Algebra 1.4: Solving Absolute Value Equations
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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?

<p>-3</p> Signup and view all the answers

What is the solution to the equation x - 4 = 6?

<p>10 and -2</p> Signup and view all the answers

What is the solution to the equation 3x + 1 = -5?

<p>No solution</p> Signup and view all the answers

What is the solution to the equation |3x + 9| - 10 = -4?

<p>-5</p> Signup and view all the answers

What is the solution to the equation |3x - 4| = |x|?

<p>2 and 1</p> Signup and view all the answers

What is the solution to the equation |4x - 10| = 2|x + 1|?

<p>4/3 and -6</p> Signup and view all the answers

Write an absolute value equation representing the minimum and maximum lines for a poetry contest with lengths of 16 to 32 lines.

<p>|x - 24| = 8</p> Signup and view all the answers

Write an absolute value equation that represents the essay length of 375 to 425 words.

<p>|x - 400| = 25</p> Signup and view all the answers

What is an extraneous solution?

<p>An apparent solution that must be rejected.</p> Signup and view all the answers

What is the solution to the equation |3x + 12| = |4x|?

<p>6 and -2</p> Signup and view all the answers

What is the solution to the equation |x + 5| = |x + 11|?

<p>No solution</p> Signup and view all the answers

What is the solution to the equation |2x - 7| = |2x + 9|?

<p>No solution</p> Signup and view all the answers

What is the solution to the equation |20x| = |4x + 16|?

<p>1 and repeated 0.6</p> Signup and view all the answers

What is the solution to the equation |4q + 9| = |2q - 1|?

<p>-5/3</p> Signup and view all the answers

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.

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