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Introduction |
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A statement is an assertion or a sentence that is either true or false, but not both. |
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A variable is a symbol that may represent any member of a specified set called replacement set or domain of that variable. |
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Some general rules of inequalities |
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In this section, you will learn how so solve inequalities. "Solving" an inequality means finding all of its solutions. A "solution" of an inequality is a number which when substituted for the variable makes the inequality a true statement.
Rule 1. Adding/subtracting the same number on both sides.
Rule 2. Switching sides and changing the orientation of the inequality sign.
Rule 3a. Multiplying/dividing by the same POSITIVE number on both sides.
Rule 3b. Multiplying/dividing by the same NEGATIVE number on both sides AND changing the orientation of the inequality sign. |
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Linear inequations |
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An inequation is said to be linear if each term of the algebraic expression (or expressions) of the inequation contains first degree variables (not the product of variables). |
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ax + by ³
0, ax + by £
0, ax + by ³ c, ax +
by £ c where a, b not equals to
0 are linear inequations of two variables, (x, y) of degree 1. ax
³ c, ay
³ k are also linear
inequations. They are single variable inequations. |
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Graphs of linear inequations |
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Consider the equation x = 0, x = 1, y = 0, y = -2.
i) x = 0 represents y-axis.
ii) x = 1 represents line parallel to y-axis.
iii) y = 0 represents x-axis.
iv) y = -2 represents line parallel to x-axis.
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Simultaneous inequations |
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Two inequalities, containing the same unknowns, are called equivalent, if they are valid at the same values of the unknowns. The same definition is used for the equivalence of two systems of simultaneous inequalities. Solving of inequalities is a process of transition from one inequality to another, equivalent inequality.
To solve the system of simultaneous inequalities it is necessary to solve each of them and to compare their solutions. This comparison results to one of two possible cases: either the system has a solution as a whole or does not. |
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Summary |
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A half-plane in the x-y plane is called a closed half-plane if the points on the line separating the half-planes are also included in the half-plane. |
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Two or more linear inequations are said to constitute a system of linear inequations. |
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The solution set of a system of linear inequations is defined as the intersection of solution sets of linear inequations in the system. |
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Conclusion |
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Linear inequations denote a user-friendly branch of mathematics which enables us to be very comfortable with the number line and properties of numbers.
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