171S3.5p Solving Equations and Inequalities with Absolute Value
October 16, 2012
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value
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/
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Some Media for this Section
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.
This program graphs an Absolute Value Equation. The solution is the x­values for points of intersection.
http://cfcc.edu/mathlab/geogebra/absolute_value.html
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171S3.5p Solving Equations and Inequalities with Absolute Value
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:
October 16, 2012
Example
Solve:
Solution: First, add one to both sides to get the expression in the form | X | = a.
| X | = a is equivalent to
X = ­a or X = a.
Solve:
Let’s check the possible solutions –2 and 8.
Check x = 8:
Check x = –2:
Solution:
The solutions are –5 and 5.
To check, note that –5 and 5 are both 5 units from 0 on the number line.
TRUE
TRUE
The solutions are –2 and 8.
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More About Absolute Value Equations
When a = 0, | X | = a is equivalent to X = 0.
Inequalities with Absolute Value
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 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
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.
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171S3.5p Solving Equations and Inequalities with Absolute Value
October 16, 2012
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Solve
285/2 . |x| = 4.5
Example
Solve: Solve and graph the solution set:
Solution:
Example
Solve: Solve and graph the solution set:
Solution:
Solve
285/6 . |x| = ­3/2
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Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Solve
285/7 . |x| = ­10.7
285/14 . |x ­ 7| = 5
Solve
285/8 . |x| = 12
285/16 . |x + 5| = 1
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171S3.5p Solving Equations and Inequalities with Absolute Value
October 16, 2012
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
285/18 . |7x ­ 4| = 8
285/24 . |x ­ 4| + 3 = 9
285/28 . |5x + 4| + 2 = 5
285/20 . |(1/3)x ­ 4| = 13
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Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html
286/30 . 9 ­ |x ­ 2| = 7
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3.5 Solving Equations and Inequalities with Absolute Value
Graphing One­Variable Inequality http://cfcc.edu/faculty/cmoore/ti­inequality­1.htm
Solve and write interval notation for the solution set. Then graph the solution set.
286/42 . |5x| < 4
286/32 . 5 ­ |4x + 3| = 2
Graphing One­Variable Inequality http://cfcc.edu/faculty/cmoore/ti­inequality­1.htm
Solve and write interval notation for the solution set. Then graph the solution set.
286/46 . |x + 6| < 10
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171S3.5p Solving Equations and Inequalities with Absolute Value
October 16, 2012
3.5 Solving Equations and Inequalities with Absolute Value
Graphing One­Variable Inequality http://cfcc.edu/faculty/cmoore/ti­inequality­1.htm
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, ∞).
Graphing One­Variable Inequality http://cfcc.edu/faculty/cmoore/ti­inequality­1.htm
Solve and write interval notation for the solution set. Then graph the solution set.
288/60. |(2x ­ 1) / 3| > 5/6
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, ∞).
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