 OR x > -1
For every value that x can take, is this statement true? Yes. For every value that x can take it is "'less than 5" or "more than -1".
All our values are actually "less than -5". We need x to satisfy either "x < -5" or "x > -1".

Hi Bunuel can we think of 2nd statement as a Big set of ( 0,1,2,3,4,5 and also -6,-7,-8...) and what we need for must be true is a subset (-6, -7, -8...) ?
Is this correct?

We are not given that x is necessarily an integer.

For |x + 3| > 2 to hold true, x < -5 or x > -1 must be true.

We know that x < -5, so |x + 3| > 2, will be true.

Check other similar questions from Trickiest Inequality Questions Type: Confusing Ranges (part of our Special Questions Directory).

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

Simplifying the given inequality we have:

4 < (7 - x)/3

12 < 7 - x

5 < -x

-5 > x

Since x is less than -5, we see that |x+3| > 2.

Also, since x < -5, we see that x + 5 will always be negative and thus -(x+5) will always be positive.

Answer: D

Hi Bunuel :

i had a question about the above reasoning. From statement II we get two ranges i.e. x< -5 (which conforms to the inequality given in the question stem) and x>-1 which is a new one based on the option. My observations here is that despite two ranges of X coming off option II, we can only take the one that conforms to the constraint (x<-5) and see if the entire expression (| x + 3| > 2 ) holds true, which it does when we consider any value of X <-5 .

Is this correct ?

Thanks & Regards,
Deeuce

Yes. We are given that x < -5 and the question asks: is |x + 3| > 2 true? For any x which is more than -5, |x + 3| > 2 IS true. So, |x + 3| > 2 must be true given that x < -5.

Simplify the expression in the question stem:
\(4 < \frac{7-x}{3}\)
\(12 < 7-x\)
\(x < -5\)
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.

Since the range presented in the Option [II] is x<-5 OR x>-1, how can the expression x<-5 be a MUST BE TRUE condition?
Shouldn't it make more sense with a Could be true statement?