Question

Help please!
Question with inequalities (I've tried cases and squaring both sides but it doesn't work)

Answer 1

(1/(|x|-1))<=2/|x|

Answer 2

\[1\div \left( \left| x\right| -1\right)\le 2\div \left| x \right|\]

Answer 3

let right and left be equal
\[\frac{ 1 }{ \left| x \right|-1 } = \frac{ 2 }{ x}\]
cross multiply
\[\left| x \right| = 2\left| x \right|-1\]
transpose
\[\left| x \right|=1\]

then substitute the value of x by 1

\[0\le2\]

Answer 4

did you move everything to the left side, and combine the fractions ?

Answer 5

cross multiply

Answer 6

you should get
\[ \frac{2-|x| }{|x|(|x|-1)}\le 0 \]

|x| is always positive.
we need to figure out the signs for the other two factors.

Answer 7

for the fraction to be negative, we require either
1) 2-|x| positive and |x|-1 negative or
2) 2-|x| negative and |x|-1 positive

case 1)
2-|x| >= 0
|x|-1<0 (not =0 , because we do not want divide by 0)

the first line gives: 2 >= |x| or |x| <= 2
second line gives: |x| < 1
for both to be true, we must have |x| < 1
this means: x <1
or -x < 1 --> x > -1
thus : -1 < x < 1 (note x \(\ne\) 0 because no divide by 0 allowed)

case 2)
2-|x| <= 0 ---> |x| >= 2
|x| -1 > 0 ----> |x| >1

for both to be true at the same time, we use |x| >= 2
x <= -2 or x>= 2

Answer 8

on a number line