Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: 3|2x + 1| > −2 analytically, we first isolate the absolute value before applying Theorem 2.3. To that end, we get −3|2x + 1| > −6 or |2x + 1| < 2. Rewriting, we now have 3 −2 < 2x + 1 < 2 so that − 3 < x < 1 . In interval notation, we write − 2 , 1 . Graphically we 2 2 2 1 see the graph of y = 4 − 3|2x + 1| is above y = −2 for x values between − 3 and 2 . 2 y 4 3 2 1 −2 −1 1 2 x −1 −2 −3 −4 3. Rewriting the compound inequality 2 < |x − 1| ≤ 5 as ‘2 < |x − 1| and |x − 1| ≤ 5’ allows us to solve each piece using Theorem 2.3. The first inequality, 2 < |x − 1| can be re-written as |x − 1| > 2 and so x − 1 < −2 or x − 1 > 2. We get x < −1 or x > 3. Our solution to the first inequality is then (−∞, −1) ∪ (3, ∞). For |x − 1| ≤ 5, we combine results in Theorems 2.1 and 2.3 to get −5 ≤ x − 1 ≤ 5 so that −4 ≤ x ≤ 6, or [−4, 6]. Our solution to 2 < |x − 1| ≤ 5 is comprised of values of x which satisfy both parts of the inequality, and so we take what’s called the ‘set theoretic intersection’ of (−∞, −1) ∪...
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