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Section 4.4 Solving Absolute Value Equations and Inequalities

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4.4 Lecture Guide: Solving Absolute Value Equations and Inequalities Objective 1: Use absolute value notation to represent intervals.

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AlgebraicallyVerbally The distance from 0 to x is d units. The distance from 0 to x is __________ than d units. The distance from 0 to x is __________ than d units. Numerical Example Graphical Example if or if and if or 33 3 0 ( ) 33 3 0 ( ) 33 3 0 Absolute Value Expressions For any real number x and any nonnegative real number d:

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Use interval notation to represent the real numbers that are solutions of these inequalities. 1.2.

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Write an absolute-value inequality to represent each set of points. 3. 4.

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Write an absolute-value inequality to represent each set of points. 5. The points between 6 and 6. 6. 22044668810246810

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Use absolute value to represent the distance between these real numbers. 7. 8. 9. y and x x and 10 x and

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10. Write an absolute value equation indicating that the distance from to a isunits.

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Objective 2: Solve absolute value equations and inequalities.

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AlgebraicallyVerbally x is either d units _________ or __________ of a. x is _________ than d units from a. Equivalent Expressions Graphically a da + d a ( ) a ( ) a Solving Absolute Value Equations and Inequalities For any real number x and any nonnegative real number d: a da + d a da + d

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Similar statements can also be made about the order relations less than or equal to and greater than or equal to Expressions with d negative are examined in the group exercises at the end of this section.

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Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 11.

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Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 12.

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Solve each equation or inequality algebraically. 13.

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Solve each equation or inequality algebraically. 14.

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Solve each equation or inequality algebraically. 15.

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Solve each equation or inequality algebraically. 16.

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Solve each equation or inequality algebraically. 17.

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Solve each equation or inequality algebraically. 18.

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Solve each equation or inequality algebraically. 19.

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Solve each equation or inequality algebraically. 20.

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If, then a and b are equal in magnitude but their signs can either agree or disagree. Thus is equivalent to ______________ or ______________. Use this result to solve the next two problems.

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21. Solve.

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22. Solve.

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23. Use the graph to solve each equation or inequality. (a) (b) (c)

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24. Use the table to solve each equation or inequality. (a) (b) (c)

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 25.

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 26.

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 27.

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 28.

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 29.

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Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 30.

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Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities. 31. 32.

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Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities. 33.34.

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35. The correct torque setting for the lug bolts on a race car is 85 foot-pounds with a tolerance of 3 foot-pounds. (a) Express the acceptable torque setting as an absolute value inequality. (b) Express the acceptable torque setting as a compound linear inequality.

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35. The correct torque setting for the lug bolts on a race car is 85 foot-pounds with a tolerance of 3 foot-pounds. (c) Determine the lower and upper limits of the interval.