Someone help me out ya'll

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Kueh wrote on 2011-08-27 05:14

I'm convinced this was a trick question, and no one I've asked got the same answer I got.

Solve this inequality:

|2x+3|<1

Even at first glance, I thought it was pretty obvious what the answer was, and I got what I thought I would when I actually solved it. But no one got what I got. T~T

I'm just nervous cause this was a question on a 5 question quiz, so it's like 20%.
Cathaoir wrote on 2011-08-27 05:23

Set it up as two equations. Flip the inequality and switch the signs of the "1." You'll get "2x+3<1" and "2x+3>-1."
Kueh wrote on 2011-08-27 05:26

Quote from Cathaoir;567514:
Set it up as two equations. Flip the inequality and switch the signs of the "1." You'll get "2x+3<1" and "2x+3>-1."

Actually, you're supposed to switch the signs of whats in the absolute value sign, so it'd be

2x+3<1
and
-2x-3<=1, or 2x+3>=1

What I ended up getting was "No solutions" but I asked around and pretty much everyone had something different.
Cathaoir wrote on 2011-08-27 05:29

No it's not. Think about it this way. When you get an answer to an absolute value, it's positive right? You have positive 1 as the answer. What's the absolute value of negative one? It's one. Trust me. I tutor Pre-calculus and Trigonometry at my school. Solve it out and tell me if it's right or not.

Edit: I decided to solve it out. "2x+3<1" will give you "x<-1" and "2x+3>-1" will give you "x>-2." If you put them together, you'll get the values between

-2<x<-1

Since that is true. Any number between -2 and -1 will work. To test it out, let's choose "-1.5"

So you'll need to plug it into the original equation.

|2(-1.5)+3|<1
|-3+3|<1
0<1

The above is what happens when you plug in "-1.5" into the original equation. You get "0" as the answer, which is less than one.

You're welcome.
Kueh wrote on 2011-08-27 05:54

Quote from Cathaoir;567525:
Solve it out and tell me if it's right or not.

If those two equations are right or not? Well even though those are close to what I got, that's still not solved. The answer is the entire set of numbers who make the original inequality true, and the teacher wants it in interval notation.

Also, now that I'm reviewing my work, I'm not sure that I did it right, could you explain the process by which you got your equations? I'll show mine, too.

Absolute Value |x| is defined as {x, if x>1; -x, if x<=1}
So for values where x is positive, the equation for absolute value |2x+3| is 2x+3.
For values where x is zero or negative, the equation is -(2x+3) or -2x-3.

So then I plugged those into two inequalities:
2x+3<1 and -2x-3<1

First equation:
2x+3<1
2x<-2
x<-1
So all values where x is less than -1 is the solution for this half, but this half only exists while x is already positive, so this half has no solutions.

Second equation (I'm re-solving this, it's different from what I thought at first):
-2x-3<1
-2x<4
x>-2
So all values where x is more than or equal to -2, so this half of the equation's solution is [-2,0] when you take into account that this is only the equation when x is negative or zero.

So the final answer is [-2,0]

Now I'm really doubting what both what I put first and what I have now for the second equation.

[edit: reading your edit]

[edit edit: But when I solve it by graphing, my original answer makes the most sense! I'm confused. Also, still reading your edit.]
Cathaoir wrote on 2011-08-27 06:03

Quote from Lyre;567550:
If those two equations are right or not? Well even though those are close to what I got, that's still not solved. The answer is the entire set of numbers who make the original inequality true, and the teacher wants it in interval notation.

Also, now that I'm reviewing my work, I'm not sure that I did it right, could you explain the process by which you got your equations? I'll show mine, too.

Now I'm really doubting what both what I put first and what I have now for the second equation.

[edit: reading your edit]

Say you're trying to get the absolute value of an equation, not an inequality. You get..

|2x+3|=1
which will give you the two equations:
2x+3=1 and 2x+3=-1.

That's the basis how I got my two inequalities.
Since you have to change the inequality when the signs change, that is why the 2nd inequality's sign changed.

To check my equations:

2x+3=1 (-1)
2(-1)+3=1
-2+3=1
1=1

2x+3=-1 (-2)
2(-2)+3=-1
-4+3=-1
-1=-1

The interval notation of the answer I got would be...

(-2,-1)

You have to make sure that you put parentheses, because you are not including the answers, rather the values greater than or less than.
Kueh wrote on 2011-08-27 06:09

Okay, it seems like I graphed it super wrong, and omitted the first equation's solution when I should have kept it.

Thanks for the help!

(But could you tell me why I should keep the first equation? That equation is only for x values more than 0, so how can any negative solutions work?)
Cathaoir wrote on 2011-08-27 06:24

Quote from Lyre;567565:
Okay, it seems like I graphed it super wrong, and omitted the first equation's solution when I should have kept it.

Thanks for the help!

(But could you tell me why I should keep the first equation? That equation is only for x values more than 0, so how can any negative solutions work?)

The solutions of an absolute value is different that the answer of the absolute value itself. It's ok to have a solution to an absolute value problem that is negative. If the answer of the absolute value equation is negative, then it is no solution.

I'll give you examples with the solutions I got from this equation.
When you plug in the negative values, no matter what sign the outcome is, it will turn positive due to the absolute value.

|2(-1)+3|=1
|-2+3|=1
|1|=1
1=1

|2(-2)+3|=1
|-4+3|=1
|-1|=1
1=1

Get it now?
Kueh wrote on 2011-08-27 06:26

Quote from Cathaoir;567580:
Get it now?

Not really, lol. I know what and absolute value does, what I don't understand is how I can have negative solutions for a set of only positive numbers. But you helped me enough already, so thanks. I'll find out later.
Cathaoir wrote on 2011-08-27 06:38

Quote from Lyre;567581:
Not really, lol. I know what and absolute value does, what I don't understand is how I can have negative solutions for a set of only positive numbers. But you helped me enough already, so thanks. I'll find out later.

Ehh. I got nothing to do atm. The reason why negative values account for this inequality is because when plugged into the inequality, it will be a positive number due to the absolute value. The answer itself isn't a negative, since the answer to the inequality is "1."

The solutions to "x" are negative for this specific inequality because those are the values that will give "less than" 1 when put into the absolute value.