If x is not equal to 0, is |x| less than 1?

[niitsm]

Hi

can you please tell me whether I am allowed to do this or not

X/|x|<x

X<|x|x. We can do that b/c x is pos

0< x|x| -x

0<x (|x| -1)

X>0 &|x| -1> 0

Thanks

[tim]

I'm assuming you mean X and x to be the same variable. Your first step you can do because |x| is positive, NOT because "x is pos" as you said. Your final step is incorrect as well - not only would you not use "&" in such a case to denote multiple solutions ("or" is preferred), but you also can't use the factors of zero trick (if ab=0 then a=0 or b=0) on an inequality the way you can on an equation.

[RobertoB400]

I thought absolute values were super easy until I realized that I'm getting them wrong on the GMAT. When we have a DS question with x in an absolute value do we have to check if statement 1 (and subsequently statm 2) satisfies at least one scenario of the absolute value (just the positive scenario for example) or does it have to satisfy both the positive and negative scenarios? Since x can only have one value on the GMAT problems, shall we just consider the problem totally insufficient simply because the absolute value automatically presents two potential scenarios of x (positive and negative) unless both scenarios lead to the same value such as zero? Is the logic correct?

[RonPurewal]

just remember the basics:

• you can only consider values that actually satisfy the statement(s) at hand.

• if, from those values, you can arrive at 2 or more different answers to the question, then "not sufficient".

[RonPurewal]

for the question here, there are only 2 possible answers in the whole world:
• "yes, |x| < 1"
• "no, |x| ≥ 1"

so, if you can make both of these happen, then that's "not sufficient".

it's easy to forget this goal, so just make a device somewhere on your scratch paper to help you remember.
e.g., two rows that say "|x| < 1?" and "|x| ≥ 1?", a column for statement 1, and a column for statement 2. if you can check off both things in a column, then "not sufficient". (if you reach the point of testing the statements together, just add another column for that.)

[ErikM411]

statement (2) means that x is negative.
this is not enough information to tell whether |x| is less than 1.
insufficient.

--

to interpret statement (1), note that the fraction x/|x| is equal to 1 for any positive value of x, and equal to -1 for any negative value of x.
therefore, to solve this equation, and just consider the positive and negative cases separately.
if x is a positive number, then this inequality can be rewritten as 1 < x.
if x is a negative number, then this inequality can be rewritten as -1 < x. since this only applies to negative values, we can amend this to give -1 < x < 0.

therefore, statement (1) means that EITHER x > 1 OR -1 < x < 0.
for the first possibility, |x| is greater than 1; for the second, |x| is less than 1. insufficient.

--

together:
the only interval that satisfies both statements is -1 < x < 0, in which all numbers satisfy |x| < 1.
sufficient.

I don't understand the "x<0". All I get out of this is that -1 < x. How do I conclude that x is less than 0?

[RonPurewal]

in that part, the reasoning is restricted to negative values, since those are the only values for which the fraction |x|/x is equal to -1. so, the situation is "-1 < x and x is negative".

in other words, that derivation doesn't apply to positive values (since the fraction no longer equals -1 for such values), so positive values are automatically excluded from the result.

[EricH836]

The explanation provides an algebraic solution, not a testing cases solution. I've noticed that for these more difficult inequality problems, going the theoretical route is easier to understand and faster to use than testing cases (e.g., if x>0, then Y and if x<0 then Z...then see which statements satisfy). Is there any guideline for using theory vs. testing cases for these inequality problems, or does it all come down to personal preference?

[RonPurewal]

The explanation provides an algebraic solution, not a testing cases solution. I've noticed that for these more difficult inequality problems, going the theoretical route is easier to understand and faster to use than testing cases (e.g., if x>0, then Y and if x<0 then Z...then see which statements satisfy). Is there any guideline for using theory vs. testing cases for these inequality problems, or does it all come down to personal preference?

^^
• do stuff
• don't NOT do stuff

• whatever comes to mind first... TRY IT

• NEVER EVER EVER sit there wondering how long something will take—just TRY IT

• if you're not making clear progress... STOP what you're doing, and TRY SOMETHING ELSE

...that's it.