Solving (x+1)/(2x-3) > 2; why do we take 2x-3>0 while obtaining the solution?

[gikiian]

I mean it's okay to take 2x-3≠0 as a necessary condition, but I can't actually grasp the fact that we have to take the denominator to be '>0' and not merely '≠0'.
A little guidance will be very much appreciated :)

 

[Stephen Tashi]

Solving (x+1)/(2x-3) > 2; why do we take 2x-3>0

If you solve the inequality in a deductive manner, you don't only look at the case 2x -3 > 0. You look at all the possible cases for the sign of the numerator and denominator of the fraction (x+1)/(2x+3). You say that the cases when one is positive and the other is negative can't povide any solutions since the left hand side would be negative, so less than 2. In another case, both the numerator and denominator are positive. In that case x+1 > 0 and 2x+3 > 0, so x> -1 and 2x > -3, x > -3/2.

You have to look at all the cases to work the problem in a systematic manner. Perhaps the materials you are looking at know some clever shortcut.

 

[checkitagain]

If you solve the inequality in a deductive manner, you don't only look at the case 2x -3 > 0.
You look at all the possible cases for the sign of the numerator and denominator of
the fraction (x+1)/(2x+3).
You say that the cases when one is positive and the other is negative can't povide
any solutions since the
left hand side would be negative, so less than 2. In another case, both the
numerator and denominator are positive.
In that case x+1 > 0 and 2x+3 > 0, so x> -1 and 2x > -3, x > -3/2.

You have to look at all the cases to work the problem in a systematic manner.
Perhaps the materials you are looking at know some clever shortcut.

The above is not one of the correct approaches.


Mine:


Subract 2 from each side, and that gives a new fraction that is greater
than 0. Then find the critical numbers to establish the candidate
intervals from which to choose for the solution:


After subtracting 2 and simplifying the fraction:


$\dfrac{-3x + 7}{2x - 3} > 0$


Critical numbers:

-3x + 7 = 0

3x = 7

x = 7/3

-------------------

2x - 3 = 0

2x = 3

x = 3/2

-------------------


Then the candidate intervals to choose from for the solution are:

(-oo, 3/2), (3/2, 7/3), and (7/3, oo).


Upon checking test values, it is shown that the solution is:


(3/2, 7/3),


or equivalently, as


3/2 < x < 7/3

 

[Deveno]

since the fraction on the left is greater than 2 > 0, both the numerator and the denominator must have the same sign.

let's look at the case where both the numerator and the denominator are < 0:

x+1 < 0 => x < -1.
2x - 3 < 0 => x < 3/2.

so for both these to be true, we take the more restrictive requirement: x < -1.

since with that assumption, we have 2x - 3 < 0 as well, we have:

x + 1 < 4x - 6 (when we multiply by a negative number, we reverse the inequality)
1 < 3x - 6
7 < 3x
7/3 < x

but this is a contradiction, x can't be BOTH < -1 AND > 7/3.

so the case where both numerator and denominator are negative, does not lead to any valid solutions.

so we may assume, without loss of generality (in THIS particular case) that both numerator and denominator are > 0.

 

[CellCoree]

I can understand your confusion about why we take the denominator to be greater than 0 when solving this inequality. Let me try to explain it in a simple way.

Firstly, we need to understand that when we solve an inequality, we are looking for all the possible values of the variable that make the inequality true. In this case, we are looking for the values of x that make the expression (x+1)/(2x-3) greater than 2.

Now, let's consider the denominator 2x-3. If this denominator is equal to 0, then the expression (x+1)/(2x-3) becomes undefined. This means that for any value of x that makes the denominator equal to 0, the inequality will not hold true. Therefore, we need to exclude those values of x that make the denominator equal to 0 from our solution set.

Now, why do we take the denominator to be greater than 0 and not just not equal to 0? This is because when we divide by a negative number, the direction of the inequality changes. For example, if we have the inequality 2x < 4, and we divide both sides by -2, we get x > -2. So, when we divide by a negative number, the direction of the inequality changes. In this case, if we divide both sides by 2x-3, we get x+1 > 4, which is not the same as the original inequality. Therefore, we need to make sure that the denominator is positive to maintain the direction of the inequality.

In summary, we take the denominator to be greater than 0 while obtaining the solution to ensure that the inequality holds true for all values of x that make the denominator positive and exclude the values that make the denominator equal to 0, while also maintaining the direction of the inequality. I hope this helps clarify the importance of taking the denominator to be greater than 0 in solving this inequality.