|x-1| < |2x+3|
I'm just wondering if somebody could verify that I've done this right.
Case 1: x < -3/2
-(x-1) < -(2x+3)
x < -4 [Valid because it satisfies the case]
Case 2: -3/2 <= x < 1
-(x-1) < 2x+3
x > -2/3 [Valid because it satisfies the case]
Case 3: x >= 1
x-1 < 2x+3
x > -4 [Not valid because it doesn't satisfy x >= 1]
Therefore {x < -4} U {x > -2/3}
On
your 8th and 11th lines are wrong. You should combine the constraints on X, which you imposed, with the results you got.
8th line should be $-2/3 < x < 1$
11th line should be $x \geq 1$
Now, when you combine all your results, you get the answer, which matches yours.
Another way to help yourself is to simply plot the functions on the XY-plane.
Hint: Assuming $$x\geq 1$$ we gert $$x-1<2x+3$$ In the next case we have $$-\frac{3}{2}\le x<1$$ and we get $$-x+1<2x+3$$ And for $$x<-\frac{3}{2}$$ we get $$-x+1<-2x-3$$