SECTION A.9 Interval Notation; Solving Inequalities A81 As the examples that follow illustrate, inequalities can be solved by using many of the same steps that would be used to solve equations. In writing the solution of an inequality, either set notation or interval notation may be used, whichever is more convenient. -2 -3 -6 -4 -5 -1 Figure 32 x 5 ≥− EXAMPLE 7 Solving an Inequality Solve the inequality x x 4 7 2 3, + ≥ − and graph the solution set. Solution + ≥ − + − ≥ − − ≥ − − ≥ − − ≥ − ≥ − ≥ − x x x x x x x x x x x x x 4 7 2 3 4 7 7 2 3 7 4 2 10 4 2 2 10 2 2 10 2 2 10 2 5 Subtract 7 from both sides. Simplify. Subtract x2 from both sides. Simplify. Divide both sides by 2. (The direction of the inequality symbol is unchanged.) Simplify. The solution set is x x 5 { } ≥ − or, using interval notation, all numbers in the interval 5, . [ ) − ∞ See Figure 32 for the graph. Now Work problem 59 Solving a Combined Inequality Solve the inequality x 5 3 2 1, − < − < and graph the solution set. Solution EXAMPLE 8 Recall that the inequality x 5 3 2 1 − < − < is equivalent to the two inequalities x x 5 3 2 and 3 2 1 − < − − < Solve each of these inequalities separately. − < − − + < − + − < − < − < x x x x x 5 3 2 5 2 3 2 2 3 3 3 3 3 3 1 − < − + < + < < < x x x x x 3 2 1 3 2 2 1 2 3 3 3 3 3 3 1 Add 2 to both sides. Simplify. Divide both sides by 3. Simplify. 1 2 -1 0 -2 -3 Figure 33 x 1 1 − < < The solution set of the original pair of inequalities consists of all x for which x x 1 and 1 − < < This may be written more compactly as x x 1 1 . { } − < < In interval notation, the solution is 1, 1 . ( ) − See Figure 33 for the graph. 4 Solve Combined Inequalities Now Work problem 79
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