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Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
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2.7 Absolute Value Inequalities
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Linear Inequalities An inequality is a statement that contains one of the symbols:, ≤ or ≥. EquationsInequalities x = 3x > 3 12 = 7 – 3y12 ≤ 7 – 3y
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Graphing Solutions A solution of an inequality is a value of the variable that makes the inequality a true statement. The solution set of an inequality is the set of all solutions.
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Graphing Solutions
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Graph each set on a number line and then write it in interval notation. a. b. c. Example
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Solve: x – 2 < 5 Graph the solution set. Example
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Solve: Graph the solution set. Example
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Solve: Graph the solution set. Example
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Solve: Graph the solution set. Example The inequality symbol is reversed since we divided by a negative number.
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Step 1: Clear the equation of fractions or decimals by multiplying both sides of the inequality by the appropriate nonzero number. Step 2: Use distributive property to remove grouping symbols such as parentheses. Step 3: Combine like terms on each side of the inequality. Step 4: Use the addition property of inequality to rewrite the inequality as an equivalent inequality with variable terms on one side and numbers on the other side. Step 5: Use the multiplication property of inequality to get the variable alone on one side of the inequality. Solving Linear Inequalities
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If a is a positive number, then X < a is equivalent to a < x < a. Absolute Value Inequalities
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Solve x + 4 < 6 6 < x + 4 < 6 6 – 4 < x + 4 – 4 < 6 – 4 10 < x < 2 ( 10, 2) Example
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Solve x 3 + 6 7 x 3 1 1 x 3 1 2 x 4 [2, 4] Example
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Solve 8x 3 < 2 No solution. An absolute value cannot be less than a negative number, since it can’t be negative. Example
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If a is a positive number, then X > a is equivalent to X > a or X < a. Absolute Value Inequalities
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Example Solve for x: The sign means “less than or equal to.” The absolute value of any expression will never be less than 0, but it may equal 0. The solution set is { 1}.
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Solve 10 + 3x + 1 > 2 10 + 3x > 1 10 + 3x 1 3x 9 x 3 ( , ) ( 3, ) Example
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Solve x + 2 0 The solution is all real numbers, since all absolute values are non-negative. Any value for x we substitute into the inequality will give us a true statement. Example
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Solve ( , 15] [1, ) Example Graph of solution ( , 15] [1, )
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