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  • Solve for x the inequalities in | x-1|-1 /|x-1|-2<0

Solve for x, the inequalities in \frac{\left | x-2 \right | -1}{\left | x-2 \right | -2}\leq 0

Answers (1)

Given: \frac{\left |x-2 \right |-1}{\left |x-2 \right |-2}\leq 0

Now, let us assume that:

y = |x-2|

thus, \frac{ y-1}{y-2}\leq 0

if y< 1,

y-1<0 & y-2<0 , & thus ,

y-1 /y-2> 0, viz. Not  needed

now, 

if 1\leq y< 2,

y-1\geq 0 & y-2< 0,

& hence,

y-1/y-2< 0… thus, we get the answer we need.

Now, 1\leq y< 2,

Thus,  1\leq \left | x-2 \right |< 2

Now, from this, we will get 2 cases, which are:

1\leq x-2< 2= 3\leq x< 4

& 1\leq -\left (x-2 \right )< 2= 1\leq -x+2< 2

On multiplying each term by -1, we get:

-2\leq x-2< -1

Now, we’ll add 2 to each term, we get:

0\leq x< 1

Therefore,  \left [ 0,1 \right ]\upsilon \left [ 3,4 \right ]

 

 

 

Posted by

infoexpert22

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