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# For what values of x is |x - 4| - |x - 3| >= 7?

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Math Expert
Joined: 02 Sep 2009
Posts: 60727
For what values of x is |x - 4| - |x - 3| >= 7?  [#permalink]

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02 Dec 2019, 01:30
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Difficulty:

35% (medium)

Question Stats:

71% (01:39) correct 29% (01:50) wrong based on 134 sessions

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For what values of x is $$|x-4|-|x-3|\geq7$$?

A. $$x\leq3$$

B. $$3\leq x\leq4$$

C. $$x\geq4$$

D. $$x\geq7$$

E. No values of x satisfy this inequality.

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Joined: 13 Feb 2018
Posts: 494
GMAT 1: 640 Q48 V28
Re: For what values of x is |x - 4| - |x - 3| >= 7?  [#permalink]

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02 Dec 2019, 01:43
I just tested the options

A) Let x=0; 4-3 is not more than seven. Dismiss
B) Let x=3.5; 0.5-0.5 does not satisfy the stem. Dismiss
C) Let x=100; 96-97 does not satisfy the stem. Dismiss
D) Let x=100; 96-97 does not satisfy the stem. Dismiss
E) The only option left

IMO
ans: E
Math Expert
Joined: 02 Aug 2009
Posts: 8327
For what values of x is |x - 4| - |x - 3| >= 7?  [#permalink]

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02 Dec 2019, 20:40
2
Bunuel wrote:
For what values of x is $$|x-4|-|x-3|\geq7$$?

A. $$x\leq3$$

B. $$3\leq x\leq4$$

C. $$x\geq4$$

D. $$x\geq7$$

E. No values of x satisfy this inequality.

Are You Up For the Challenge: 700 Level Questions

You can also solve it logically.
In |x-4|-|x-3|, whatever happens to x in one MOD will happen to x in 2nd MOD.
So basically x is cancelling out with each other and what you are left with is |-4|-|-3|=4-3=1
So the equation will be equal to 1 whatever be the value of x.
So it will NEVER be $$\geq{7}$$

E
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Re: For what values of x is |x - 4| - |x - 3| >= 7?  [#permalink]

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04 Dec 2019, 22:06
For what values of x is |x−4|−|x−3|≥7
consider three situations x<3 , 3<x<4 , 4<x

$$when x<3$$
-x+4-(-x+3)=1
1 is not greater than or equal to 7
So equation does not holds true for x<3

$$when 3<x<4$$
-x+4-x+3= -2*x+7
check for x=3.9
=-2*3.9+7= -0.8
-0.8 is not greater than or equal to 7
So equation does not holds true for 3<x<4

$$when 4<x$$
x-4-x+3=-1
-1 is not greater than or equal to 7
So equation does not holds true for 4<x

Therefore No value of x satisfies the inequality

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Re: For what values of x is |x - 4| - |x - 3| >= 7?  [#permalink]

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30 Dec 2019, 08:56
Can we not just say that |x-4| and |x-3| are two consecutive numbers and hence the difference can never be other than 1? This takes 5 seconds.
Re: For what values of x is |x - 4| - |x - 3| >= 7?   [#permalink] 30 Dec 2019, 08:56
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