# 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 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.* | *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 *a*_{1 } ,* a*_{2}* *, * …*,* a*_{n} are positive. The equality takes place only if all numbers are equal. Back |