help with math problem answer?

Question

this was my question:

directions: solve the following inequality algebraically.
absolute value 2x-1>3.6
so 2x-1 is an absolute value
3.6 is not an absolute value

i recieved several answers but they all kinda make sense to me....
they were
x= 2.3 and -1.3

x>2.3

x<1.3

x is either greater than 2.3 or less than -2.3

x>2.3, x<-1.3

x > 4.6/2 or |2.3|

which one is right?
answer only if u are sure
:

Answer 1

If |a|>b, then that really means either:

(a) a > b
or else
(b) a < -b

Note that "a" in this case stands for EVERYTHING within the absolute value. So...

| 2x-1 | > 3.6
... means either:

(a) The quantity inside the absolute value is positive.
2x - 1 > 3.6
2x > 4.6
x > 2.3

... or else

(b) The quantity inside the absolute value is negative.
2x - 1 < -3.6
2x < -2.6
x < -1.3

Thus...

The values of x satisfying "| 2x-1 | > 3.6 " are both "x < -1.3" and "x > 2.3". The previous respondent moved "-1" out of the absolute value, but it doesn't work that way, because the quantity in the absolute value signs can be negated.

Consider x = -2, which is in my solution (it's less than -1.3) but not in the other respondents.

| 2x - 1 | > 3.6
| (2*-2) - 1 | > 3.6
| -4 - 1 | > 3.6
| -5 | > 3.6
5 > 3.6

It is a solution.

Answer 2

absolute value 2x-1>3.6
so 2x-1 is an absolute value
3.6 is not an absolute value

I think you mean |2x-1|>3.6
What that means is
If 2x-1 is positive, it is greater than 3.6
If 2x-1 is negative, it is less than -3.6
You have to solve both of these inequalities
2x-1>3.6

Adding 1 to both sides
2x>4.6

Multiply both sides by ½
x>2.3

Now the other horn of the dilemma…
2x-1<-3.6

Adding 1 to both sides
2x< -2.6

Multiplying both sides by ½
x < -1.3

So our solution set is {x| x< -1.3, OR x> 2.3}

Let’s do a little checking…
x=-2<-1.3
|2x-1|>3.6
|2(-2)-1| = |-4-1| = |-5|=5>3.6 As expected

-1.3 ≤ x=0 ≤ 2.3
|2(0)-1| = |-1| = 1 < 3.6 As expected

x=3 > 2.6
|2(3)-1| = |6-1| = |5| = 5 > 3.6 As expected


Therefore:
x= 2.3 and -1.3 This is WRONG

x>2.3 This is half right.

x<1.3 This is wrong.

x is either greater than 2.3 or less than -2.3… Again first half is correct.

x>2.3, x<-1.3 This is CORRECT

x > 4.6/2 or |2.3| Again, half right.

Answer 3

Sakura, one remembers
|x| > a ------> x > a or x < - a

|2x-1|>3.6

2x - 1> 3,6 or 2x - 1< - 3,6

2x > 3.6 + 1 or 2x < - 3.6 + 1

2x > 4.6 or 2x < - 2.6

x > 4.6/2 or x < - 2.6/2

x > 2.3 or x < -1.3

(-oo, - 1.3) U (2.3, +oo)

Answer 4

Answer is greater than 2.3 and less than -2.3.

|2x-1|>3.6

|2x|>4.6

|x|>2.3

This means that if x is positive, it must be greater than 2.3

If x is negative is must be less than 2.3

x > 2.3

OR

x< -2.3

McFate got the alternative way to do it, and more exact.

To: McFate And thanks for teaching me something new.

Answer 5

The first answer is wrong. You must do the or before you move the -1 to the other side of the equation.

|2x-1| > 3.6
splits into:
2x-1 > 3.6 ---------- or ------- -(2x-1) > 3.6
2x > 4.6 ---------------------- -2x+1 > 3.6
x > 2.3 -------------------------- -2x > 2.6
------------------------------------------- x < -1.3

Answer: x > 2.3 or x < -1.3

Answer 6

I2x-1I>3.6 => 2x-1>3.6 and -(2x-1)>3.6
2x-1>3.6 => x>2.3
-(2x-1)>3.6 => 2x-1<-3.6 => x<-1.3