The “thing separating the numerator and denominator” is the same thing as “divide”.

If i’m reading you correctly, the example given is: “negative three is bigger than one-third times R”

In this case, you can sort of treat the inequality as an equals sign. If one-third of R is smaller than negative three, than R MUST be smaller than -9:
-9 > R
(you get this by multiplying both sides by three). For example, if R is -18, one third of R is -6, and -6 is smaller than -3. Likewise, -18 is smaller than -9

Another example:
X+2 > 4

since X plus two is greater than 4, X must be greater than 2:
X > 2
(subtract two from each side).
for instance, if X is 3, X+2 = 5, which is greater than 4. Meanwhile, X itself is greater than two, so that works.

Another one:
5*Y < 20
(five times Y is less than 20). By this same method, Y must be less than 4:
Y < 4
(divide both sides by 5). If Y is 2, 5*Y is less than 20, and Y is less than 4.

There is one exception to this, which is when negative numbers are involved. If you need to divide by a negative number, you need to FLIP the inequality sign.
example:

-2*U < 10
divide both sides by negative two, and flip the sign:
U > -5

this is because of what happens when you multiply negative numbers by eachother. If U = -3, the inequality holds, beucause -2*-3 = +6.

multiple both sides by the reciprocal. easy peazy. :)

Thanks @hannahsugs the extra examples were not all neccessary for my knolege, but others can use it i suppose, i just nvr remember the fractions. Thanks

Wow, somehow i missed the “involving fractions” part. Ooops! Guess I got a little carried away. i feel silly now

But yeah, for fractions just multiply both sides by the reciprocal. You can also think about a fraction as a multiplication and a division, x*¾ (x times three-fourths) is the same as (x*3) / 4 (x times three, divided by 4).

In a nutshell, just remember this:

Solve inequalities just like equalities except for the minus (negative) sign.

When inequalities contain a minus sign and if you want to make it positive (by multiplying/dividing by -1 on both sides) like in your example, after multiplying (or dividing) by -1, change the less than (<) sign to greater than (>), or vice versa (> to <, <= to >=, >= to <=).

For your example,
-3 > (⅓) * R
(-1) * (-3) > (-1) * (⅓) * R…(multiplying both sides by -1)
(+) 3 < (-⅓) * R…(notice the changed less than sign)

Proceed or solve then as you would do for the equalities.

Continuing your example,
3 < (-⅓) * R
3 * 3 < (-⅓) * 3 * R…(multiplying both sides by 3)
9 < (-1) * R
(-1) * 9 < (-1) * (-1) * R…(multiplying both sides by -1)
(-9) > R…(notice the changed greater than sign)

So the answer is the value of “R” is greater than -9.

Don’t panic or be afraid, just solve one by one, and you’ll do it right.
All the best!

Well, you can solve your example like below:

-3 > (⅓) * R
(-3) * 3 > (⅓) * 3 *R…(multiplying both sides by 3)
(-9) > R…same result!

I solved earlier the long way to show you how to take care of the minus sign.
Anyways, good luck!

Oops, @hannahsugs have done it better, I read it after finishing mine

And, remember, flip the inequality sign, even when there are minus signs on both sides.
For example,
-x < -5
Multiplying/dividing -1 on both sides,
x > 5…(notice the flipped inequality sign)

Take x = 6 for example. So, either 6 ( or x) > 5 or -6 (or -x) < -5.

Thanks everyone, I feel like a fool because I tried solving a good amount of Fraction Inequality’s by just making the answer look right, (Like a Whole number, or Long Steps) like a Retarded fish frog And all I actually had to do is Multiply by reciprocal.

Well thanks for helping! Everyone, Really!