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Section 4.4
Solving Absolute Value Equations and Inequalities
4.4 Lecture Guide: Solving Absolute Value Equations and Inequalities
Objective 1: Use absolute value notation to represent intervals.
Algebraically Verbally
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
−3 30
( )−3 30
()−3 30
Absolute Value ExpressionsFor any real number x and any nonnegative real number d:
3x 3x
3x 3x
3 3x 3x
3x
3x
3x 3x
x d
x d
x d
Use interval notation to represent the real numbers that are solutions of these inequalities.
1. 2.9x 9x
Write an absolute-value inequality to represent each set of points.
3. 4.
-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10
Write an absolute-value inequality to represent each set of points.
5. The points between – 6 and 6.
6.
-10 -8 -6 -4 -2 0 2 4 6 8 10
, 1 1,
−2 0−4−6−8−10 2 4 6 8 10
Use absolute value to represent the distance between these real numbers.
7.
8.
9.
y and x
x and 10
x and 10
10. Write an absolute value equation indicating that the distance from x
to a is d units.
Objective 2: Solve absolute value equations and inequalities.
Algebraically Verbally
x is either d units _________ or __________ of a.
x is _________ than d units from a.
x is _________ than d units from a.
Equivalent Expressions
Graphically
a – d a + d a
( )a
()a
Solving Absolute Value Equations and InequalitiesFor any real number x and any nonnegative real number d:
x a d
x a d
x a d
orx a d
x a d
orx a d
x a d
a – d a + d
a – d a + d
d x a d
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 groupexercises at the end of this section.
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. , 2 8, -10 -8 -6 -4 -2 0 2 4 6 8 10
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.-10 -8 -6 -4 -2 0 2 4 6 8 10
5,3
Solve each equation or inequality algebraically.
13. 3 8x
Solve each equation or inequality algebraically.
14. 3 10x
Solve each equation or inequality algebraically.
15. 2 22x
Solve each equation or inequality algebraically.
16. 2 1 9x
Solve each equation or inequality algebraically.
17.2
123x
Solve each equation or inequality algebraically.
18. 2 1 5 13x
Solve each equation or inequality algebraically.
19. 5 1 9x
Solve each equation or inequality algebraically.
20. 3 5 2 8x
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.
a ba b
21. Solve .1 2x x
22. Solve .3 1 2 3x x
23. Use the graph to solve each equation or inequality.
-2
5
-3 5
x
y(a)
(b)
(c)
1y
2y
1 2y y
1 2y y
1 2y y
24. Use the table to solve each equation or inequality.
1 2, ,
8 3 2
7 2 2
6 1 2
5 0 2
4 1 2
3 2 2
2 3 2
x y or y
(a)
(b)
(c)
1 2y y
1 2y y
1 2y y
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.
10, 10, 1 by 10, 10, 1
2 5x
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.
10, 10, 1 by 10, 10, 1
2 5x
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.
10, 10, 1 by 10, 10, 1
2 5x
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.
2 3 5x
10, 10, 1 by 10, 10, 1
28.
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.
10, 10, 1 by 10, 10, 1
29. 2 3 5x
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.
10, 10, 1 by 10, 10, 1
30. 2 3 5x
Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities.
31. 32. 1,15 , 16 6,
Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities.
33. 34. , 9 1, 21,45
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.
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.
