SECTION A.1 Algebra Essentials A5 An inequality is a statement in which two expressions are related by an inequality symbol. The expressions are referred to as the sides of the inequality. Inequalities of the form a b < or b a > are called strict inequalities , whereas inequalities of the form a b ≤ or b a ≥ are called nonstrict inequalities . Based on the discussion so far, we conclude that • a a 0isequivalentto ispositive > • a a 0isequivalentto isnegative < The inequality a 0 > is sometimes read as “ a is positive.” If a 0, ≥ then either a 0 > or a 0, = and this may be read as “ a is nonnegative.” Now Work problems 27 and 37 Graphing Inequalities (a) On the real number line, graph all numbers x for which x 4. > (b) On the real number line, graph all numbers x for which x 5. ≤ Solution (a) See Figure 8. Notice that we use a left parenthesis to indicate that the number 4 is not part of the graph. (b) See Figure 9. Notice that we use a right bracket to indicate that the number 5 is part of the graph. EXAMPLE 4 Now Work problem 43 3 Find Distance on the Real Number Line The absolute value of a number a is the distance from 0 to a on the number line. For example, 4− is 4 units from 0, and 3 is 3 units from 0. See Figure 10. That is, the absolute value of 4− is 4, and the absolute value of 3 is 3. A more formal definition of absolute value is given next. DEFINITION Absolute Value The absolute value of a real number a , denoted by the symbol a , is defined by the rules a a a a a a if 0 and if 0 = ≥ = − < For example, because 4 0, − < the second rule must be used to get 4 4 4 ( ) − = − − = Computing Absolute Value (a) 8 8 = (b) 0 0 = (c) 15 15 15 ( ) − = − − = EXAMPLE 5 Look again at Figure 10. The distance from 4− to 3 is 7 units. This distance is the difference 3 4 , ( ) − − obtained by subtracting the smaller coordinate from the larger. However, since 3 4 7 7 ( ) − − = = and 4 3 7 7, − − = − = we can use absolute value to calculate the distance between two points without being concerned about which is smaller. Figure 8 x 4 > 2 3 4 5 6 7 0 1 -2 -1 Figure 9 x 5 ≤ 2 3 4 5 6 7 0 1 -2 -1 Figure 10 -5 4 4 units 3 units 0 1 2 3 -1 -2 -3 -4
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