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# Help With Add/Subtract/Mult/Divid Multiple Inequalities

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02 Jul 2013, 22:08
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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.

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02 Jul 2013, 22:27
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amargius wrote:
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.

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 $$c<d$$ (signs in opposite direction: $$>$$ 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 an 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: $$x^2<y^2$$;

But if either of side is negative then raising to even power doesn't always work.
For example: $$1>-2$$ 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 an 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 $$-5<1$$ --> $$-5^2=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.

THEORY ON INEQUALITIES:

x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html
inequations-inequalities-part-154664.html
inequations-inequalities-part-154738.html

QUESTIONS:

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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03 Jul 2013, 15:19
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?

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03 Jul 2013, 20:39
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alphabeta1234 wrote:
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?

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.
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03 Jul 2013, 23:24
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Merging topics.
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31 Aug 2013, 06:25
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16 Sep 2014, 16:29
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01 Oct 2016, 00:17
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09 Jul 2017, 03:29
VeritasPrepKarishma wrote:
alphabeta1234 wrote:
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?

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.

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
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