Answer :

Given:


< 2, x ϵ R.


Intervals of |x - 3|


|x – 3| = -(x – 3) or (x – 3)


When |x - 3| = x – 3


x – 3 ≥ 0


Therefore, x ≥ 3


When |x - 3| = -(x – 3)


(x – 3) < 0


Therefore, x < 3


Intervals: x ≥ 3 or x < 3


Domain of < 2:


is not defined for x = 0


Therefore, x > 0 or x < 0


Now, combining intervals and domain:


x < 0 or 0 < x < 3 or x ≥ 3


For x = 0



Now, subtracting 2 from both the sides





Signs of 3 – 4x:


3 – 4x = 0


(Subtracting 3 from both the sides and then dividing both sides by -1)


3 – 4x > 0


(Subtracting 3 from both the sides and then multiplying both sides by -1)


3 – 4x < 0


(Subtracting 3 from both the sides and then multiplying both sides by -1)


Signs of x:


x = 0


x < 0


x > 0


Intervals satisfying the required condition: < 0


x < 0 or


Combining the intervals:


x < 0 or and x < 0


Merging the overlapping intervals:


x < 0


Similarly, for 0 < x < 3


x < 0 or and 0 < x < 3


Merging the overlapping intervals:


< x < 3


For, x ≥ 3



Now, subtracting 2 from both the sides





Signs of -3 – 2x:


-3 – 2x = 0


(Adding 3 to both the sides and then dividing both sides by -2)


-3 – 2x > 0


(Adding 3 to both the sides and then multiplying both sides by -1)


-3 – 2x < 0


(Adding 3 to both the sides and then multiplying both sides by -1)


Signs of x:


x = 0


x < 0


x > 0


Intervals satisfying the required condition: < 0


or x > 0


Combining the intervals:


or x > 0 and x ≥ 3


Merging the overlapping intervals:


x ≥ 3


Combining all the intervals:


x < 0 or or x ≥ 3


Merging overlapping intervals:


x < 0 and


Therefore,



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