Modulus Function?

Question

For the equation x+ /2x-1/ = 3 , the answers are -2 and 4/3.
But there is no solution for the equation 3 + /2x-1/ = x.
WHY?

Answer 1

3 + |2x-1| = x

When x > 1/2, then 3 + 2x-1 = x
or x = -2
This cannot happen as x cannot be greater than 1/2 and also equal to -2 (which is less than 1/2) at the same time.
So there is no solution when x > 1/2.

When x < 1/2, then 3 - (2x - 1) = x
or x = 4/3
This cannot happen as x cannot be less than 1/2 and also equal to 4/3 (which is greater than 1/2) at the same time.
So there is no solution when x < 1/2.

Now, check what happens when x = 1/2.
Then, 3 + 0 = 1/2 i.e. 3 = 1/2.
This is impossible.
So x cannot be greater than or less than or even equal to 1/2.
This means no real value of x can satisfy the given equation and hence it has no solution.


For the other equation, x + |2x - 1| = 3

When x > 1/2, then x + 2x-1 = 3
or x = 4/3
This can happen as x is 4/3 (which is greater than 1/2).

When x < 1/2, then x - 2x + 1 = 3
or x = -2
This can happen as x is -2 (which is less than 1/2).

Again, x = 1/2 doesn't satisfy the equation.

So the solutions are -2 and 4/3.

Answer 2

For the first equation...
when is 2x-1 >= 0? when x >=1/2
when is 2x-1 <0? when x <1/2

x + /2x-1/ = 3
/2x-1/ = 3 - x

There are two possibly solutions x>=1/2 and
2x-1 = 3 - x which implies x = 4/3
OR
x < 1/2 and
-(2x-1) = 3-x which implies x = -2

For the second equation...
when is 2x-1 >= 0? when x >=1/2
when is 2x-1 <0? when x <1/2

3 + /2x-1/ = x
/2x-1/ = x - 3

There are two possibly solutions x>=1/2 and
2x-1 = x - 3 which implies x = -2
OR
x < 1/2 and
-(2x-1) = x - 3 which implies x = 4/3

But WAIT how can x>=1/2 and x = -2 or how can x < 1/2 and x = 4/3?...

The answer is they can't, it's a contradiction, therefore the only two possible REAL NUMBER solutions are invalid.

Answer 3

Consider both cases. Suppose 2x -1 > 0.
Then 3 + 2x -1 = x
so x = -2. This is impossible because 3 + |2x -1| > 0.

Now suppose 2x-1 < 0. Then 3 - 2x +1 = x,
so x = 4/3. This is impossible because 3 + |2x-1| >=3.
Finally, x = 1/2 does not yield a solution.
So there is no solution to the given equation.

Answer 4

you mean the modulus of (2x-1) in the equation right?
Then with your notation,

-/2x-1/ = x-3
(the modulus is always non negative so the left side is always negative with that sign)
Now square both sides then,
(2x-1)^2 = (x-3)^2
4x^2 - 4x + 1 = x^2 - 6x + 9
=> 3x^2 + 2x - 9 = 0
x = [-2 +/- V(4 + 4.3.9)]/6
= [-2 +/- V(112)] / 6
So x has too solutions
(where V is the radical sign)
So there are solutions!

Answer 5

x+ /2x-1/ = 3
when x >1/2
x +2x -1 =3
3x = 4
x = 4/3.......................................i
When x<1/2
x -2x+1=3
-x=2
x = -2...........................................ii
Now try yourquestion
x+ /2x-1/ = x
when x >1/2
x +2x -1 =x
2x = 1
x = 1/2.......................................i
When x<1/2
x -2x+1=x
-2x=-1
x = 1/2...........................................ii
Since x is on left side as well as on right side which is to be cancelled therefore question will be
x+ /2x-1/ = x
/2x-1/ =0
Here only solution is zero