Max@Math Revolution wrote:Which of the following is satisfied with |x-4|+|x-3|<2?


A. 1<x<5
B. 2<x<5
C. 2.5<x<4.5
D. 2.5<x<4
E. 3<x<4


* A solution will be posted in two days.

The easiest way to deal with such questions is to check the validity of options

|x-4|+|x-3|<2

A. 1<x<5
@x=1.1 |x-4|+|x-3|= |1.1-4|+|1.1-3|= 2.9+1.9 which is NOT less than 2 Hence INCORRECT OPTION

B. 2<x<5
@x=2.1 |x-4|+|x-3|= |2.1-4|+|2.1-3|= 1.9+0.9 which is NOT less than 2 Hence INCORRECT OPTION

C. 2.5<x<4.5
@x=2.6 |x-4|+|x-3|= |2.6-4|+|2.6-3|= 1.4+0.4 which is less than 2
@x=4.4 |x-4|+|x-3|= |4.4-4|+|4.4-3|= 0.4+1.4 which is less than 2 Hence CORRECT OPTION

D. 2.5<x<4
Since the equation is also satisfied @x=4.4 which is beyond this range hence, INCORRECT OPTION

E. 3<x<4
Since the equation is also satisfied @x=2.6 which is beyond this range hence, INCORRECT OPTION

Answer: Option C

Max@Math Revolution wrote:Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5
B. 2<x<5
C. 2.5<x<4.5
D. 2.5<x<4
E. 3<x<4

D and E imply that x must be LESS than 4, while A, B and C imply that x can be GREATER than 4.
Test x=4.1.
If we plug x=4.1 into |x-4|+|x-3|<2, we get:
|4.1 - 4| + |4.1 - 3| < 2
0.1 + 1.1 < 2
1.2 < 2.
This works.
Since x=4.1 is a valid solution, eliminate any answer choice that does not include x=4.1 within its range.
Eliminate D and E.

C implies that x must be LESS than 4.5.
Test x=4.5.
If we plug x=4.5 into |x-4|+|x-3|<2, we get:
|4.5 - 4| + |4.5 - 3| < 2
0.5+ 1.5 < 2
2 < 2.
Doesn't work.
Since x=4.5 is NOT a valid solution, eliminate any answer choice that includes x=4.5 within its range.
Eliminate A and B.

The correct answer is C.

For an algebraic approach, check my posts here:
https://www.beatthegmat.com/absolute-val ... 86129.html

Another approach:

(|x-4| + |x-3|) < 2

So the average of the two is less than 1.

From here we want a number x for which

1:: The distance between x and 3
AND
2:: The distance between x and 4

sum to less than 2.

On the low end, 2.5 would have distances of 0.5 + 1.5 = 2, respectively. So our number > 2.5.

On the high end, 4.5 would have distances of 1.5 + 0.5 = 2, respectively. So our number < 4.5.

It follows that anything in the range 2.5 < x < 4.5 would have a smaller sum, so we're set.

Which of the following is satisfied with |x-4|+|x-3|<2?

A. 1<x<5 B. 2<x<5 C. 2.5<x<4.5 D. 2.5<x<4 E. 3<x<4


--> If there is addition when there are 2 absolute values, you can just ignore the middle. That is, |x-4|+|x-3|<2 -> |x-4+x-3|<2 -> |2x-7|<2, -2<2x-7<2, 5<2x<9, 5/2<x<9/2 -> 2.5<x<4.5
Therefore, the answer is C.

Max@Math Revolution wrote: --> If there is addition when there are 2 absolute values, you can just ignore the middle. That is, |x-4|+|x-3|<2 -> |x-4+x-3|<2

|x - 4| + |x - 3| = |x - 4 + x - 3| does not hold.

Suppose that x = 3.5. The left hand side is .5 + .5, and the right hand side is 0, leaving us with 1 = 0.