Mathmatics Question?? Absolute Value and Inequalities

Ok, I received some help earlier.
I'm hoping for some kind souls to help again.
Now I'm completely lost on a problem.

Solve and Express in interval notation.
1<=|x|<=4

I came up with this
-4<=x<=4 and x<=-1 and 1<=x

the book says it is
[-4,-1]U[1,4]

I know what interval notation is, but I'm lost on the how they grouped the different x values.
What step am I missing to get the books answer?

[edit1]
Do we default grouping the negative values together and the positive values together?


I'm not sure, but let me try to explain why I see it grouping in this way.

-4 is less then or equal to x which is less then or equal to 4
this covers all values from -4 to 4
x is also less then or equal to -1
x is greater than or equal to 1

-4 to 4 is my outside boundries
so
-4 to -1 is one of the boundries

1 to 4 is the upper side boundry.

Is this the way they are looking at it or is there a fail safe rule that I can use?
[/edit1]

basically, imagine this:

1 <= |x| <= 4

we get the following

1 <= +x <= 4

now, if we put -x and remove the absolute value we get

1 <= -x <= 4

which is(clearing the -)

-4 <= x <= -1

therefore

1 <= |x| <= 4 means

1 <= x <= 4 &&
-4 <= x <= -1

that is where your two intervals come from, sorry if I am being unclear.

You have to remember that you must turn it into two inequalities because of the absolute value sign. Because of the value sign you end up with that one inequality equaling both 1<=x<=4 and 1<=-x<=4. What you then do is flip the signs for the second one, making -4<=x<=-1

on the prior problem I'm not sure why it uses a connecting "or" instead of a connecting "and" in the interval notation.

here is a similar problem.
I believe I did it correctly according to what you have mentioned and what is in the book.
Unfortunately I can not check this one in the book.

0<|x-5|<=1/2

-1/2<=x-5<=1/2
9/2<=x<=11/2

0<x-5
5<x
x-5<0
x<5

therefore in interval notation

[-1/2,5)*and/or*(5,1/2]

*and/or !!not sure which or why i would use "and" over "or" in this problem!!*

I'm not sure how you got to the line -1/2<=x-5<=1/2. For the first inequality I got 0<x-5<=1/2 then add five to each side for 5<x<=5.5. For the second I got 0<-(x-5)<=1/2 then change the sign and add five so 4.5<x<=5. I'm not 100% sure on this second one though, I havent done inequalities in a couple of years.

Post 400!

>>9/2<=x<=11/2

This looks correct to me, so [9/2,11/2] I believe would be the answer.

Lets see...

0<|9/2-5|<=1/2 = 0<|-1/2|<=1/2 = 0<1/2<=1/2 <--yes

0<|11/2-5|<=1/2 = 0<|1/2|<=1/2 <---yes

Edit: Wait I don't think that's right...

Ok I did it again and got:

[9/2,5)u(5,11/2]

Oh I think thats what Draco got hehe

Concisely,

0 < abs(x - 5) <= .5

case i.
0 < x - 5 <= .5
5 < x <= 5.5

case ii.
0 < 5 - x <= .5
-5 < -x <= -4.5
5 > x >= 4.5

x in [4.5, 5) U (5, 5.5]

An easy way to check is to insert a value and watch what happens:

x = 4.7
0 < (4.7 - 5) <= .5
0 < abs(-.3) <= .5
0 < .3 <= .5 (true)

x = 5.2
0 < abs(5.2 - 5) <= .5
0 < abs(.2) <= .5
0 < .2 <= .5 (true)

x = 5
0 < abs(5 - 5) <= .5
0 < 0 <= .5 (false)

Just wondering, is this calculus? I haven't taken it yet...but this stuff seems really interesting and I'm finding I'm actually understanding it...which is really cool.

this is intermediate to advanced algebra, more of a pre-calc level

Oh, I kinda thought it was a bit simple and was wondering why I was actually understanding it, lol. I've just never seen these examples in class before so they seemed odd to me I guess.

yeah, I understand you. If this stuff is intersting for you you're going to love calculus.

this is the last college algebra class prior to trig.
these rules are probably used in calculas though.

follow up question:

link referencing the below rule
http://www.wtamu.edu/academic/anns/m...22_linineq.htm

rule: Absolute Value and Inequalities

if
|x|>1
then
x<-1
and
1<x

So does that not throw a wrench into this portion of the problem
"0 < |x - 5| "
wouldn't that disprove your calculations?

Quote:

---

case i.
0 < x - 5 <= .5
5 < x <= 5.5

case ii.
0 < 5 - x <= .5
-5 < -x <= -4.5
5 > x >= 4.5

---

I might not have seen in my book on how the rule you are using works.

Quote:

---

rule: Absolute Value and Inequalities

if
|x|>1
then
x<-1
and
1<x

---

Actually it says OR instead of AND. Think of it this way, it can't be an AND because a number can't be between 4.5 and 5 AND also be between 5 and 5.5 at the same time, it has to be OR. For example, 4.8 works when you do the math, but if it was AND then it wouldn't because when you say AND it must fit both inequalities which it doesn't. It only fits one of the inequalities. No different than if you were programming with &&'s and ||'s.

[Edited]


As for the problem
0 < |x - 5|
Remember that the absolute value always is possible. The only way this inequality can be false is when x - 5 == 0 which it is when x == 5
The solution is thus:
x != 5

As for the original problem,
1<=|x|<=4
Geometrically, |x| is the distance from the origin to x. The distance is between 1 and 4. Thus, the solution is:
1<=x<=4
OR
-4<=x<=-1

OK, perhaps I'll try to explain this a little better this time.

When you encounter an absolute value, you have to separate the equation of inequality into two different equations where you assume that the expression within || are negative and positive respectively.

Then, the solutions is always given by (i) OR (ii)

I'll post some examples:
(EDIT: litte typo here, x <= 0 should be x >= 0
[-4,1] should be [-4,-1])

Do we really need to include the
" and x>=0"
" and x<0"
in the expressions all the time?
YES!

Consider:

Another example, this time an equation:

Here it's also very important to keep track of when x<0 and when x>=0