Data Sufficiency

Hi all,

I am really struggling with questions on inequalities and absolute terms.

Questions such as following DS -

Is |x-1| < 1?

a> (x-1)^2 <= 2
b> x^2 - 1 > 0

Can someone point to good material on develop concepts on these.

Could you provide the Official Answer?

Is |x-1| < 1?

a> (x-1)^2 <= 2
b> x^2 - 1 > 0

First lets look at the Prompt:

|x-1|<1 => (x-1)<1 or (x-1) > -1 .. this gives x < 2 ; x > 0 This means x is between 0 & 2. That is what the question is asking, is x a value between 0 and 2.

Statement 1:

(x-1)^2 <= 2 this also gives 2 cases (loke x^2 = 4 gives x = + - 2); also taking sqrt(2) = 1.4

x-1 <= 1.4 which gives x <= 2.4
x-1 >= -1.4 which gives x >= -0.4

Insufficient (since the range is greater than what we want).

Statement 2:

X^2 > 1.. which gives x > 1 or x < -1

Insufficient x can be in or out of the desired range

Together,

X can be between 1 and 2.4, this again can be inside or outside the range.

So, even together they seem insufficient.

Waiting for the OA

sinsofgmat wrote:Hi all,

I am really struggling with questions on inequalities and absolute terms.

Questions such as following DS -

Is |x-1| < 1?

a> (x-1)^2 < 2
b> x^2 - 1 > 0

Can someone point to good material on develop concepts on these.

Question : Is |x-1| < 1?
Question : Is -1 < x-1 < 1?
Question Rephrased : Is 0 < x < 2?

Statement 1) (x-1)^2 < 2
i.e. -√2 < (x-1) < √2
i.e. -1.4 < (x-1) < 1.4
i.e. -0.4 < x < 2.4
NOT SUFFICIENT

Statement 2) x^2 - 1 > 0
i.e. x^2 > 1
i.e. x > 1 OR x < -1
NOT SUFFICIENT

Combining the Two statements
x > 1 OR x < -1 AND -0.4 < x < 2.4
NOT SUFFICIENT

Answer: Option E

sinsofgmat wrote: Is |x-1| < 1?

a> (x-1)^2 <= 2
b> x^2 - 1 > 0

Number line approach:

|a-b| = the distance between a and b on the number line.
Thus:
|x-1| = the distance between x and 1 on the number line.

Draw a number line:
<-----0-----1-----2----->

The question stem asks whether the distance between x and 1 is less than 1.
In order for x to be less than 1 place away from 1, x must within the range in red.
In other words, x must be between 0 and 2.
Question stem, rephrased:
Is 0 < x < 2?

Both statements are satisfied if x=1.1:
Statement 1: (1.1 - 1)² ≤ 2
Statement 2: (1.1)² - 1 > 0
In this case, x is between 0 and 2.

Both statements are satisfied if x=2:
Statement 1: (2 - 1)² < 2
Statement 2: 2² - 1 > 0
In this case, x is NOT between 0 and 2.

Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.