If |x| < 20 and |x – 8| > |x + 4|, which of the following expresses th

B for me.

|x| < 20 means that -20<x<20.

Now the expression: |x – 8| > |x + 4|
Squaring both sides we get (x-8)^2>(x+4)^2
=>x<2
So, combining the 2 ranges, we get –20 < x < 2.

Given: |x| < 20 and |x – 8| > |x + 4|,

|x| < 20, so this gives -20 < x < 20 ------> A

|x – 8| > |x + 4|

Here we have 3 possibilities:

(i) x < -4

-x+8 > - x + 4, as x's cancel out, we don't get a range here.

(ii) -4 < x < 8

-x+8 > x+4
x< 2 -------------> B

(iii) x > 8

x-8 > x+4, , as x's cancel out, we don't get a range here.

Therefore, by combining the ranges A & B, we get -20 < x < 2. Answer: B

mikemcgarry, could you please confirm if the solution is correct?

Dear quantumliner,

I'm happy to respond.

Your approach is 100% correct BUT long & detailed & time-consuming. It is an extremely reductionist and left-brain approach, algebraic & formulaic rather than pattern-based. See the original post, linked with the problem, for a more pattern-based approach to these problems.

It may not have taken very long on this relatively straightforward problem, but having only this approach will create problems for you on some advanced problems. See this post:
How to do GMAT Math Faster
My friend, you obviously have great algebraic skill. If you can combine these organization & detail management skills with intuition and pattern-matching, you will take your math performance to a whole other level.

Does all this make sense?
Mike
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Hello Mr Mike the way approached this problem was;

|x|= 20 is x<20 & x>-20

therefore -20<x<20

now since mode has two solutions |x-8|>x+4= x-8> x+4 would lead to x=0 we'll be only left with

-x+8>x+4 which results in x<2

therefore our answer is -20<x<2 which is B

Is it right??

Given :

1. |x| < 20

-20 < x < 20 --------------- Range -1

2. |x – 8| > |x + 4|

Case 1 : \(x - 8 > x +4\)

\(-8 > 4\) ------ Not a valid case

Case 2 : \(x - 8 < -x - 4\)

\(2x < 4\)

\(x < 2\) ------- Range - 2

Combine range 1 & 2, we get

\(-20 < x < 2\)

Hence B.
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Hi mikemcgarry, I'd like to ask you. maybe it is novice question haha. There are only three approaches of +=+, -=- and -=+. Why can't I try to solve for +=- giving me the range x>2? I'm getting lost here. Could you please explain?

I'd like to ask you. maybe it is novice question haha. There are only three approaches of +=+, -=- and -=+. Why can't I try to solve for +=- giving me the range x>2? I'm getting lost here. Could you please explain?[/quote]

I dont understand this either, would be good to know!

Using the Distance Approach:


(1st) Given the Inequality Expression:

[X - 8] > [X + 4] ---- this can be re-written as:

[X - 8] > [X - -(4)]


The Distance between X and +8 must be GREATER THAN the Distance between X and -4


--------------- (-)4-------------------- +2---------------------------- +8 ----------------------


+2 is the Mid-Point between (-)4 and +8 --- when X = +2, then X will be Equi-Distant from -(4) and +8



Looking at the Number Line, if X is Placed Anywhere to the RIGHT of +2 (such that X > +2) ----

then X will ALWAYS be Closer to +8 and Further away from -(4)......in contrast to the Given Inequality.



If X is Placed Anywhere to the LEFT of +2 (such that X < +2 ) ,

then X will Always be Closer to -(4) and FURTHER from +8




Thus, when X < +2:

the Distance from X to +8 will always be GREATER THAN > the Distance from X to -(4)

This Satisfies the Given Inequality


X < (+)2------ (1st Inequality)



(2nd)Given: [X] < 20

This Inequality is better solved using Algebra.



Rule: Given --- [X] < K ---- where K = Some Number Value

then it is true that we can Open the Modulus in the following way:
-(K) < X < +(K)


Thus, [X] < 20 becomes:

-(20) < X < +(20) ----- (2nd Inequality)



(3rd) Combining the 2 Inequalities:

X < +2

-(20) < X < +(20)


Combined: -(20) < X < +2


Answer -B-

\(|x-8|>|x+4|\)
Since both sides are non-negative, we can square them.
\(|x-8|^2>|x+4|^2\)
\(\implies (x-8)^2-(x+4)^2>0\)
\(\implies (x-8+x+4)(x-8-x-4)>0\)
\(\implies (2x-4)(-12)>0\)
\(\implies 2(x-2)(-12)>0\)
\(\implies -24(x-2)>0\)
\(\implies (x-2)<0\) (Inequality gets reversed since we're dividing both sides by -24, i.e. a negative number)
\(\implies x<2\)
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I had the same question as you because I was getting x<2 and x>2 solutions after solving the equation |x – 8| > |x + 4|.

Solution x>2 doesn't work because if you plug in x=3,4,5 etc in the equation, you will see that the equation does not hold. I think that is why x>2 does not work as a valid solution for this equation.

Hi mikemcgarry, can we solve this by plugging in the answer options?

By that, I mean using a point from the given range to see if it satisfies the situation and eliminates the ans options?

for eg. in this question for the last 3 options, I chose 10 as an x and it did not satisfy the situation so all three were eliminated basis that, and for the remaining 2 I checked the extreme values to check which one does not satisfy the inequality.

can this be a viable way to solve in terms of the real-life GMAT exam?

Took me 2.5 minutes to solve this.
Plugged in a couple of numbers & eliminated options.

First checked with x=15 (Satisfies C, D, & E).
\(|15-8| \ngtr |15+4|\)

This eliminates C, D & E.

Then checked with x=11
\(|11-8| \ngtr |11+4|\)
This eliminates A.

Hence B.

Update: I think it makes more sense to square both sides of the inequality and solve it that way. Should be quicker to solve that way.
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Hi, Can you explain how did you get x<2?

Thanks mikemcgarry !! I went through your post and it makes sense. Visual understanding of a problem and to understand what exactlyis happening indeed makes solving the problem faster and easier. I will try to use this method to get a better grip of this technique.

Dear mikemcgarry,
Could you please show us a shorter/ faster approach please, thanks. I did refer How to do GMAT Math Faster.
Cheers.

Hello Amulya,

You solved using a short method. Can you kindly explain the idea of squaring the two sides ? What are the situations I can use this method ?

Thanks.

JeffTargetTestPrep, could you walk me on a detailed solution to |x – 8| > |x + 4|? Thanks in advance!

mikemcgarry so hwo to do this question fast??

|x| < 20

-20 < x < 20 ------- (1)

|x – 8| > |x + 4|

Squaring both sides we get

\(x^2 + 64 - 16x > x^2 + 16 + 8x\)

24x < 48

x < 2 ------ (2)

Common Range of (1) and (2)

-20 < x < 2

Option B
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