Text of Section 4.4 Solving Absolute Value Equations and Inequalities
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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.