Questions and Answers
What is the result of simplifying the expression -|-2|?
-2
What is the result of simplifying the expression |-7| - |7|?
0
What is the result of simplifying the expression |-3 x 2|?
6
What is the result of simplifying the expression -15/5?
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What is the solution to the equation x - 4 = 6?
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What is the solution to the equation 3x + 1 = -5?
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What is the solution to the equation |3x + 9| - 10 = -4?
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What is the solution to the equation |3x - 4| = |x|?
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What is the solution to the equation |4x - 10| = 2|x + 1|?
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Write an absolute value equation representing the minimum and maximum lines for a poetry contest with lengths of 16 to 32 lines.
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Write an absolute value equation that represents the essay length of 375 to 425 words.
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What is an extraneous solution?
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What is the solution to the equation |3x + 12| = |4x|?
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What is the solution to the equation |x + 5| = |x + 11|?
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What is the solution to the equation |2x - 7| = |2x + 9|?
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What is the solution to the equation |20x| = |4x + 16|?
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What is the solution to the equation |4q + 9| = |2q - 1|?
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Study Notes
Absolute Value Basics
- Absolute value |x| represents the non-negative value of x, ignoring its sign.
- Example: |-2| = 2; therefore, -|-2| = -2.
Simplifying Absolute Value Expressions
- |-7| - |7| simplifies to 0 (7 - 7).
- |-3 x 2| simplifies to 6 (|-6| = 6).
- -15/5 equals |-3| which simplifies to 3.
Solving Absolute Value Equations
- Equations have two cases due to the properties of absolute values.
- Example: For x - 4 = 6, solutions are x = 10 and x = -2.
- Example: For 3x + 1 = -5, no solution exists as it leads to contradictory results.
Examples of Solving Equations with Absolute Values
- For |3x + 9| - 10 = -4,
- Rearranging gives |3x + 9| = 6, leading to two equations: 3x + 9 = 6 and 3x + 9 = -6.
- Solutions are x = -1 and x = -5.
Solving with Two Absolute Values
- Example |3x - 4| = |x| involves breaking into separate cases:
- Case 1: 3x - 4 = x results in x = 2.
- Case 2: 3x - 4 = -x results in x = 1.
Absolute Value with equations
- For |4x - 10| = 2|x + 1|,
- Break into two cases for solution.
- First case 4x - 10 = 2x + 2, leads to x = 6.
- Second case 4x - 10 = -2x - 2, leads to x = 4/3.
Monitoring Practice Problems
- Absolute value equations depict constraints, e.g., the length of a poem between 16 and 32 lines represented as |x - 24| = 8.
- For essays, word limits can be represented as |x - 400| = 25.
Extraneous Solutions
- An extraneous solution is one that does not satisfy the original equation, often appearing during the solving process.
Identifying Solutions' Validity
- Example: for |2x + 12| = |4x|, solving gives x = 6 and x = -6 where x = 6 is valid but x = -6 may not fit original equation.
Additional Practice
- Practice solving various absolute value equations is crucial to understand the concepts.
- Exploring cases involving no solutions or contradictions illustrates the complexity of some absolute value equations.
Summary of Important Concepts
- Every absolute value equation can result in two cases due to positive and negative scenarios.
- Verifying solutions against the original equations is essential to avoid extraneous solutions.
- Use strategies like isolating the absolute value before expanding into cases for easier solving.
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Description
This quiz focuses on solving absolute value equations as detailed in Algebra I, Section 1.4. Test your understanding of simplifying expressions involving absolute values through various exercises and examples.
