Inequalities:
common information
Inequality. Signs of inequalities. Identical inequality.
Strict inequality.
Non-strict inequality. Solving of inequality or system of simultaneous
inequalities. Main properties of inequalities. Some important
inequalities.
Two
expressions ( numerical or literal ones ), connected by
one of the signs: “more”
(>) , “less” (<) , “more than or equal to” ( ), “less than or equal
to” ( ) form
an inequality ( numerical or literal ). Any
valid inequality is called an identical inequality.
For example, the following
inequalities are identical: 3 · 7 – 20 > 2 · 4 -10, a 2 0, | - 5 | > 3. ( Why ? ) Depending
on the sign of inequality we have a strict inequality
( > , < ) , or a nonstrict
inequality ( , ). The record 5a 4b means,
that either 5a is less
than 4b or is
equal to it. Literal values, included in an inequality, can
be both known values
and unknowns. To solve an inequality means to find
the bounds, within which
values of unknowns must be contained, so as the inequality
will be valid. To
solve a system of simultaneous inequalities means
to find the bounds, within which
values of unknowns must be contained, so as all inequalities,
containing in the system will be valid simultaneously.
Main
properties of inequalities.
| 1. |
If a < b, then b > a ; or if a > b, then b < a .
|
| 2. |
If a > b, then a + c > b + c ; or if a < b, then a + c < b + c . That
is, one can add (or subtract) the same value to both sides of inequality.
|
| 3. |
If a > b and c > d,
then a + c > b + d . That is, inequalities
of the same sense ( with the same sign > or < ) can be added term by term. Note,
that inequalities of the same sense cannot be subtracted term by term one
from another, because
the result can be both correct and incorrect.
|
| 4. |
If a > b and c < d, then a
– c > b – d . Or if a < b and c > d,
then a
– c < b – d . That is, inequalities of the opposite
sense can be subtract
one from another, and the sign of the resulting inequality
is the same as of the minuend inequality.
|
| 5. |
If a > b and m > 0,
then ma > mb and a/m > b/m . That is, both
sides of
any inequality can be multiplied or divided by the same
positive number; the
inequality sense is the same.
|
6. |
If a > b and m < 0, then ma < mb and a/m < b/m
. That is, both sides of any inequality can be multiplied or divided by the same
negative number, but the inequality sense changes to the opposite one. |
Some
important inequalities.
1. | a + b | | a | + | b | . Modulus of sum is not more than sum of the
modules.
2. a + 1 / a 2, ( a – a positive number ). An equality is valid only at a =
1.
 ( a and b
– positive numbers ). An equality takes place
only if a = b.
Geometric average is not more than
arithmetic average.
In general case this
inequality has the following shape:
Numbers a1 , a2 , …, an are
positive. The equality takes place only if all numbers
are equal.
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