Algebra
posted by Andrea .
The (absolute value) x20 (absolute value) is less than or equal to 6. I did it first with six positive and got x is lesss than or equal to 26. Then I made the six negative and got x is less than or equal to 14. But this is impossible for these to both be true, right? How come my book says no solution isn't an answer? (It has 14 is less than or equal to x which is less than or equal to 26 as the answer). How did they get this?

abs(x20)<=6
but absolute anything is greater than zero, so
0<=abs(x20)<=6
which now means x can go to 14 to 26, right?
Now, why in the world did you solve it for
abs(x20)>=6 If that had be the question, you would be right, however, it is not the problem. 
The answer is not a single number, because it is an inequality. Inequalities in the real domain (ℝ) has infinite number of solutions, and is expressed in the form of an interval.
When dealing with the absolute value function, it makes life easier to split the inequality into two equations. After that, the two solution sets will be intersected to give the final solution.
For example, to solve
x3 ≤ 2
we write
x3 ≤ 2 ....(1), and
(x3) ≤2 ....(2).
The solution of (1) gives
x≤5
and the solution of (2) gives
x≥1
So the answer would be
1≤x≤5, or expressin interval notation,
x ∈ [1,5]