171S3.5q Solving Equations and Inequalities with Absolute Value

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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

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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.

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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

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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.

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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

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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:

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285/2. |x| = 4.5

285/6. |x| = ­3/2

Solve

Solve

Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html

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285/7. |x| = ­10.7Solve

285/8. |x| = 12Solve

Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html

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285/14. |x ­ 7| = 5

285/16. |x + 5| = 1

Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html

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285/18. |7x ­ 4| = 8

285/20. |(1/3)x ­ 4| = 13

Absolute Value Equation.http://cfcc.edu/mathlab/geogebra/absolute_value.html

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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

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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