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171S3.5q Solving Equations and Inequalities with Absolute Value
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March 03, 2013
Oct 49:22 PM
3.1 The Complex Numbers3.2 Quadratic Equations, Functions, Zeros, and Models3.3 Analyzing Graphs of Quadratic Functions3.4 Solving Rational Equations and Radical Equations3.5 Solving Equations and Inequalities with Absolute Value
CHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus AlgebraDr. Claude Moore
Cape Fear Community College
This is a good 6minute video to solve two problems: | 3y + 9 | ≥ 6 and | 3x + 5 | 8 < 5. http://www.youtube.com/watch?v=Jad08Q4puOc
Instructions for graphing onevariable inequality with TI calculator.http://cfcc.edu/faculty/cmoore/tiinequality1.htm
Go to SAS Curriculum Pathways, use Subscriber Login and User name: able7oxygen. Use "Exploring Graphs of Absolute Value Equations and Inequalities" by using Quick Launch # 1442 at
http://www.sascurriculumpathways.com/
Oct 49:27 PM
1. Absolute Value Equation short video (by Dr. Moore) demonstrating the solution of absolute value equations. http://cfcc.edu/faculty/cmoore/AbsoluteValueEquations1.wmv
2. Absolute Value Inequality 1 short video (by Dr. Moore) demonstrating the solution of simple inequalities. http://cfcc.edu/faculty/cmoore/AbsoluteInequality1.wmv
3. Absolute Value Inequality 2 short video (by Dr. Moore) demonstrating the solution more complex inequalities. http://cfcc.edu/faculty/cmoore/AbsoluteInequality2.wmv
NOTE: These videos are in the Technology on the Important Links webpage.
Some Media for this Section
This program graphs an Absolute Value Equation. The solution is the xvalues for points of intersection.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:22 PM
3.5 Solving Equations and Inequalities with Absolute Value
• Solve equations with absolute value.• Solve inequalities with absolute value.
Equations with Absolute Value
For a > 0 and an algebraic expression X:
| X | = a is equivalent to
X = a or X = a.
Solve:
The solutions are –5 and 5.
Solution:
To check, note that –5 and 5 are both 5 units from 0 on the number line.
Oct 49:22 PM
Example
Solve:
Let’s check the possible solutions –2 and 8.
Solution: First, add one to both sides to get the expression in the form | X | = a.
Check x = –2: Check x = 8:
TRUE
The solutions are –2 and 8.
TRUE
Oct 49:22 PM
More About Absolute Value Equations
When a = 0, | X | = a is equivalent to X = 0.
Note that for a < 0, | X | = a has no solution, because the absolute value of an expression is never negative.The solution is the empty set, denoted
Inequalities with Absolute Value
Inequalities sometimes contain absolutevalue notation. The following properties are used to solve them.
For a > 0 and an algebraic expression X:| X | < a is equivalent to a < X < a.| X | > a is equivalent to X < a or X > a.
Similar statements hold for | X | < a and | X | > a.
Oct 49:22 PM
Inequalities with Absolute Value
For example,
| x | < 3 is equivalent to 3 < x < 3
| y | ≥ 1 is equivalent to y ≤ 1 or y ≥ 1
| 2x + 3 | ≤ 4 is equivalent to 4 < 2x + 3 < 4
171S3.5q Solving Equations and Inequalities with Absolute Value
2
March 03, 2013
Oct 49:22 PM
Example
Solve: Solve and graph the solution set:
Solution:
Example
Solve: Solve and graph the solution set:
Solution:
Oct 49:44 PM
285/2. |x| = 4.5
285/6. |x| = 3/2
Solve
Solve
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:44 PM
285/7. |x| = 10.7Solve
285/8. |x| = 12Solve
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:44 PM
285/14. |x 7| = 5
285/16. |x + 5| = 1
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:44 PM
285/18. |7x 4| = 8
285/20. |(1/3)x 4| = 13
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:44 PM
285/24. |x 4| + 3 = 9
285/28. |5x + 4| + 2 = 5
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
171S3.5q Solving Equations and Inequalities with Absolute Value
3
March 03, 2013
Oct 49:44 PM
286/30. 9 |x 2| = 7
286/32. 5 |4x + 3| = 2
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Oct 49:44 PM
3.5 Solving Equations and Inequalities with Absolute Value
Solve and write interval notation for the solution set. Then graph the solution set.286/42. |5x| < 4
Solve and write interval notation for the solution set. Then graph the solution set.286/46. |x + 6| < 10
Graphing OneVariable Inequality http://cfcc.edu/faculty/cmoore/tiinequality1.htm
Graphing OneVariable Inequality http://cfcc.edu/faculty/cmoore/tiinequality1.htm
Oct 49:44 PM
3.5 Solving Equations and Inequalities with Absolute Value
Solve and write interval notation for the solution set. Then graph the solution set.286/56. |5 2x| > 10
Since x = 2.5 gives y = 0 and x = 7.5 gives y = 0, the inequality is false for 2.5 and 7.5. Thus, we have open circles at these two values. So, the solution is (∞, 2.5) U (7.5, ∞).
Solve and write interval notation for the solution set. Then graph the solution set.286/60. |(2x 1) / 3| > 5/6
Graphing OneVariable Inequality http://cfcc.edu/faculty/cmoore/tiinequality1.htm
Since x = 0.75 gives y = 1 and x = 1.75 gives y = 1, the inequality is true for 0.75 and 1.75. Thus, we have closed circles at these two values. So, the solution is (∞, 0.75] U [1.75, ∞).
Graphing OneVariable Inequality http://cfcc.edu/faculty/cmoore/tiinequality1.htm
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