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

Question

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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sasyaharry · Accepted Answer

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.

chetan2u · Answer

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

LevanKhukhunashvili · Answer

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

Ansh777 · Answer

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

Answer: E

NHiromoto · Answer

x is not integer here.
So, x can be 3.5 and, in that case, the values of |x-4| and |x-3| are 0.5.
Just to be precise, these are not consecutive values although the answer will never be ≥7.

anuspn · Answer

Consider, X-4=A

The equation will become |A| - |A+1| ≥ 7

Case 1: A>0
A-A+1≥7

Case 2: -1<A<0

-A-A+1≥7
-2A+1≥7
-2A≥6
A≤-3

X-4≤-3
X≤1

Check the condition, -1<X-4<0
3<X<4

No Solution in case 2

Case 3 A<-1
-A-(-A-1)≥7
-A+A+1≥7

No Solution

Answer is E

chacinluis · Answer

Real line approach

<-----A---------(-4)--B---(-3)-------C---------------------->

|x-4| is distance of x from -4
|x-3| is distance of x from -3

If x is in A then distance of x to -4 is 1 less than distance of x to -3. Therefore the subtraction is -1
If x is in C then distance of x to -4 is 1 more than distance of x to -3. Therefore the subtraction is 1
If x is in B then distance of x to -4 plus the distance of x to -3 is are most one, so the subtraction is less than 1.

Therefore for no x-values |x-4|-|x-3| will be greater than or equal to 7, b/c the subtraction is at most 1.

Fdambro294 · Answer

You can Solve the Question pretty quickly by understanding the Distance Interpretation of Absolute Value:

Given:   -   >/= 7

We need to find Values of X where:

(Distance from X to +4) - (Distance from X to +3) = AT LEAST 7 Units

However, if you draw the Number Line, there is NO WHERE we can Place X such that this is met.

For Instance, if X = +10

(Distance from +10 to +4 = 6)
-
(Distance from + 10 to +3 = 7)

= -(1)

X definitely can NOT be on the Positive Side of 4 and 3 because X will always be Closer to +4 and it will result in a (-)Negative Value

If X = -20

(Distance from -20 to +4 = 24 Units) 
-
(Distance from -20 to +3 = 23 Units)

=

(+)1

In other words, since +4 and +3 are only 1 Unit Apart on the Number Scale, no matter which Value we Pick for X that is to the RIGHT of +4 or to the LEFT of +3 ---- the result of the Expression will always be EITHER = (+)1 or (-)1

-E- No Values of X can Satisfy this Expression

PriyanshuR · Answer

x is |x−4|−|x−3|≥7

So the ranges are x ≤ 3 and x ≥ 4.
Nothing satisfies the value for x.

Thus,  E

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

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