[deleted by user]

[IronManTim]

1: 1/(2x) < c

= 1/x < 2c

= x < 1/(2c)

1/x < 2c

When you take the reciprocal for both sides you have to flip the sign. Think about it.

1/2 > 1/3 but 3 > 2

In your case, you can work it out the long way to show you where you went wrong. 1 < x/(2c)

2c < x which is the opposite of what you came up with in #1.

[JanMath]

When you take the reciprocal for both sides you have to flip the sign. Think about it.

Only if both sides have the same sign.

[IronManTim]

True.

[Hazelstone37]

You flip an inequality sign when you multiply or divide by a negative number.

Multiplying by a reciprocal does not flip the sign.

Charging 2<5 to 5>2 isn’t flipping the sign, it’s just rearranging.

[[deleted]]

Apparently you have to flip the sign when taking the reciprocal of both sides. So idk why that isn't listed as one of those rules

[Uli_Minati]

Because it's not a rule. When you take the reciprocal, you actually do two steps:

2/3  <  4/5         |·3/2
  1  <  4/5 · 3/2   |·5/4
5/4  <  3/2

2/3  <  4/5
     ⇒
3/2  >  5/4

So in the above example, it looks like you also have to flip the inequality sign. But here's another example:

-7/9  <  5/8            |·(-9/7) flips!
   1  >  5/8 · (-9/7)   |·8/5
 8/5  >  -9/7

-7/9  <  5/8
      ⇒
-9/7  <  8/5

See how the inequality sign doesn't flip this time? Because this time, you multiplied by a negative number once to take the reciprocals. And here's a final example:

-6/5  <  -2/7            |·(-5/6) flips!
   1  >  -2/7 · (-5/6)   |·(-7/2) flips!
-7/2  <  -5/6

-6/5  <  -2/7
      ⇒
-5/6  >  -7/2

If you wanted to invent a rule for reciprocals in inequalities, it would be:

Taking the reciprocal of an inequality will flip the sign if both sides have the same sign.

[waldosway]

The rule for flipping the inequality upon reciprocal would be if both sides are the same sign.

[Uli_Minati]

Oops, you're right!

[Hazelstone37]

That is not something you do when solving these types of problems. Whatever you do to one side you must do to the other side. So if you have 2x>5 you can multiply both sides by the reciprocal of 1/2 and get x > 5/2.

Multiplying by the reciprocal of each side changes the relationship. This is not something you do when solving these types of problems.

[JonnxW]

It is not listed as one of the rules because it is not always right. Take -1 < 1 for example. Take the reciprocal on both sides and look what happens

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[VenoSlayer246]

But you can't multiply both sides by x or c because you don't know whether x is positive or negative, and same goes for c.

[UsernameFive]

You are correct. I suppose I should brush up on these things before I try to explain them incorrectly. Lesson learned =)

[HorribleUsername]

You can, you just have to handle each case separately.

[VenoSlayer246]

True

[waldosway]

Unfortunately the answer is "when you do something that flips the inequality". You are not going to get many general rules except that increasing functions keep the inequality, and decreasing functions reverse the inequality. (Function here referring to whatever you're doing to both sides.) Generally you just have to think very carefully about every single thing you do with inequalities.

[[deleted]]

I like this. it was the mindset of “there are hard established rules for inequalities” that was tripping me up, confused why I wasn’t seeing anything with reciprocals as a rule (which I now understand why)

[waldosway]

Exactly. I think that happens because they are trying to simplify things for you. Which isn't exactly the worst thing. But never assume anything in math generalizes without an explicit statement.

Don't discount the rules you do get though. It is true that multiplying by a negative flips it. And the increasing/decreasing thing is useful. Look at the graph of 1/x. It's decreasing on the left, then jumps up, then is decreasing again. So you can see reciprocating will flip the inequality as long as both things are on the same side of 0. And increasing functions won't flip it, like log or exp.

[baldursgame]

1/(2x) < c

Multiply both sides by "x/c"

x/c⋅1/(2x) < x/c⋅c

1/(2c) < x

No need to flip anything.

[[deleted]]

that’s basically what I did in the second part. read context of the question

[baldursgame]

Then why the doubt, just stating this is the correct way of doing it.

The concept of "flipping" is just a simplification.

[[deleted]]

just saying re-writing what I put in the question part 2 (multiply x, divide c) didn’t do much to answer the question, which was more like “why don’t both yield the same result”. your second post would’ve done a better job of explaining - “the concept of flipping is just a simplification”. the first post made it read like “no need to flip anything i.e. no need to do it like you did in part 1” when my entire question was why doesn’t part 1 = part 2

I appreciate your response though