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### Course: Algebra (all content)>Unit 13

Lesson 12: Rational inequalities

# Rational inequalities: both sides are not zero

Sal solves the rational inequality (x-3)/(x+4)≥2. Created by Sal Khan.

## Want to join the conversation?

• In the previous Rational Inequalities video the solution was x>1 and x<-2. Why is that a valid answer while the first example in this video ( x>-4 and x<-11 ) is NOT a valid answer?
• Hello Cyrus,

I can see that your confusion is with the *"and"* and *"or"*.

If there is a word *"and"* then you must see if they have common numbers: if they don't have, then the solution is invalid.

If there is a word "or" then you don't need to determine if they have common numbers.
• This is really helpful. However, how you would go about in solving (x+1)(x+3) / x-1 > 0 ?

I tried solving using the method Khan uses:
The numerator and denominator can be either both positive or negative.

When they are both positive:
x+1>0 ---- x>-1
x+3>0 ---- x>-3
x-1>0 ---- x>1
So x>1 satifies this situation.

When they are both negative:
x<-1
X<-3
x<1
So x<-3 satifies this approach.

So x could either be x>1 or x<-3. However, this is not the answer. The answer is -3<x<-1 OR x>1

You say "Since (x + 3) will always be greater than (x + 1), it will have to positive, and (x + 1) will have to be negative."
Why is that?
• At Sal says the second method only works when a/b > 0 and then rearranges the equation as so. Why does this only work when one side of the inequality is a zero? Surely in the original problem a/b > 2 the same logic applies, both a and b must be positive or negative to reach a number greater than 2?
• It is because inequalities have to have zero to balance the equation.
• I'm not quite sure why the inequality was swapped at ? I understand that when you divide by a negative this is done but I don't see the negative-- is there a video that goes over this?
• If x < -4,

When we multiply the inequality by (x + 4), we are actually multiplying the inequality by a negative number.

How? ...

When we assume x < -4 then the greatest value that can be attributed to x is -5. Substituting we have: (x + 4) ---> (-5 + 4) = -1
We are multiplying the inequality by a negative number.

Hope this helps!
• I am taking a Pre-Calc class and the process they give to solve Polynomial and Rational inequalities is totally different from both of these ways. They have us find the zeros and take sections of the number line around those. Then create a table and determine which are positive and which are negative. Does any one know what I'm talking about and give some tips on this?
• I see that your question is about a year old. Yes - I call this a sign chart. The focus is on the zeros and asymptotes and the questions is what makes something positive or negative. I haven't found that method explained yet on KA but that's what I'm most familiar with.
• This is Rational Inequalities 2... Where is Rational Inequalities 1?
• I guess they haven't put it up on the site, here is the YouTube link: http://www.youtube.com/watch?v=ZjeMdXV0QMg
• I don't understand why x+4 can't be less than OR equal to. Why is it just less than?
• Hi - the reason that x+4 cannot be equal to zero is that it will make the equation undefined as you cannot have the denominator equal to zero to create either a m/0 situation (where m is a number not equal to zero at the same value of x).
Use sign chart to solve it..
x^2 + 4 over 2x^2 -18 is greater than 0
It has complex roots. So ia cant understand how to use these rooys in sign chart..