Answer :

Given:


1, x ϵ R. – {–2, 2}


Intervals of |x|:


x ≥ 0, |x| = x and x < 0, |x| = -x


Domain of 1


1 is undefined at x = -2 and x = 2


Therefore, Domain: x < -2 or x > 2


Combining intervals with domain:


x<-2, -2<x<0, 0 ≤ x<2, x>2


For x < - 2


1


Subtracting 1 from both the sides


1 -1


0


0


0


Signs of -1 -x:


-1 – x = 0 x = -1


(Adding 1 to both the sides and then dividing by -1 on both the sides)


-1 – x > 0 x < -1


(Adding 1 to both the sides and then multiplying by -1 on both the sides)


-1 – x < 0 x > -1


(Adding 1 to both the sides and then multiplying by -1 on both the sides)


Signs of 2 + x:


2 + x = 0 x = -2 (Subtracting 2 from both the sides)


2 + x > 0 x > -2 (Subtracting 2 from both the sides)


2 + x < 0 x < -2 (Subtracting 2 from both the sides)


Intervals satisfying the required condition: ≥ 0


-2 < x < 1 or x = -1


Merging overlapping intervals:


-2 < x ≤ 1


Combining the intervals:


-2 < x ≤ 1 and x < -2


Merging the overlapping intervals:


No solution.


Similarly, for -2<x<0:


1


Therefore,


Intervals satisfying the required condition: ≥ 0


-2 < x ≤ 1 and x < -2


Merging overlapping intervals:


-2 < x ≤ 1


Combining the intervals:


-2 < x ≤ 1 and -2 < x < 0


Merging the overlapping intervals:


-2 < x ≤ 1


For 0 ≤ x < 2


1


Subtracting 1 from both the sides


1 -1


0


0


0


Signs of x -1:


x - 1 = 0 x = 1(Adding 1 to both the sides)


x - 1 > 0 x > 1(Adding 1 to both the sides)


x - 1 < 0 x < 1(Adding 1 to both the sides)


Signs of 2 + x:


2 + x = 0 x = -2 (Subtracting 2 from both the sides)


2 + x > 0 x > -2 (Subtracting 2 from both the sides)


2 + x < 0 x < -2 (Subtracting 2 from both the sides)


Intervals satisfying the required condition: ≥ 0


1 < x < 2 or x = 1


Merging overlapping intervals:


1 ≤ x < 2


Combining the intervals:


1 ≤ x < 2 and 0 ≤ x < 2


Merging the overlapping intervals:


1 ≤ x < 2


Similarly, for x >2:


1


Therefore,


Intervals satisfying the required condition: ≥ 0


1 < x < 2 or x = 1


Merging overlapping intervals:


1 ≤ x < 2


Combining the intervals:


1 ≤ x < 2 and x > 2


Merging the overlapping intervals:


No solution.


Now, combining all the intervals:


No solution or -2 < x ≤ 1 or 1 ≤ x < 2


Merging the overlapping intervals:


-2 < x ≤ 1 or 1 ≤ x < 2


Thus, x є (-2, -1] υ [1,2)


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