linear_inequalities

# linear_inequalities - Solving linear inequalities...

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Solving linear inequalities Introductory examples Example A : Find the set of all real numbers x such that - x 4 > 0. Solution : Wherever x is located on the number line, - x 4 is 4 units to the left of x . Hence, in order for - x 4 to be greater than 0, we need x to be greater than 4. The statement - x 4 > 0 is equivalent to x > 4. The set of all real numbers which satisfy the inequality - x 4 > 0 is { x | x > 4 }, or ( ) , 4 using interval notation. This is the solution set of the inequality. The solution set is illustrated on the real number line by the following diagram. Example B : Find the set of all real numbers x such that 2 x > 6. Solution : Wherever x is located on the number line, 2 x is at a distance from the origin which is twice that of x . Hence, in order for 2 x to be greater than 6, we need x to be greater than 3. The statement 2 x > 6 is equivalent to x > 3. The set of all real numbers which satisfy the inequality 2 x > 6 is { x | x > 3 }, or ( ) , 3 using interval notation. This is the solution set of the inequality. The solution set is illustrated on the real number line by the following diagram. Some rules for solving inequalities An inequality in a single real number variable x is a statement of the form ( ) f x R ( ) g x in which R is one of the relations < (is less than), (is less than or equal to), > (is greater than), > (is greater than or equal to). Note that A > B is equivalent to < B A , and A > B is equivalent to B A . The problem of solving an inequality in x is that of identifying, in the simplest possible way, the real numbers x that satisfy the inequality.

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One can apply steps similar to those which are used to solve equations. (1) Adding or subtracting a real number to or from each side of an inequality gives an equivalent inequality.
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linear_inequalities - Solving linear inequalities...

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