Mission2012 wrote:Q. 1 < X < 9, which equation represents this best -

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

|a-b| = the DISTANCE between a and b.

To illustrate:
|x-3| < 2 means the distance between x and 3 is less than 2.
On the number line, this means that x can be up to 2 places to the left of 3 or 2 places to the right of 3:
1...3...5
In other words, x must be between 1 and 5:
1 < x < 5.

In |x-3| < 2, notice the following:
The value in red is equal to the MIDPOINT of the range.
The value in blue is equal to HALF the total range.

Applying this reasoning to 1 < x < 9:
Midpoint = (1+9)/2 = 5.
Half the total range = (9-1)/2 = 4.
Thus, the correct answer choice must look like this:
|x-5| < 4.

The correct answer is D.

Mission2012 wrote:Q. 1 < X < 9, which equation represents this best -

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

An alternate approach is to DISPROVE the answer choices.
The correct answer choice must include EVERY VALUE BETWEEN 1 AND 9 but cannot include any value OUTSIDE this range.

Test an EXTREME value between 1 and 9.
Let x=8.
The correct answer choice must include x=8 within its range.

A. |x| < 3
|8| < 3
8 < 3.
Doesn't work.
Eliminate A.

B. |x + 5| < 4
|8+5| < 4
13 < 4.
Doesn't work.
Eliminate B.

C. |x-1|<9
|8-1| < 9
7 < 9.
This works.
Hold onto C.

D. |-5+x|<4
|-5+8| < 4
3 < 4.
This works.
Hold onto D.

E. |3+x|<5
|3+8|<5
11<5
Doesn't work.
Eliminate E.

In the remaining answer choices, test a value OUTSIDE 1 < x < 9.
Let x=0, which is OUTSIDE 1 < x < 9.
The correct answer CANNOT include x=0 within its range.

Answer choice C: |x-1|<9
|0-1| < 9
1 < 9.
This works.
Since answer choice C includes x=0 within its range, eliminate C.

The correct answer is D.