Question

Explain, please:
|x| < a means x < a or -x < -a
|x| < a means -a < x and x > a
or we say -a < x < a is same as |x| < a

Answer 1

|x| means the absolute value of x, or the distance of x away from 0.
Therefore, if x<0, |x| returns -x
if x > 0 , |x| returns x

Answer 2

ok but i must compare the two hamburgers by comparing the volume of the beef in each burger. the volume V of a burger is V= Ah, whee A is the area and h is the height. the area of the lite burger is nr^2, where r is the radius (half the distance across the middle) of the burger. for the lite burger, r= 2 inches, so its area is

Answer 3

In other words or at least more words: positive numbers return themselves
negative numbers turn positive
\[\left| -2 \right| = 2 and \left| 2 \right| = 2\]

Answer 4

Thank you, I understand that. But then how come for a question like this |5x -2| < 4 has two answers, x < 6/5 or x > -2/5 ?

Answer 5

This is easier to see on a number line so let me try something easier |x| > 4
the answer is x<-4 or x>4....
does it make sense that you have to go more than 4 units away from zero in either direction because the || symbols ignore positive/negative?

Answer 6

who says |5x-2|<4 has two answers (solutions)? first of all there are an infinite number of solutions, but they are contained in one interval. the interval is
-2/5 < x < 6/5 which is an AND statement: x is greater than -2/5 and less than 6/5

Answer 7

Oh okay, I understand that now. But then how do we define whether x is > or < ? This is my initial question, maybe before I was framing it wrong.

Answer 8

let me give you a simple example. start with | x -1 | < 2. this says x is within 2 units of 1. so if you draw a number line you can see you can go up to three and down to -1, so the solution is
-1 < x < 3

Answer 9

wheres if we say | x - 1| > 2 this means x is greater than two units from 1. so we can be below -1 or above 3. the answer is therefore two intervals, all numbers less than -1 or all numbers greater than 3. you have to write two intervals:
x < -1 or x > 3

Answer 10

if you draw a number line you can see it easily.
| x - a | < b will be one interval (assuming b is greater than or equal to zero, otherwise it makes no sense) and
| x - a | > b will be two intervals.

Answer 11

it is really not that complicated if you remember that | x - a | is the distance between x and a. so if i am on a highway that runs east west (like the number line) and I say i am within 5 miles from mile marker 100, that means i must be somewhere between mile marker 95 and mile marker 105. i.e. the solution to | x - 100 | < 5 is
95 < x < 105
but if i say i am further than 5 miles from mile marker 100 i must be either east of marker 105 or west of marker 95 i.e.
|x -100| > 5 means
x < 95 or x > 105

Answer 12

I understand this too. But when it comes to complex questions, I get confused. Like for example: |1-3x| > 8. So in the solution set, how do we determine that x is < -7/3 and x > 3 when we started with 1-3x >8 and 1-3x < -8 respectively?

Answer 13

ok first of all i hope it is clear that |1-3x| = |3x-1|
try it with numbers, or just know that |a-b|=|b-a|

Answer 14

Okay I get that.

Answer 15

so there is no difference between solving |1-3x|>8 and |3x-1|>8

Answer 16

second of all, as for my example above, you know that if you are solving |x-a|<b you will have one interval (not one solution) and if you have | x - a| > b you will have two solutions.

Answer 17

and something they often forget to mention is that there is really know difference between solving |x - a| < b and |x - a | > b
after all, if i know one i know the other. for example if the solution to |x - a| < b is
5 < x < 10 then i know the solution to
|x - a|> b is
x < 5 or x > 10 after all those are the logical choices right? either greater than or less than.

Answer 18

so how would i solve
|1-3x|> 8 ? first of all i hate having to remember to switch the inequality when dividing by a negative number so i would rewrite it as
|3x-1|>8

Answer 19

then i have a choice. i could solve |3x-1|< 8 and take everything else, like this:
-8 < 3x -1 < 8
-7 < 3x < 9
-7/3 < x < 3 and that is exactly what i don't want so i say
x < -7/3 or x > 3

Answer 20

or i could do what they teach you and say
|3x - 1|> 8 means two intervals (we know that) therefore two inequalities to start:
3x-1< -8 or 3x - 1 > 8 and solve these separately.
3x -1 < -8
3x < -7
x < -7/3
OR
3x-1 > 8
3x > 9
x > 3
either way you need to know when you will have one interval (compound inequality) or two

Answer 21

(just to be sure) ..and that is when |x-a| < b there will be one interval and when |x-a| > b two intervals?

Answer 22

yes exactly. if you can remember that you will be in good shape

Answer 23

thank you so much. this explains much better!

Answer 24

|5x+17|>9 two intervals
|3x+7|<12 on interval

Answer 25

welcome

Answer 26

and while i have your attention one more thing"
Like for example: |1-3x| > 8. So in the solution set, how do we determine that x is < -7/3 and x > 3 when we started with 1-3x >8 and 1-3x < -8 respectively?
in this sentence your "ands" should be "OR"s
x < 2 OR x > 7
there is not such thing as x < 2 and x > 7

Answer 27

Lol oh yeah, thanks.