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SOLVING ABSOLUTE-VALUE EQUATIONS You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x | 8 has two solutions: 8 and 8. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative. Solving an Absolute-Value Equation Solve | x 2 | 5 SOLUTION The expression x 2 can be equal to 5 or 5. x 2 IS IS POSITIVE POSITIVE x 2 IS NEGATIVE || xx 22 || 55 xx 22 5 5 |x2|5 xx 77 x 2 5 x 3 The equation has two solutions: 7 and –3. CHECK |72||5|5 | 3 2 | | 5 | 5 Solving an Absolute-Value Equation Solve | 2x 7 | 5 4 SOLUTION Isolate the absolute value expression on one side of the equation. 2x 7 IS IS POSITIVE POSITIVE 2x 7 IS IS NEGATIVE NEGATIVE | 2x 7 | 5 4 | 2x 7 | 5 4 | 2x 7 | 9 | 2x 7 | 9 2x 7 +9 2x 16 x8 2x 7 9 9 2x 2 TWO SOLUTIONS x 1 Solving an Absolute-Value Equation Recall that |xx is | isthe thedistance distancebetween betweenxxand and0.0.IfIf x | x | 8,8,then then any number between 8 and 8 is a solution of the inequality. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 You can use the following properties to solve absolute-value inequalities and equations. 7 8 SOLVING ABSOLUTE-VALUE INEQUALITIES SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES | ax b | c means ax b c and a x b c. means valueais | a xWhen b | an c absolute x less b than c and a x the b c. a number, inequalities are connected by and. When an absolute means | a xvalue b | iscgreater a x b cthe inequalities or a x bare c. than a number, connected by or. | ax b | c means ax b c or a x b c. | ax b | c means ax b c or a x b c. Solving an Absolute-Value Inequality Solve | x 4 | < 3 x 4 IS POSITIVE |x4|3 x 4 3 x7 x 4 IS NEGATIVE |x4|3 x 4 3 x1 Reverse inequality symbol. The solution is all real numbers greater than 1 and less than 7. This can be written as 1 x 7. Solving an Absolute-Value Inequality Solve 1 | 3 6 and graph 2x + 1| 2x IS POSITIVE 2x +the 1 ISsolution. NEGATIVE | 2x 1 | 3 6 | 2x 1 | 3 6 2x + 1 IS POSITIVE | 2x|2x1 | 13| 69 2x + 1 IS NEGATIVE 1 | 69 | 2x|2x1 | 3 2x 1 9 | 2x 1 | 9 2x 10 2x 8 2x 1 9 2x 1 +9 x4 x 5 2x 10 2x 8 The solution is all real numbers greater than or equal x4 x 5 to 4 or less than or equal to 5. This can be written as the compound inequality x 5 or x 4. Reverse 2x11| +9 | 2x 9 inequality symbol. 6 5 4 3 2 1 0 1 2 3 4 5 6 Writing an Absolute-Value Inequality You work in the quality control department of a manufacturing company. The diameter of a drill bit must be between 0.62 and 0.63 inch. a. Write an absolute-value inequality to represent this requirement. b. Does a bit with a diameter of 0.623 inch meet the requirement? Writing an Absolute-Value Inequality The diameter of a drill bit must be between 0.62 and 0.63 inch. a. Write an absolute-value inequality to represent this requirement. Let d represent the diameter (in inches) of the drill bit. Write a compound inequality. 0.62 d 0.63 Find the halfway point. 0.625 Subtract 0.625 from each part of the compound inequality. 0.62 0.625 d 0.625 0.63 0.625 0.005 d 0.625 0.005 Rewrite as an absolute-value inequality. | d 0.625 | 0.005 This inequality can be read as “the actual diameter must differ from 0.625 inch by no more than 0.005 inch.” Writing an Absolute-Value Inequality The diameter of a drill bit must be between 0.62 and 0.63 inch. b. Does a bit with a diameter of 0.623 meet the requirement? | d 0.625 | 0.005 | 0.623 0.625 | 0.005 | 0.002 | 0.005 0.002 0.005 Because 0.002 0.005, the bit does meet the requirement.