Inequality Problem : 10 seconds solution requested

[ygdrasil24]

If 1<x<9, then which of the following represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

:roll:

[Zarrolou]

The answer is D.

|-5+x|<4
-5+x>=0, x>=5
So we can split the absolute value in 2 cases:
1)Condition: x>=5 (x is positive)
-5+x<4 x<9
2)Condition: x<4 (x is negative)
+5-x<4 x>1

Thus 1<x<9

[Anju@Gurome]

If 1<x<9, then which of the following represents this condition?

All the options include absolute value.
That is our clue that we have to think in terms of distance on the number line.

As x lies between 1 and 9, the distance of x from the midpoint of 1 and 9 on the number line must be less than the distance of the midpoint from either 1 or 9.

Hence, |x - midpoint of 1 and 9 on the number line| < |9 - midpoint of 1 and 9 on the number line|
-----> |x - (1 + 9)/2| < |9 - (1 + 9)/2|
-----> |x - 5| < |9 - 5|
-----> |x - 5| < 4

The correct answer is D.

[GMATGuruNY]

If 1<x<9, then which of the following represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

An alternate approach:

Since the desired range is 1<x<9, every value between 1 and 9 must work in the correct answer choice.
Let x=8:
A. |8|<3
B. |8+5|<4
C. |8-1|<9
D. |-5+8|<4
E. |3+8|<5
Since x=8 does not work in A, B, or E, eliminate A, B and E.

Since the desired range is 1<x<9, no value less than 1 can work in the correct answer choice.
Let x=0:
C. |0-1|<9
Since x=0 works in C -- and x=0 is OUTSIDE the desired range -- eliminate C.

The correct answer is D.

[saurav.jha]

In problems of this kind just follow the approach..|x| < n implies -n < x < n .
Use this and get the answer..Answer shall be option D.

[Brent@GMATPrepNow]

If 1<x<9, then which of the following represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

We can take 1 < x < 9 and create equivalent inequalities be adding or subtracting the same amount to/from all 3 parts.
For example, 1 < x < 9 is the same as 1+3 < x+3 < 9+3 (or 4 < x+3 < 12)
Similarly, 1 < x < 9 is the same as 1-2 < x-2 < 9-2 (or -1 < x-2 < 7)

IMPORTANT: Our goal is to use the fact that x < |k| is the same as -k < x < k (for positive x and k)
So, we want to create an equivalent inequality such that the variable part (the part with x in it) is between two values with the same magnitude (e.g., 3 and -3)

Now notice that 1 < x < 9 is the same as 1-5 < x-5 < 9-5, which simplifies to be -4 < x-5 < 4
Since -4 < x-5 < 4 can be rewritten as |x-5| < 4, the correct answer is B

Cheers,
Brent

[Lifetron]

If 1<x<9, then which of the following represents this condition?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

With mod questions, the flow is this

|x|<2
x<2 and x>-2

Now,
the question has the upper limit as 9, which is the "less-than" limit. So, jus take that limit.
Only D gives 9. The answer could have been a smaller range but, when you check the "greater-than" limit. It is 1.
Hence, D !

[Gurpreet singh]

Test each given answer with both the conditions.


-4<|-5+x|<4
add 5
1<|x|<9

correct answer is D.

[Matt@VeritasPrep]

10 second solution:

1 < x < 9

5-4 < x < 5+4

-4 < x - 5 < 4

|x - 5| < 4

[Matt@VeritasPrep]

20 second solution:

x = 2 and x = 8 must work, x = 1 and x = 9 must not.

Testing all the answers, only D obeys these conditions.