A80 APPENDIX Review Procedures That Leave the Inequality Symbol Unchanged • Simplify both sides of the inequality by combining like terms and eliminating parentheses: ( ) + + > + + + > + x x x x x Replace by 2 6 2 5 1 8 7 5 • Add or subtract the same expression on both sides of the inequality: ( ) − < − + < + x x Replace by 3 5 4 3 5 5 4 5 • Multiply or divide both sides of the inequality by the same positive expression: x x Replace 4 16 by 4 4 16 4 > > Procedures That Reverse the Sense or Direction of the Inequality Symbol • Interchange the two sides of the inequality: x x Replace 3 by 3 < > • Multiply or divide both sides of the inequality by the same negative expression: x x Replace 2 6 by 2 2 6 2 − > − − < − 3 Solve Inequalities An inequality in one variable is a statement involving two expressions, at least one containing the variable, separated by one of the inequality symbols: <, , >, ≤ or .≥ To solve an inequality means to find all values of the variable for which the statement is true. These values are called solutions of the inequality. For example, the following are all inequalities involving one variable x : + < − ≥ − ≤ + − > x x x x x 5 8 2 3 4 1 3 1 2 0 2 As with equations, one method for solving an inequality is to replace it by a series of equivalent inequalities until an inequality with an obvious solution, such as x 3, < is obtained. Equivalent inequalities are obtained by applying some of the same properties that are used to find equivalent equations. The addition property and the multiplication properties for inequalities form the basis for the following procedures. For example, we know 5 3. > By Property 4(c), 1 3 1 5 , > which, of course, is true. Reciprocal Properties for Inequalities • a a If 0, then 1 0. > > (4a) • a a If 0, then 1 0. < < (4b) • b a a b If 0, then 1 1 0. > > > > (4c) • a b b a If 0, then 1 1 0. < < < < (4d)
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