Absolute Value Equations & Inequalities as Solution Sets?

write solutions in terms of "distance," change absolute value notation to other notations & vice versa (e.g, write lxl<4, lx-5l less than or equal to 6, lxl greater than or equal to 9 as number lines, as words in terms of distance, as intervals, & in set notation; write : [-8,8], (-4,6) as absolute values).

For [-8,8]:
The easiest way to think of how to write this as an absolute value statement is to find the halfway point between -8 and 8. That is the average of the two numbers. (-8 + 8)/2 = 0. 0 then is the point equally far from -8 and 8. So as an absolute value statement: abs(x - 0) <= 8. This gives the inequality statement: -8 <= x - 0 <= 8; -8 <= x <= 8. In interval notation [-8, 8]. The square brackets indicate that -8 and 8 are included in the solution.

For (-4, 6):
Again find the point midway between -4 and 6. (-4 + 6)/2 = 1. 6 and -4 must be equally far away from this point. Both are 5 units away. So the absolute value statement reflecting this is abs(x - 1) < 5. The inequality statement would then be
-5 < x -1 < 5, and the solution is -4 < x < 6. In interval notation this becomes (-4, 6). Parentheses indicate that -4 and 6 are not included in the solution.