1<x<9. What inequality represents this condition?

[RonPurewal]

problem courtesy of a student, from an app called "GMAT Toolkit":

1<x<9
What inequality represents this condition?

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


2 ways to solve this one.

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1) PLUG IN NUMBERS
the original inequality is very good for plug-in, because it's so straightforward -- you can just plug in a bunch of numbers between 1 and 9 and see which choices work. then, if necessary, you can plug in numbers outside the interval 1<x<9 and see which choices don't work.

try x = 2. (this value is in the interval 1<x<9, so we want a choice for which it works.)
(a) 2 < 3 ... true, keep it for now.
(b) 7 < 4 ... false, eliminate.
(c) 1 < 9 ... true, keep it for now.
(d) 3 < 4 ... true, keep it for now.
(e) 5 < 5 ... falso, eliminate.
so now we're down to a, c, d.

try x = 8. (this value is in the interval 1<x<9, so we want a choice for which it works.)
(a) 8 < 3 ... false, eliminate.
(c) 7 < 9 ... true, keep it for now.
(d) 3 < 4 ... true, keep it for now.
so now we're down to c and d.

try x = 0. (this value is OUTSIDE the interval 1<x<9, so we want a choice for which it DOES NOT work.)
(c) 1 < 9 ... this works, but it shouldn't, so eliminate.
(d) 5 < 4 ... this doesn't work (and it shouldn't!) keep it.

answer (d).

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2) KNOW THE MEANING OF ABSOLUTE VALUE DIFFERENCES

know this:
FACT:
|A - B| is the DISTANCE between quantity 'A' and quantity 'B'.

for instance, |x - 2| is the distance between x and 2. similarly, |x + 2| is the distance between x and -2 (because x + 2 is the same as x - (-2)).

in this case, the interval is 1 < x < 9.
the CENTER point of this interval is x = 5; the interval is 4 units away from 5 on either side.
therefore, the interval may be re-expressed as "x is less than 4 units away from 5".
or, "the distance between x and 5 is less than 4".
or, |x - 5| < 4.

(d).

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3) SOLVE THE INEQUALITIES IN THE ANSWER CHOICES

remember that
|quantity| < #
may be rephrased as
-# < quantity < #
(as long as "#" is positive).

so:

(a)
-3 < x < 3
incorrect

(b)
-4 < x + 5 < 4
-9 < x < -1
incorrect

(c)
-9 < x - 1 < 9
-8 < x < 10
incorrect

(d)
-4 < -5 + x < 4
1 < x < 9
correct

done

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for a very similar problem, which is almost certainly the inspiration for this one, see OG12 #130.

note that the turning around of |x - 5| into |-5 + x| is not something that the GMAT would be likely to do; the GMAT more or less always expresses mathematical expressions in a way that's as conventional/standard as possible.

[Milanproda1]

Thanks Ron nice explanation.

[ambikasrinivas]

thank you for the explanation - just a quick question:

(c)
-9 < x - 1 < 9
-10 < x < 8
incorrect

should this be -8< x < 10 instead?

[jnelson0612]

thank you for the explanation - just a quick question:

(c)
-9 < x - 1 < 9
-10 < x < 8
incorrect

should this be -8< x < 10 instead?

Nice catch! I'm going to edit the post above. Thank you!