A78 APPENDIX Review Writing Intervals Using Inequality Notation Write each interval as an inequality involving x . (a) 1, 4 [ ) (b) 2, ( )∞ (c) 2, 3 [ ] (d) , 3 ( ] −∞ − Solution EXAMPLE 2 (a) 1, 4 [ ) consists of all real numbers x for which x 1 4. ≤ < (b) 2, ( )∞ consists of all real numbers x for which x 2. > (c) 2, 3 [ ] consists of all real numbers x for which x 2 3. ≤ ≤ (d) , 3 ( ] −∞ − consists of all real numbers x for which x 3. ≤ − Addition Property of Inequalities (a) If x 5, < − then x 5 5 5 + < − + or x 5 0. + < (b) If x 2, > then x 2 2 2 ( ) ( ) + − > + − or x 2 0. − > EXAMPLE 3 NOW WORK problems 13, 25, and 33 2 Use Properties of Inequalities The product of two positive real numbers is positive, the product of two negative real numbers is positive, and the product of 0 and 0 is 0. For any real number a , the value of a2 is 0 or positive; that is, a2 is nonnegative. This is called the nonnegative property . When the same number is added to both sides of an inequality, an equivalent inequality is obtained. For example, since 3 5, < then3 4 5 4 + < + or 7 9. < This is called the addition property of inequalities. Now Work problem 41 Nonnegative Property For any real number a , a 0 2 ≥ (1) In Words The square of a real number is never negative. Addition Property of Inequalities For real numbers a , b , and c , • a b a c b c If , then . < + < + (2a) • a b a c b c If , then . > + > + (2b) In Words The addition property states that the sense, or direction, of an inequality remains unchanged if the same number is added to both sides.
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