A82 APPENDIX Review Observe in Example 8 that solving each of the two inequalities required exactly the same steps. A shortcut to solving the original inequality algebraically is to deal with the two inequalities at the same time, as follows: Add 2 to each part. Simplify. Divide each part by 3. Simplify. − < − < − + < − + < + − < < − < < − < < x x x x x 5 3 2 1 5 2 3 2 2 1 2 3 3 3 3 3 3 3 3 3 1 1 Now Work problem 89 5 Solve Inequalities Involving Absolute Value Using a Reciprocal Property to Solve an Inequality Solve the inequality x4 1 0, 1 ( ) − > − and graph the solution set. Solution EXAMPLE 9 Recall that x x 4 1 1 4 1 . 1 ( ) − = − − Reciprocal Property (4a) states that if a 0, > then its reciprocal is greater than zero. x x x x x 4 1 0 1 4 1 0 4 1 0 4 1 1 4 1 ( ) − > − > − > > > − Reciprocal Property (4a) Add 1 to both sides. Divide both sides by 4. The solution set is x x 1 4 , { } > that is, all real numbers in the interval 1 4 , . ( )∞ Figure 34 gives the graph. 1 0 –1 4 Figure 34 x 1 4 > Solving an Inequality Involving Absolute Value Solve the inequality x 3, > and graph the solution set. Solution EXAMPLE 11 We are looking for all points whose coordinate x is a distance greater than 3 units from the origin. Figure 36 gives the graph.Any number x less than 3− or greater than 3 satisfies the condition x 3. > The solution set consists of all numbers x for which x 3 < − or x 3, > that is, all x in , 3 ( ) −∞ − ∪ 3, . ( )∞ * *Recall that the symbol ∪stands for the union of two sets. Refer to page A2 if necessary. Solving an Inequality Involving Absolute Value Solve the inequality x 4, < and graph the solution set. Solution EXAMPLE 10 We are looking for all points whose coordinate x is a distance less than 4 units from the origin. See Figure 35 for the graph. Because any x between 4− and 4 satisfies the condition x 4, < the solution set consists of all numbers x for which x 4 4, − < < that is, all x in the interval 4, 4 . ( ) − Figure 35 x 4 < 0 1 2 3 4 O 21 23 22 25 24 Less than 4 units from origin O Figure 36 > x 3 0 1 2 3 4 21 23 22 25 24
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