Inequalities trick

[rainmaker]

If -5 <= x <= 3, then of the following is correct?

(A) |x| <= 3
(B) |x| <= 5
(C) |x - 2| <= 3
(D) |x - 1| <= 4
(E) |x +1| <= 4

Please explain how you arrived at the answer and if possible a strategy/trick to solve such problems.

Thanks

[this_time_i_will]

Just remember two simple points for inequalities:

I. if |x| <= b & b>0 then -b <= x <= b...that is x lies between -b & b, iclusive
II. if |x| >= b & b>0 , then either x <= -b or x >= b...that is x lies outside the range -b to b, exclusive.

in the backdrop of above information, lets attack answer choices.

given: -5 <= x <= 3

A. -3 <= x <= 3....fits the bill, inside given range

B. -5 <=x <= 5....wrong option, some positive values outside range

C. -3 <= x-2 <= 3 => -1 <= x <= 5....wrong option

D. -3 <= x <= 5 ....wrong option

E.-5 <= x <= 3...better than A.

[thephoenix]

imo e
its the rephrase of q

[eaakbari]

Options


A - If x = -5 then it does not satisfy ;Incorrect
B - the given inequality is a subset of this inequality, Hence all values satisfy. Maybe Correct
C - iF X = -4 . Incorrect
D - If x = -5 . Incorrect
E - All values of x satisfy. Maybe Correct


I cant see how which one is more correct then the other from B and E. Do help

[kstv]

If -5 <= x <= 3, then of the following is correct? (A) |x| <= 3
(B) |x| <= 5 (C) |x - 2| <= 3 (D) |x - 1| <= 4 (E) |x +1| <= 4
Please explain how you arrived at the answer and if possible a strategy/trick to solve such problems.
Thanks

Using a number line and plugin the extreme values possible under each option.
-5 <= x <= 3 so the number line will be

-5 ]-------------------------------0--------------------------[3
x is any value between -5 and +3
Pick values to prove/disprove each option
A. if x = -4 |x| > 3 Eliminate B. if x = -5 |x| < = 5 but if x = 3 |x| < = 5
C. if x = -4 |x -2| > 6 Eliminate D. if x = -5 |x-1| > 6 Eliminate
E. if x = 3 |x+1| <= 4 but if x = -5 |x+1| is still <= 4
notice B and E are the same expression.
I am editing/ highlighting it as from the Expert reply it is obvious that this is an error. B and E are not the same expression.

[phuonghang44]

If -5 <= x <= 3, then of the following is correct?

(A) |x| <= 3
(B) |x| <= 5
(C) |x - 2| <= 3
(D) |x - 1| <= 4
(E) |x +1| <= 4

Please explain how you arrived at the answer and if possible a strategy/trick to solve such problems.

Thanks

Hi,

I think E will be correct.
|x +1| <=4
<=> x+1<=4 and x+1=> -4 (mathematic formula)
<=> x<= 3 and x=>-5
Hope that it helps.

[eaakbari]

@ kstv
So what is your final answer?
@ phuonghang44
Why not B

@ any1 else reading this
How do you cut down from options B and E

[stephen@knewton]

Hey guys,

A couple of strategy points on this one:

- Sometimes in dealing with inequalities, it's helpful to consider the boundaries. You can do this by considering only the "or equal to" portion of the inequality (or, if it's not an inclusive boundary, just pretend it is!). As long as you are careful about which regions make the inequality true (in this case, inside the boundaries), this technique can make things very quick and easy.

STEP 1: The boundaries of the prompt are given to us fairly straighforwardly: they are -5 and 3.

STEP 2: Let's have a look at the answer choices. We'll try to figure out what values of X would make a similar EQUALITY true:

(In case anyone needs a refresher, absolute value means the distance from zero of what's inside the bars. In other words, negatives become positive, positives stay positive).

(A) |x| <= 3 ... the values of X that make this true at the boundaries would be -3 and 3.
(B) |x| <= 5 ... likewise -5 and 5
(C) |x - 2| <= 3 ... -1 and 5
(D) |x - 1| <= 4 ... -3 and 5
(E) |x +1| <= 4 ... -5 and 3

STEP 3: As the first response to this thread noted, some of these answer choices include regions that are "not allowed" by the prompt. These are answer choices B, C and D, and they can be eliminated. By looking at the boundaries in this way, you'll also notice that choices (B) and (E) are actually not describing the same values of X. Remember that we're not allowed to add or subtract to both sides of an inequality from within a absolute value!

STEP 4: Choosing between (A) and (E), both of which describe regions which make the prompt true, is the hardest part. We need to know what the GMAT expects of us. In questions worded as this one is, they are looking for another equation that COMPLETELY and ACCURATELY describes the allowable values for X. Answer choice (A) is wrong because it is incomplete; it fails to describe the region between -5 and -3 that satisfies the prompt.

One last cool absolute value trick, for fun: the form of inequality presented in (C), (D) and (E) describe something very useful. Generically: |x-a|=b describes the two values that are B away from A. In other words, C is all the numbers less than or equal to 3 away from 2. D is all the numbers less than or equal to 4 away from 1. And E is all the numbers less than or equal to 4 away from -1 ... our answer! When you see that form of inequality on the GMAT, this can a great shortcut ... just don't get it backwards!!

Hope this helps a little ...

Cheers, Steve P.

[stephen@knewton]

... and since I also teach sentence correction, I should correct my last paragraph. The form DESCRIBES.

[harshavardhanc]

One last cool absolute value trick, for fun: the form of inequality presented in (C), (D) and (E) describe something very useful. Generically: |x-a|=b describes the two values that are B away from A. In other words, C is all the numbers less than or equal to 3 away from 2. D is all the numbers less than or equal to 4 away from 1. And E is all the numbers less than or equal to 4 away from -1 ... our answer! When you see that form of inequality on the GMAT, this can a great shortcut ... just don't get it backwards!!

Soooooooooper!!!!!

[phuonghang44]

@ kstv
So what is your final answer?
@ phuonghang44
Why not B

@ any1 else reading this
How do you cut down from options B and E

Hallo eaakbari,

They give you a calculation which is A,B,C,D,E choice. Your task is to solve it to find out the x value. After u find out, then check it with the criteria which is the value of x given in the theory above which is -5<=x<=3. E is the most correct therefore E is the best choice.

Also I would like to mention the formula:
1. |x|=a
<=> x=+/-a
2. |x|<=a
<=> x<=a AND x>= - a (it is And not Or).

You apply no.2 in solving A, B, C, D, E then you will find out what I mean. For such calculation, when you get familiar, you will easily do mental calculation and can come up with the answer in just a few second.

hope that it helps.

Also I saw a very useful post after your post. You can refer for further info.

[rainmaker]

Thanks guys for the great tips!