Combining inequalities

[Stockmoose16]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

[cramya]

I see that question

x>8
x<17
x+5<19

x>8 is the same as 8 < x


The range of x is 8<x<14 (satisfies all three conditions given)

Any value of x > 14 would affect the x+5<19 inequality

Hence the range is 8<x<14 (it goes on the say choose the most restrictive upper or lower limits)

Does this help any??

[cramya]

see that question

x>8
x<17
x+5<19

x>8 is the same as 8 < x


The range of x is 8<x<14 (satisfies all three conditions given)

Any value of x >= 14 would affect the x+5<19 inequality

Hence the range is 8<x<14 (it goes on the say choose the most restrictive upper or lower limits)

Does this help any??

[Stockmoose16]

see that question

x>8
x<17
x+5<19

x>8 is the same as 8 < x


The range of x is 8<x<14 (satisfies all three conditions given)

Any value of x >= 14 would affect the x+5<19 inequality

Hence the range is 8<x<14 (it goes on the say choose the most restrictive upper or lower limits)

Does this help any??

Cramya,

It doesn't make sense, because Ron Purewal has stated in several different posts that you can add inequalities as long as the signs face the same way. In this set of equations, I simply multiplied X>8 by -1 to ensure all the inequalities were facing the same way.

Read this thread where Ron discusses adding inequalities:

https://www.beatthegmat.com/ds-weakness- ... 14388.html

I don't understand why he says you can add them, and yet, I did exactly that and got the wrong answer.

[cramya]

I see where u are coming from now.

I am not sure if adding inequalities can be applied to a range problem like this .I am sure we can request Ron to respond and clarify this for us.

P.S : By the way You always bring intersting questions to the table

[Stockmoose16]

I see where u are coming from now.

I am not sure if adding inequalities can be applied to a range problem like this .I am sure we can request Ron to respond and clarify this for us.

P.S : By the way You always bring intersting questions to the table

Thanks. I just wish I didn't confuse myself with all these problems. What makes a range problem different from a DS inequality problem? I don't understand why the rules would change for a range. I really wish my GMAT wasn't 2 weeks away... I'll never score 700 at this rate.

[cramya]

Dont worry ; u will do good!

2 weeks is better than a day before. Have confidence; you are bound to do well, buddy!

[logitech]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

Try to picture in your mind my friend. Don't just apply the rules, which you can not picture!

[Stockmoose16]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

Try to picture in your mind my friend. Don't just apply the rules, which you can not picture!

Logitech,

Your picture makes sense to me, but I still don't understand the logic behind the posts that Ron and other GMAT instructers have submitted, saying that you can add inequalities.

I hope an expert weighs in on this topic.

[logitech]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

Stockmoose,

Yes you can add inequalities as long as they have the same signs. What does this mean ?

Lets say a > 0 ( positive ) and b > 0 ( positive)

and you can definetly say that adding to positive number yields to another positive number

a+b>0

Another example

5 > 3 ( a > b )

2 > 1 ( c > d )

If we add these 2 ineq

7 > 4

or a + c > b + d

Lets go back to your question and find out what causes you trouble:

X>8
X<17
X<14

well - as you can see here, there is ONLY one variable and you are trying to find the set up numbers which hold all of these THREE inequalities. This is what I described in my previous post. Since there is only one variable, X, lets say we want to add the 2nd and the 3rd equation:

2x < 31

or x < 15.5

:!:

well as you can see there is something wrong here because we know that X is less than 14! If I am younger than 100 and younger than 35 ..well I can't be between 35 and 100 right ? so you need to find out the limiting equation here which is X < 14

but if the 3rd equation were Y < 14, then you would have :

X + Y < 31 --> WHICH IS CORRECT, since there are two variables.

this is why YOU CAN NOT ADD THOSE THREE INEQs in this question, since we are talking about only ONE variable.

-X<-8
X<17
X<14
------------
X>23

how come X can be greater than 23 if it is less than 14 :

Hope this helps.

P.S. Try NOT to subtract the INEQs...Bunch of signs will flip and it will get complicated.

Hope this helps, and let me know if I can help you more.

[Ian Stewart]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

What Ron has said about adding inequalities is true (though it's definitely not true that you should 'always' add them, and I don't think he said that; what is true is that you can add them, as long as the inequality sign is facing in the same direction in each). When you do add them, the inequality sign in the result must face in the same direction as the inequalities that you are adding, so there is a small error in your work above, highlighted in red: the conclusion should read "x < 23".

That's a perfectly valid conclusion -- x must be less than 23 -- but we actually already knew that x must be less than 23; in fact, we knew that x<14. To find the range of possible values of x, we want to use the most restrictive conditions provided. Since we know, for example, that

x < 14 *and* x < 17 *and* x < 23

then x must be less than 14 for all three of these to be true. The only inequality that helps us to get a minimum value for x is this:

x > 8

so 8 < x < 14.

I like logitech's age analogy above- you can interpret these inequalities in words. If x is Xavier's age, and we know all of the following:

x < 14 *and* x < 17 *and* x < 23 *and* x > 8

we know that Xavier is less than 14 years old, less than 17 years old, less than 23 years old, and greater than 8 years old. So Xavier must be strictly between the ages of 8 and 14. I'd add, however, that logitech reached an incorrect conclusion above: you can add three inequalities as long as all are facing in the same direction.

Adding inequalities can be useful, but on GMAT problems I only find myself doing it occasionally- usually only in situations with more than one unknown.

[logitech]

I like logitech's age analogy above- you can interpret these inequalities in words. If x is Xavier's age, and we know all of the following:

x < 14 *and* x < 17 *and* x < 23 *and* x > 8

we know that Xavier is less than 14 years old, less than 17 years old, less than 23 years old, and greater than 8 years old. So Xavier must be strictly between the ages of 8 and 14. I'd add, however, that logitech reached an incorrect conclusion above: you can add three inequalities as long as all are facing in the same direction.

Adding inequalities can be useful, but on GMAT problems I only find myself doing it occasionally- usually only in situations with more than one unknown.

Thanks Ian, you are absolutely right. Great post.

[aj5105]

Thanks Ian.

Now, when i combine x-y>-2 and x-2y <-6, i get y>4

I also get x>3 ; -x > -4 --> x < 4
Hence x > 3 but less than 4.

Am I right in infering this?

[Stockmoose16]

Hello,

I'm confused about how to combine inequalities. There was a post by Ron from Mgmat that said you should always add inequalities. I tried to do so with the following equations, and got the wrong answer:

X>8
X<17
X<14

I then multiplied X>8 by -1 so that all the inequality signs would match. So now, I have:

-X<-8
X<17
X<14
------------
X>23

The Mgmat book says the answer is 8<X<14 ...

That doesn't make sense if you can simply add the equations, as I did above. Any thoughts?

What Ron has said about adding inequalities is true (though it's definitely not true that you should 'always' add them, and I don't think he said that; what is true is that you can add them, as long as the inequality sign is facing in the same direction in each). When you do add them, the inequality sign in the result must face in the same direction as the inequalities that you are adding, so there is a small error in your work above, highlighted in red: the conclusion should read "x < 23".

That's a perfectly valid conclusion -- x must be less than 23 -- but we actually already knew that x must be less than 23; in fact, we knew that x<14. To find the range of possible values of x, we want to use the most restrictive conditions provided. Since we know, for example, that

x < 14 *and* x < 17 *and* x < 23

then x must be less than 14 for all three of these to be true. The only inequality that helps us to get a minimum value for x is this:

x > 8

so 8 < x < 14.

I like logitech's age analogy above- you can interpret these inequalities in words. If x is Xavier's age, and we know all of the following:

x < 14 *and* x < 17 *and* x < 23 *and* x > 8

we know that Xavier is less than 14 years old, less than 17 years old, less than 23 years old, and greater than 8 years old. So Xavier must be strictly between the ages of 8 and 14. I'd add, however, that logitech reached an incorrect conclusion above: you can add three inequalities as long as all are facing in the same direction.

Adding inequalities can be useful, but on GMAT problems I only find myself doing it occasionally- usually only in situations with more than one unknown.

Ian,

As always, thank you very much for your clear and concise explanations. You are a real asset to this board.

Also, Logitech, thank you for the time and effort you put in to answer my posts.

Both of you are incredibly helpful!