SECTION A.9 Interval Notation; Solving Inequalities A79 Multiplying an Inequality by a Positive Number Express as an inequality the result of multiplying both sides of the inequality 3 7 < by 2. Solution EXAMPLE 4 Begin with 3 7 < Multiplying both sides by 2 yields the numbers 6 and 14, so we have 6 14 < Multiplying an Inequality by a Negative Number Express as an inequality the result of multiplying both sides of the inequality 9 2 > by 4. − Solution EXAMPLE 5 Begin with 9 2 > Multiplying both sides by 4− yields the numbers 36 − and 8, − so we have 36 8 − < − Multiplication Property of Inequalities (a) If x2 6, < then x 1 2 2 1 2 6 ⋅ < ⋅ or x 3. < (b) If x 3 12, − > then x 3 3 3 12 ( ) − − < − ⋅ or x 36. < − (c) If x4 8, − < − then x4 4 8 4 − − > − − or x 2. > (d) If x 8, − > then x 1 1 8 ( )( ) ( ) − − < − ⋅ or x 8. < − EXAMPLE 6 Now let’s see what happens when both sides of an inequality are multiplied by a nonzero number. Note that the effect of multiplying both sides of 9 2 > by the negative number 4− is that the direction of the inequality symbol is reversed. Examples 4 and 5 illustrate the following general multiplication properties for inequalities: Multiplication Properties for Inequalities For real numbers a , b , and c , • a b c ac bc If and if 0, then . < > < (3a) • a b c ac bc If and if 0, then . < < > • a b c ac bc If and if 0, then . > > > (3b) • a b c ac bc If and if 0, then . > < < Now Work problem 47 In Words Multiplying by a negative number reverses the inequality. In Words The multiplication properties state that the sense, or direction, of an inequality remains the same if both sides are multiplied by a positive real number, whereas the direction is reversed if both sides are multiplied by a negative real number.
RkJQdWJsaXNoZXIy NjM5ODQ=