Absolute Value Equations
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Questions and Answers

What does the absolute value of a number or expression represent?

  • The direction from 0 on the number line
  • The opposite of the number
  • The distance from 0 on the number line (correct)
  • The magnitude of the number

    • What is the absolute value of a product?

  • The sum of the absolute values of the factors
  • The difference of the absolute values of the factors
  • The quotient of the absolute values of the factors
  • The product of the absolute values of the factors (correct)

    • What is the first step in solving an absolute value equation?

  • Split the equation into two possible cases
  • Combine the solutions
  • Solve each case separately
  • Isolate the absolute value expression on one side of the equation (correct)

    • What is true about the absolute value of a number?

    <p>It is always non-negative or zero</p> Signup and view all the answers

    What is the purpose of splitting an absolute value equation into two possible cases?

    <p>To account for the possibility of the expression inside the absolute value being non-negative or negative</p> Signup and view all the answers

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

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

    Study Notes

    Absolute Value Equations

    Definition

    • An absolute value equation is an equation that involves the absolute value of a variable or expression.
    • The absolute value of a number is its distance from 0 on the number line.

    Notation

    • The absolute value of a number or expression is denoted by | |, for example: |x| or |2x - 3|.
    • The absolute value can be thought of as the "distance" from 0, so |x| = a can be read as "x is a units away from 0".

    Properties

    • The absolute value of a number is always non-negative (or zero).
    • The absolute value of a product is the product of the absolute values: |ab| = |a| |b|.
    • The absolute value of a sum is not necessarily the sum of the absolute values: |a + b| ≠ |a| + |b|.

    Solving Absolute Value Equations

    • To solve an absolute value equation, isolate the absolute value expression on one side of the equation.
    • Split the equation into two possible cases: one where the expression inside the absolute value is non-negative, and one where it is negative.
    • Solve each case separately, and then combine the solutions.

    Examples

    • Solve the equation: |x| = 5
      • Case 1: x = 5
      • Case 2: x = -5
      • Solution: x = 5 or x = -5
    • Solve the equation: |2x - 3| = 7
      • Case 1: 2x - 3 = 7
      • Case 2: 2x - 3 = -7
      • Solution: x = 5 or x = -2

    Absolute Value Equations

    • An absolute value equation involves the absolute value of a variable or expression, which is the distance from 0 on the number line.

    Notation

    • The absolute value of a number or expression is denoted by | |, such as |x| or |2x - 3|.
    • The absolute value can be thought of as the "distance" from 0, so |x| = a can be read as "x is a units away from 0".

    Properties

    • The absolute value of a number is always non-negative (or zero).
    • The absolute value of a product is the product of the absolute values: |ab| = |a| |b|.
    • The absolute value of a sum is not necessarily the sum of the absolute values: |a + b| ≠ |a| + |b|.

    Solving Absolute Value Equations

    • To solve an absolute value equation, isolate the absolute value expression on one side of the equation.
    • Split the equation into two possible cases: one where the expression inside the absolute value is non-negative, and one where it is negative.
    • Solve each case separately, and then combine the solutions.

    Examples

    • The equation |x| = 5 has two solutions: x = 5 and x = -5.
    • The equation |2x - 3| = 7 has two solutions: x = 5 and x = -2.

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    Quiz Team

    Description

    Learn about absolute value equations, notation, and properties in algebra. Understand the concept of absolute value and its application in equations.

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