Help With Add/Subtract/Mult/Divid Multiple Inequalities

Question

Trying to remember the rules. Anyone know a good comprehensive place to find the rules?

I remember that you can add them so long as you keep the inequality sign the same way, but can't remember the other rules.

Bunuel · Accepted Answer

ADDING/SUBTRACTING INEQUALITIES: You can only add inequalities when their signs are in the same direction: If a>b and c>d (signs in same direction: > and > ) --> a+c>b+d . Example: 3<4 and 2<5 --> 3+2<4+5 . You can only apply subtraction when their signs are in the opposite directions: If a>b and c and < ) --> a-c>b-d (take the sign of the inequality you subtract from). Example: 3<4 and 5>1 --> 3-5<4-1 . RAISING INEQUALITIES TO EVEN/ODD POWER: A. We can raise both parts of an inequality to a positive even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). For example: 2<4 --> we can square both sides and write: 2^2<4^2 ; 0\leq{x}<{y} --> we can square both sides and write: x^2-2 if we square we'll get 1>4 which is not right. So if given that x>y then we can not square both sides and write x^2>y^2 if we are not certain that both x and y are non-negative. B. We can always raise both parts of an inequality to a positive odd power (the same for taking an odd root of both sides of an inequality). For example: -2<-1 --> we can raise both sides to third power and write: -2^3=-8<-1=-1^3 or -5<1 --> -5^2=-125<1=1^3 ; x we can raise both sides to third power and write: x^3https://gmatclub.com/forum/x2-4x-94661.html#p731476 https://gmatclub.com/forum/inequalities-trick-91482.html https://gmatclub.com/forum/data-suff-ine ... 09078.html https://gmatclub.com/forum/range-for-var ... 09468.html https://gmatclub.com/forum/everything-is ... 08884.html https://gmatclub.com/forum/graphic-appro ... 68037.html https://gmatclub.com/forum/inequations-i ... 54664.html https://gmatclub.com/forum/inequations-i ... 54738.html QUESTIONS: All DS Inequalities Problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=184 All PS Inequalities Problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=189 700+ Inequalities problems: https://gmatclub.com/forum/inequality-an ... 86939.html Hope it helps.

KarishmaB · Answer

Multiplication - Yes but very constrained.
If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

x < a
y < b
xy < ab
Given x, y, a, b are all positive.

Otherwise 
-2 < -1
10 < 30
Multiply: -20 < -30 (Not correct)
or
-2 < 7
-8 < 1
Multiply: 16 < 7 (Not correct)

For division, this may not hold.
e.g
2 < 10
4 < 40
Divide: 1/2 < 1/4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

x < a
y > b

x/y < a/b (given all x, y, a, b are positive)
The final inequality takes the sign of the numerator. Take examples.

alphabeta1234 · Answer

Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!

Supermaverick · Answer

We can divide two negative inequalities as well provided that all variables have negative signs:
 Given : x ,y, a and b are all negative

x < a
y > b
 
x/y > a/b ( always)

The final inequality takes the sign of the  DENOMINATOR . Take examples

IanStewart · Answer

I generally don't recommend memorizing the more obscure inequality rules, since they're very easy to misapply. It's usually best just to think conceptually about unusual inequality situations, because if you aren't aware of all of the exceptions, you'll very likely make a mistake.

For these rules about powers, we don't only need to know whether our bases are positive or negative. We also need to know if the exponent is positive or negative. So what I quote above is not strictly correct. It's true, for example, that 3 > 2. If we raise both sides to a positive integer power, we'll get something true; 3^2 > 2^2, for example. But if we raise both sides to the even power 0, what we get is no longer true: 3^0 = 1 is not greater than 2^0 = 1. Similarly, if we raise both sides to a negative power, we'll get something untrue: 3^(-2) = 1/9 is not greater than 2^(-2) = 1/4.

But as I said above, I don't suggest studying inequalities by learning all of these 'rules' and their various exceptions.

yppaul · Answer

I always go to the GBAT Club chat sheets download and keep those handy

Abdullah9097 · Answer

Can we add or subtract variables in both sides of inequality?

Example: 2x>x+3

Bunuel · Answer

Yes. You can add or subtract the same expression (even one with variables) on both sides of an inequality without changing its direction.

Example:

2x > x + 3

Subtract x from both sides: x > 3.

Help With Add/Subtract/Mult/Divid Multiple Inequalities

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