2-7 Solving Absolute-Value Inequalities
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- Clemence Griffith
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1 Warm Up Solve each inequality and graph the solution. 1. x + 7 < x x > 1
2 When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality x < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between 5 and 5, so x < 5 can be rewritten as 5 < x < 5, or as x > 5 AND x < 5.
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4 Additional Example 1A: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. x 3 < 1 x 3 < x < 2 x > 2 AND x < 2 Since 3 is subtracted from x, add 3 to both sides to undo the subtraction. Write as a compound inequality. 2 units 2 units
5 Additional Example 1B: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. x 1 2 x 1 2 AND x x 1 AND x 3 Write as a compound inequality. Solve each inequality. Write as a compound inequality
6 Helpful Hint Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities.
7 Check It Out! Example 1a Solve the inequality and graph the solutions. 2 x 6 2 x x 3 x 3 AND x 3 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. Write as a compound inequality. 3 units 3 units
8 x Check It Out! Example 1b Solve each inequality and graph the solutions. x x x AND x x 15 AND x 9 Since 4.5 is subtracted from x + 3, add 4.5 to both sides to undo the subtraction. Write as a compound inequality. Subtract 3 from both sides of each inequality
9 The inequality x > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than 5 or greater than 5. The inequality x > 5 can be rewritten as the compound inequality x < 5 OR x > 5.
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11 Additional Example 2A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. x x x 5 x 5 OR x 5 Since 14 is added to x, subtract 14 from both sides to undo the addition. Write as a compound inequality. 5 units 5 units
12 Additional Example 2B: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. 3 + x + 2 > x + 2 > Since 3 is added to x + 2, subtract 3 from both sides to undo the addition. x + 2 > 2 Write as a compound inequality. x + 2 < 2 OR x + 2 > 2 Solve each inequality x < 4 OR x > 0 Write as a compound inequality
13 Check It Out! Example 2a Solve each inequality and graph the solutions. x x x 2 x 2 OR x 2 Since 10 is added to x, subtract 10 from both sides to undo the addition. Write as a compound inequality. 2 units 2 units
14 Check It Out! Example 2b Solve the inequality and graph the solutions. Since is added to x + 2, subtract from both sides to undo the addition. Write as a compound inequality. Solve each inequality. OR x 6 x 1 Write as a compound inequality.
15 Check It Out! Example 2b Continued Solve the inequality and graph the solutions
16 When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions.
17 Additional Example 4A: Special Cases of Absolute- Value Inequalities Solve the inequality. x > 8 x > Add 5 to both sides. x + 4 > 3 All real numbers are solutions. Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
18 Additional Example 4B: Special Cases of Absolute- Value Inequalities Solve the inequality. x < 7 x < x 2 < 2 The inequality has no solutions. Subtract 9 from both sides. Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
19 Remember! An absolute value represents a distance, and distance cannot be less than 0.
20 Check It Out! Example 4a Solve the inequality. x 9 11 x x 2 All real numbers are solutions. Add 9 to both sides. Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
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A rational inequality is an inequality that contains one or more rational expressions. x x 3 by using a graph and a table. Use a graph. On a graphing calculator, Y1 = x and Y = 3. x The graph of Y1 is
Quadratic and Rational Inequalities
Quadratic and Rational Inequalities Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax 2 + bx + c < 0 ax 2 + bx + c > 0 ax 2 + bx + c