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, a2 ≥ 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.