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In mathematics, an inequality is a statement how the relative size or order of two objects, or about whether they are the same or not (See also: equality). The notation a < b means that a is less than b. The notation a > b means that a is greater than b. The notation a â‰ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size. In each statement above, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b". In contrast to strict inequalities, there are two types of inequality statements that are not strict: The notation a â‰¤ b means that a is less than or equal to b (or, equivalently, not greater than b) The notation a â‰¥ b means that a is greater than or equal to b (or, equivalently, not less than b) An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude. The notation a b means that a is much less than b. The notation a b means that a is much greater than b. If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. Properties Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding nonstrict inequality sign (â‰¤ and â‰¥). Transitivity The transitivity of inequalities states: For any real numbers, a, b, c: If a > b and b > c; then a > c If a < b and b < c; then a < c If a > b and b = c; then a > c If a < b and b = c; then a < c Addition and subtraction The properties that deal with addition and subtraction state: For any real numbers, a, b, c: If a < b, then a + c < b + c and a âˆ’ c < b âˆ’ c If a > b, then a + c > b + c and a âˆ’ c > b âˆ’ c i.e., the real numbers are an ordered group Multiplication and division The properties that deal with multiplication and division state: For any real numbers, a, b, and nonzero c If c is positive and a < b, then ac < bc and a/c < b/c If c is negative and a < b, then ac > bc and a/c > b/c More generally this applies for an ordered field, see below. Additive inverse The properties for the additive inverse state: For any real numbers a and b If a < b then −a >−b If a > b then −a <−b Multiplicative inverse The properties for the multiplicative inverse state: For any nonzero real numbers a and b that are both positive or both negative If a < b then 1/a > 1/b If a > b then 1/a < 1/b If either a or b is negative (but not both) then If a < b then 1/a < 1/b If a > b then 1/a > 1/b Applying a function to both sides Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function. For a nonstrict inequality (a â‰¤ b, a â‰¥ b): Applying a monotonically increasing function preserves the relation (â‰¤ remains â‰¤, â‰¥ remains â‰¥) Applying a monotonically decreasing function reverses the relation (â‰¤ becomes â‰¥, â‰¥ becomes â‰¤) As an example, consider the application of the natural logarithm to both sides of an inequality: 0 < a < b \Leftrightarrow \ln(a) < \ln(b). This is true because the natural logarithm is a strictly increasing function. Ordered fields If (F, +, ×) is a field and â‰¤ is a total order on F, then (F, +, ×, â‰¤) is called an ordered field if and only if: a â‰¤ b implies a + c â‰¤ b + c; 0 â‰¤ a and 0 â‰¤ b implies 0 â‰¤ a × b. Note that both (Q, +, ×, â‰¤) and (R, +, ×, â‰¤) are ordered fields, but â‰¤ cannot be defined in order to make (C, +, ×, â‰¤) an ordered field, because −1 is the square of i and would therefore be positive. The nonstrict inequalities â‰¤ and â‰¥ on real numbers are total orders. The strict inequalities < and > on real numbers are . Chained notation The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. Hence, for example, a < b + e < c is equivalent to a âˆ’ e < b < c âˆ’ e. This notation can be generalized to any number of terms: for instance, a1 â‰¤ a2 â‰¤ ... â‰¤ an means that ai â‰¤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai â‰¤ aj for any 1 â‰¤ i â‰¤ j â‰¤ n. When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 â‰¤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x â‰¥ −1 respectively, which can be combined into the final solution −1 â‰¤ x < 1/2. Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b = c â‰¤ d means that a < b, b = c, and c â‰¤ d. This notation exists in a few programming languages such as Python. Inequalities between means There are many inequalities between means. For example, for any positive numbers a1, a2, â€¦, an we have where Power inequalities Sometimes with notation "power inequality" understand inequalities that contain ab type expressions where a and b are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations. Examples If x > 0, then x^x \ge \left( \frac{1}{e}\right)^{1/e}.\, If x > 0, then x^{x^x} \ge x.\, If x, y, z > 0, then (x+y)^z + (x+z)^y + (y+z)^x > 2.\, For any real distinct numbers a and b, \frac{e^be^a}{ba} > e^{(a+b)/2}. If x, y > 0 and 0 < p < 1, then (x+y)^p < x^p+y^p.\, If x, y, z > 0, then x^x y^y z^z \ge (xyz)^{(x+y+z)/3}.\, If a, b, then a^b + b^a > 1.\, This result was generalized by R. Ozols in 2002 who proved that if a1, ..., an, then a_1^{a_2}+a_2^{a_3}+\cdots+a_n^{a_1}>1 (result is published in Latvian popularscientific quarterly The Starry Sky, see references). Wellknown inequalities See also list of inequalities. Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names: Azuma's inequality Bernoulli's inequality Boole's inequality Cauchyâ€“Schwarz inequality Chebyshev's inequality Chernoff's inequality CramÃ©r–Rao inequality Hoeffding's inequality HÃ¶lder's inequality Inequality of arithmetic and geometric means Jensen's inequality Kolmogorov's inequality Markov's inequality Minkowski inequality Nesbitt's inequality Pedoe's inequality PoincarÃ© inequality Triangle inequality Complex numbers and inequalities The set of complex numbers \mathbb{C} with its operations of addition and multiplication is a field, but it is impossible to define any relation â‰¤ so that (\mathbb{C},+,\times,\le) becomes an ordered field. To make (\mathbb{C},+,\times,\le) an ordered field, it would have to satisfy the following two properties: if a â‰¤ b then a + c â‰¤ b + c if 0 â‰¤ a and 0 â‰¤ b then 0 â‰¤ a b Because â‰¤ is a total order, for any number a, either 0 â‰¤ a or a â‰¤ 0 (in which case the first property above i
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Answers:Why do you need the zeros or intercepts? You take your y= graph and shade the upper or lower part depending on the direction of the inequality....
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Answers:In both cases, rewrite the inequality to have the functional form f(x) either less than or greater than zero. [ f(x) > 0 or f(x) < 0 ] Plot the line y=f(x), then select the regions of the curve where f(x) is greater than or less than zero respectively. [if f(x) is greater than zero, it lies above the xaxis. if f(x) is less than zero, it lies below the xaxis.] Noting that the inequalities are strictly less than or greater than, the points on the xaxis should be represented by open circles (rather than solid dots.)
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