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Solving Absolute Value Inequalities: Step-by-Step Guide and Interval Notation

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This guide provides a detailed approach to solving absolute value inequalities step-by-step. It covers rewriting inequalities as conjunctions or disjunctions, negating terms, reversing inequality signs, and graphing solutions. Examples include both "and" and "or" statements, demonstrating how to solve and express solutions in interval notation. Understand the crucial steps needed to isolate variables and find solutions, with a focus on avoiding common pitfalls. Perfect for students needing reinforcement in this area of mathematics!

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Solving Absolute Value Inequalities: Step-by-Step Guide and Interval Notation

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  1. Do Now: Solve, graph, and write your answer in interval notation. • CAUTION… you must THINK through your final answer!!! • 1) a + 2 > -2 or a – 8 > 1 • 2) b – 3 > 2 and b + 3 < 4

  2. Solving Absolute Value Inequalities Section: 1-6 Page 40 in your textbook

  3. Solving an Absolute Value Inequality • Step 1: Rewrite the inequality as a conjunction or a disjunction. • If you have a you are working with a conjunction or an ‘and’ statement. Remember: “Less thand” • If you have ayou are working with a disjunction oran ‘or’ statement. Remember: “Greator” • Step 2: In the second equation you must negate the right hand side and reversethe direction of the inequality sign. • Step 3: Solve as a compound inequality.

  4. -4 3 Example 1: This is an ‘or’ statement. (Greator). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. Write the solution in interval notation. • |2x + 1| > 7 • 2x + 1 > 7 or 2x + 1 >7 • 2x + 1 >7 or 2x + 1 <-7 • x > 3 or x < -4 • (-inf., -4) U (3, +inf.)

  5. 2 8 Example 2: This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. Write the solution in interval notation. • |x -5|< 3 • x -5< 3 and x -5< 3 • x -5< 3 and x -5> -3 • x < 8 and x > 2 • (2, 8)

  6. |x |≥ -2 x ≥ -2 or x ≤ 2 Solution: All Real Numbers ( -inf., +inf.) This is an ‘or’ statement. (Greator). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. Write the solution in interval notation. -3 -2 -1 0 1 2 3 Example 3:

  7. |2x| < -4 2x < -4 and 2x > 4 x < -2 and x > 2 NO SOLUTION!! This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. Write the solution in interval notation. -3 -2 -1 0 1 2 3 Example 4:

  8. Solve and Graph • 1) |y – 3| > 1 • 2) |p + 2| < 6 • 3) | g | - 2 < -4

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