Inequalities: common information
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 5 a ≤ 4 b means, that either 5 a is less than 4 b 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. |