1.7B Solve Absolute Value Inequalities Goal Solve absolute value (equations) and inequalities. VOCABULARY Absolute value, Extraneous solution ABSOLUTE VALUE INEQUALITIES You can solve an absolute value inequality by rewriting it as a ___compound___ ___inequality___ and then solving ___each___ ___part___. In the inequalities below, c > 0. Inequality Equivalent Form |ax + b| _<_ c c < ax + b < c |ax + b| _≤_ c c ≤ ax + b ≤ c |ax + b| _>_ c ax + b < c or ax + b > c |ax + b| _≥_ c ax + b ≤ c or ax + b ≥ c Graph of Solution Example 1: Solve an inequality of the form |ax + b| > c Solve |2x + 5 | > 3. Then graph the solution. The absolute value inequality is equivalent to 2x + 5 < _3_ or 2x + 5 > _3_ . First Inequality Second Inequality 2x + 5 < _3_ Write inequalities. 2x + 5 > _3_ 2x < _8_ Subtract _5_ from each side. 2x > _2_ 2x < _4_ Divide each side by _2_ . x > _1_ The solutions are all real numbers less than _4_ or greater than _1_. Example 2: Solve an inequality of the form |ax + b| c Solve | 3x - 7 | 5 . Then graph the solution. The absolute value inequality is equivalent to 2x + 5 < _3_ or 2x + 5 > _3_ . First Inequality 3x - 7 ≤ _-5_ Write the inequalities. 3x ≤ _2_ Add _7_ to each side. x ≤ _2/3_ Divide each side by _3_ . Second Inequality 3x -7 ≥ _5_ 3x ≥ _12_ x ≥ _4_ The solutions are all real numbers less than or equal to _2/3_ or greater than or equal to _4_. You Try: Solve the inequality. Then graph the solution. 1. |x 2| 7 x 5 or x 9 Example 3: Solve an inequality of the form | ax + b | < c Solve | 2x + 1 | < 9 . Then graph the solution. First Inequality Second Inequality 2x + 1 < _9_ Write inequalities. 2x + 1 > _-9_ 2x < _8_ Subtract _1_ from each side. 2x > _10_ x < _4_ Divide each side by _2_ . x > _5_ The solution is between _-5_ and _4_, inclusive. -5 ≤ x ≤ 4 Example 4: Solve an inequality of the form | ax + b | c Solve |x 1.5| 4.5. Then graph the solution. First Inequality x – 1.5 ≤ _4.5_ Write inequalities. x ≤ _6_ Add _1.5_ to each side. Second Inequality x - 1.5 ≥_-4.5_ x ≥ _-3_ The solution is between _-3_ and _6_, inclusive. -3 ≤ x ≤ 6 You Try: Solve the inequality. Then graph the solution. 2. |4x 1| < 9 2 < x < 2.5 Example 5: Solve Word Problems Using Absolute Value Inequalities A food manufacturer specifies that every family-sized box of cereal should have a net weight of 25 ounces, with a tolerance of 1.2 ounces. Write and solve an absolute value inequality that describes the acceptable net weights for the cereal in a family-size box. Let w = weight of the cereal box |w 25| 1.2 First Inequality x – 25 ≤ _1.2_ Write inequalities. x ≤ _26.2__ Add _25_ to each side. Second Inequality x - 25 ≥ _-1.2_ x ≥ _23.8_ The solution is between _23.8_ and _26.2_, inclusive. 23.8 ≤ x ≤ 26.2 You Try: You have found that your new winter coat is comfortable to wear when the outdoor temperature is between 10°F and 42°F, inclusive. Write an absolute value inequality for this temperature range, where t represents the temperature in degrees Fahrenheit. Given: 10 ≤ x ≤ 42 Let t = temperature |t 26| 16