Solving Absolute Value Equations

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Questions and Answers

What is the first step in solving the absolute value equation |3x + 5| - 7 = 10?

  • Subtract 10 from both sides.
  • Add 7 to both sides. (correct)
  • Split the equation into two separate equations.
  • Divide both sides by 3.

How many solutions does the absolute value equation |5x - 2| = 0 have?

  • Two
  • No solutions
  • One (correct)
  • Three

Solve for x in the absolute value equation: |x - 4| = 7

  • x = -3 only
  • x = 3 or x = -11
  • x = 11 only
  • x = 11 or x = -3 (correct)

Which of the following is a valid algebraic manipulation when solving for x in the equation |2x + 1| - 5 = 4?

<p>Add 5 to both sides, then split the equation into two paths. (A)</p> Signup and view all the answers

Flashcards

Absolute Value Equation

An equation that contains absolute value expressions.

Isolate Absolute Value

Move all terms to get the absolute value alone on one side of the equation.

Split into Two Equations

Remove absolute value bars by creating two separate cases, positive and negative.

Solve for x

Find the values of x that satisfy both equations formed from the absolute value.

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Number of Solutions

An absolute value equation may have 1, 2, or no solutions depending on the case.

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Study Notes

Solving Absolute Value Equations

  • When solving absolute value equations, there are possibilities of 1, 2, or no solutions.

Solving Process

  • Isolate the absolute value bars: Get the absolute value expression by itself on one side of the equation.
  • Remove the absolute value bars and split the equation: Create two separate equations. One equation will keep the expression inside the absolute value bars the same, and the other will change the expression to its opposite.
  • Solve for x: Solve each equation independently to find the solutions for x.

Example

  • Original Equation: |2x - 3| - 20 = -15

  • Isolate absolute value bars: Add 20 to both sides to get |2x - 3| = 5

  • Create two equations:

  • 2x - 3 = 5

  • 2x - 3 = -5

  • Solve first equation:

  • 2x = 8

  • x = 4

  • Solve second equation:

  • 2x = -2

  • x = -1

  • Solutions: x = 4, x = -1

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