Question

|2x -4 | > |1 - x|
I am still having trouble with these questions in which there are two absolute value terms. Would someone mind giving me an explanation (both in math, and in English to make it more intuitive) as to how you are going about with solving for x in this absolute value inequality.

I would REALLY appreciate it, as these are the types of problems that are throwing me off for a quiet a while now.

Answer 1

i can say it in english, but i don't know how much that will help

Answer 2

first of all there is no difference between
\[|1-x| \]and
\[|x-1|\]

Answer 3

so your are looking at
\[|2x-4|>|x-1|\]

Answer 4

now to remove the absolute values,
\[|2x-4|=2x-4\] so long as x > 2 otherwise
\[|2x-4|=-2x+4\]

Answer 5

and likewise
\[|x-1|=x-1\text { if } x>1\] otherwise
\[|x-1|=-x+1\]

Answer 6

so you have cases to consider. whether
\[x<1, 1<x<2, x> 2\]

Answer 7

Since |1-x| =|x-1| (as pointed out by satellite73), let's work with |2x - 4 | > |x - 1| instead of |2x - 4 | > |1 - x|.

Let's recall that by definition of absolute value
|2x - 4| = 2x - 4 if x>=2 &
|2x - 4| = -(2x - 4) if x<2
and
|x - 1| = x - 1 if x>=1 &
|x - 1| = -(x - 1) if x<1

So we need to consider 3 cases :

Case #1 - when x>=2
Case #2 - when x>=1 and x<2
Case #3 - when x<1

Now, according to the above definitions of absolute value:

For Case #1 : |2x - 4| = 2x - 4 and |x - 1| = x - 1.
For Case #2 : |2x - 4| = -(2x - 4) and |x - 1| = x - 1
For Case #3 : |2x - 4| = -(2x - 4) and |x - 1| = -(x - 1)

For each case we will solve an inequality :

Case #1 : solve 2x - 4 > x - 1
Case #2 : solve -(2x - 4) > x - 1
Case #3 : solve -(2x - 4) > -(x - 1)

Case #1 solution is : x > 3
Case #2 solution is : x < 5/3
Case #3 solution is : x < 3

So our solution to the original inequality is x < 5/3 or x > 3. (Note : we used x < 5/3 instead of x < 3 because values in the range 5/3 <= x < 3 would satisfy Case #3 but would not satisfy Case #2)