When |x|>1 and |x-3|>1, which of the following is contained in the scope of x?

A. -0.5
B. 0
C. 0.5
D. 1.5
E. 2.5

Contained in the scope of x in simple terms means ' Which of the choices fit in the range of x'.

Now |x|>1 means x<-1 and x>1
|x-3|>1 means x-3>1..x>4 or 3-x>1 or x<2..

Common range is x<-1, 1<x<2 and x>4..
Only 1.5 falls in one of the range above 1<x<2

Use OPTIONS

Fit in the two inequalities |x|>1 and |x-3|>1 to check what fits in.

A. -0.5……|-0.5|>1……NO
B. 0 ……|0|>1……NO
C. 0.5 ……|0.5|>1……NO
D. 1.5 ……|1.5|>1……YES and |1.5-3|>1……YES
E. 2.5 ……|2.5|>1……YES and |2.5-3|>1……NO

Analyse on a NUMBER line

|x|>1 means x has to be more than one away from 0. So discard anything between -1 and 1, both inclusive.
A, B and C are gone.

|x-3|>1 means x will be more than one away from 3. So cannot be between 2 and 4, both inclusive.
E is gone

so D
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Going by the answer options, Only x = 1.5(Option D) is contained in the scope of x

When x=1.5, |x| = 1.5 > 1 and |x-3| = |1.5 - 3| = |-1.5| > 1
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by simplifying modulus, x has following solutions:

X >1
X< -1
X<2
X>4

choice D satisfies 2nd and 3rd solution, so D.
try number line and plot if solutions make it messy.

==> |x|>1 becomes x<-1 or 1<x, and for |x-3|>1, you get x-3<-1 or 1<x-3 --> x<2 or 4<x. Then, what satisfies both becomes 1<x<2 or 4<x.

The answer is D.
Answer: D
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can anyone please explain what scope of x means and how 1.5 was considered.. i am confused

Drawing out on number line makes 1.5 obvious choice

MathRevolution Hi,

Doesn't, x<-1 satisfies both conditions?

Hello,

That is one of the inequalities that satisfies but doesn't satisfy the range.

Combine all possible conditions to get the range or scope.

Thanks
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