Question

Solve the inequality 2|x - 3| <= 2 + x.
Hence, find x that satisfies 2|x+3|<= 2-x

I solved first one and got the answer as x: 4/3<= x <=8

Then i moved to the second.

By putting x = -x in the first I got,
2|(-x) + 3)| <= 2 + (-x)
===> 2|3 -x | <= 2 - x
===> 2|x-3| <= 2 - x ,
Which gives the range of x;
x:4/3<=x<=8 ,
Is this also the range that of x,
Which satisfies the inequality ,
2|x+3| <= 2-x ???

Answer 1

4/3<=x<=8 is correct
but when you break the absolute value into two choices, you no longer need the || sign.
2|x - 3| <= 2 + x means that
2(x-3) <= 2+x if x-3 >= 0
2(3-x) <= 2+x if x-3 < 0

Answer 2

I use the following method, it never fails.

2|x+3| ≤ 2-x
Initial condition based on definition of ||,
2-x ≥ 0
-x ≥ -2
x ≤ 2

Now for the split,
+2(x+3) ≤ 2-x <b>AND</b> -2(x+3) ≤ 2-x , (had it been ≥, I would have used OR)
3x ≤ -4 AND -x ≤ 8
x ≤ -4/3 and x ≥ -8
-8 ≤ x ≤ -4/3 , which also satisfies the important initial x ≤ 2 condition