If 6

Question

If  6 < \frac{4 - x}{5} , which of the following must be true?

I.  x < 26 
II.  |x + 19| > 7 
III.  |x| = -x

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

JonShukhrat · Accepted Answer

Note that we are asked to determine which MUST be true, not could be true.

\(6 < \frac{(4-x)}{5}\) next \(30<4-x\) next \(x<-26\)

So we know as a fact that \(x<-26\). Now, keeping this in mind we should find out which of the following will be true for \(x<−26\)

In other words, wtether any number less than \(-26\) will make the following true?

I. \(x<26\) Yes for this interval. Whatever less than \(-26\) would also be less than \(26\). Or we can just take any number from \(x<-26\) and check whether it is true for \(x<26\). For example, \(-50\) is less than \(-26\), now is \(-50\) also less than \(26\)? Yes.

II. \(|x+19|>7\) means that we have two cases. Case 1: \(x+19>7\) or simplfied \(x>-12\). Case 2: \(-x-19>7\) or simplified \(x<-26\). We are given that the second range is true (\(x<−26\)), so this inequality holds true.

III. \(|x|=−x\) means that \(x\leq{0}\). Whatever less than \(-26\) is also less than \(0\). Thus true. We can see that all the three hold true for \(x<-26\).

Hence E

geezus24x7 · Answer

6<4-x/5 or x<-26

I. not possible
2. x>-12 or x<-26. Possible
3. Since x<-26 then |x|=-x. Possible.

Hence IMO C is the answer

firas92 · Answer

4-x > 30
-x > 26

x < -26

Thus all 3 must be true

Answer is (E)

mira93 · Answer

6<4-x/5 (multiply by 5)
30<4-x
4-x>30
-x>26
x<-26, we need to find for which of the following statements this subset falls into.
1) x<26, in this statement, all negative numbers are included, including anything lower than negative 26, thus this one is sufficient, must be true
2) |x+19|>7, we have two cases to solve
2.1) if x is positive, x+19>7, x>-12, but our x must be positive, hence no solution.
2.2) if x is negative, -(x+19)>7, -x-19>7, x<-26, this is exactly what we need to find if x is less than negative 26, and according to this statement, yes, x is less than negative 26. Must be true
3) x|=−x, from this we know that x is not positive, because any number on left will give positive result and to be equal to right, that number must be negative, so that two negatives cancel out and we have both sides positive or equal to 0. Since any number in this set is not positive, then anything less than negative 26 is also in this statement. Must be true

E is our answer

santorasantu · Answer

IMO E:

solving the inequality we get x< -26.
This implies from I, x<26 
as x<-26, which means that x<26 must be true
from 2: |x+19|>7, given x<-26, distance between -26 and 9 is 7, and this will alwayys be greaater then 7, as x<-26.

from 3: |x| = - x, as x<0, this statement must be true.

Therefore all must be true.

E

nick1816 · Answer

6< (4-x)/5
x<-26

I. x will always smaller than 26
Must be true

II. Distance of x, on a number line, from -19 will always be greater than 7 
Must be true

III As x is negative, |x|= -x
Must be true

IMO E

prashanths · Answer

Hi Bunuel , Not sure I agree with this solution. This question is similar to the following GOT question: https://gmatclub.com/forum/if-x-3-1-the ... 99375.html In that question, |x-3|>1 simplifies to x>4 or x<2 However, the options were I. |x|>4 II. x2>16 III. x>4 In this case since x>4 and x<2, implies x>4 and x<-4 as well So all the options should be true. However, None of the options was the answer for that question. Presumably because, x=1 satisfies the given condition but not the options. Coming back to this question. the equation simplifies to x<-26. If so, how can x<26 be true, considering that x=25 will not satisfy the condition. So none of the options "must be true".

Bunuel · Answer

We go that x < -26. Any x which is less that -26 for sure will be less than 26.

As for your doubt x CANNOT be 25 because we know that x < -26.

prashanths · Answer

In that case, for the other question: https://gmatclub.com/forum/if-x-3-1-the ... 99375.html x>4 and x<2 All the options imply x > 4 or x < -4. So all the options must be true, shouldn't they?

Bunuel · Answer

In that question we have that x < 2 or x > 4.

I.  |x| > 4 . Not always true. Consider x = 0.
II.  x^2 > 16 . Not always true. Consider x = 0.
III.  x > 4 . Not always true. Consider x = 0.

prashanths · Answer

Okay,

similarly in this question.
x<-26

I. x<26 -- not always true since x=25 satisfies this condition but does not satisfy our original question
II. |x+19|>7 -- not always true since x = 0 satisfies this condition but does not satisfy our original question
III. |x|=−x ---- not always true x = 0 same as above.

Bunuel · Answer

I'try again.

GIVEN AS A FACT: x < -26.

I. x<26. x CANNOT be 25 because we know that x < -26.
II. |x+19|>7. x CANNOT be 0 because we know that x < -26.
III. |x|=−x. x CANNOT be 0 because we know that x < -26.

To understand the underline concept better practice other  Trickiest Inequality Questions Type: Confusing Ranges .

nazii · Answer

Dear Friends I write this for those in future maybe have the same as my problem in understanding this question
Before this question I was solving another question, which I share the link here
https://gmatclub.com/forum/if-x-1-2-whi ... l#p3116795
I had the problem with these two.
Here is the simple logic
for such questions this is the solution I learned from my mistakes
First: find the proper region of x from the question stem
Second and the most important things to remember: do not solve the equations (in(l),(ll), (lll)) for finding the region of X, instead, try to replace different value of X based on the appropriate region you found from the question stem. Then check that do the values you assigned for each part, make that statement always true or not. If not, Omit that answer choice.
My own problem before was that I tried to solve other equations in the answer choices and then find the intersection between each and the proper region of X from question stem. This is very confusing for me.
The important note is that the proper value of X is determined by question stem (which you have to solve to find it). In answer choices you should look that , do the value from X , obtained by the question stem, satisfy the equation always or not
I hope it was clear

Danush649 · Answer

HI  Bunuel  ScottTargetTestPrep 
Im not able to understand the question
In the given stem, it says IF x<-26 then Which must be true
Then shouldn't it mean that the THREE conditions MUST FALL UNDER THE RANGE OF X<-26 for them to be true
But the solution says if the GIVEN condition fall under the RANGES OF (I)(II)(III) they are true.
for example III says X<0, shouldn't we say it is false because -1 or -2 doesn't fall In the range of X<-26

ScottTargetTestPrep · Answer

It should actually be the other way around, x < -26 should fall under the range of the three conditions. The question tells you for a fact that x is less than -26, and in I, it is asking whether x is less than 26. The answer is yes, because if you know some number is less than -26, then the same number is also less than 26. The answer would be not necessarily if it was the other way around; i.e. if you knew x < 26 and you were trying to determine whether x < -26 as well. The answer could be yes for certain values of x such as -27 or -30 or -45, but the answer could also be no if x is -20 or 0 or 10. The same thing goes for II and III. For II, take any number x less than -26 and substitute in |x + 19|, you'll see that the result you get will be greater than 7, which is why this statement must be true as well. For III, take any number x less than -26 and substitute in |x|. You'll see that the expression will evaluate to -x. In this statement, the fact that -1 or -2 does not satisfy x < -26 does not mean the statement is false; you can only choose values which are less than -26 for x. It is perfectly okay for x < 0 to contain values which do not satisfy x < -26, the statement would only be false if there were some values which satisfied x < -26 but did not satisfy x < 0, and there are no such values.

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If 6 < (4 - x)/5, which of the following must be true?

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