Solving Absolute Value Equations Made Easy

In this guide, we'll explore how to solve absolute value equations step by step. Whether you're new to algebra or need a refresher, you'll find clear explanations and practical examples to help you master this topic.
By Taylor

What is Absolute Value?

The absolute value of a number is its distance from zero on the number line, regardless of direction. For example:

  • The absolute value of 5 is 5.
  • The absolute value of -5 is also 5.

This is written as |5| = 5 and |-5| = 5.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two scenarios because an absolute value can equal a positive or a negative value.

Example 1: Simple Absolute Value Equation

Equation:
\[ |x - 3| = 4 \]

Step 1: Set up two equations.

  1. \[ x - 3 = 4 \]
  2. \[ x - 3 = -4 \]

Step 2: Solve each equation.

  1. For the first equation:
    \[ x - 3 = 4 \]
    Add 3 to both sides:
    \[ x = 4 + 3 = 7 \]
  1. For the second equation:
    \[ x - 3 = -4 \]
    Add 3 to both sides:
    \[ x = -4 + 3 = -1 \]

Solutions:
\[ x = 7 \text{ or } x = -1 \]

Example 2: More Complex Absolute Value Equation

Equation:
\[ |2x + 1| = 5 \]

Step 1: Set up two equations.

  1. \[ 2x + 1 = 5 \]
  2. \[ 2x + 1 = -5 \]

Step 2: Solve each equation.

  1. For the first equation:
    \[ 2x + 1 = 5 \]
    Subtract 1 from both sides:
    \[ 2x = 5 - 1 \]
    \[ 2x = 4 \]
    Divide by 2:
    \[ x = 2 \]
  1. For the second equation:
    \[ 2x + 1 = -5 \]
    Subtract 1 from both sides:
    \[ 2x = -5 - 1 \]
    \[ 2x = -6 \]
    Divide by 2:
    \[ x = -3 \]

Solutions:
\[ x = 2 \text{ or } x = -3 \]

Example 3: Absolute Value with Variables

Equation:
\[ |3(x - 2)| = 9 \]

Step 1: Simplify the absolute value.
Divide by 3:
\[ |x - 2| = 3 \]

Step 2: Set up two equations.

  1. \[ x - 2 = 3 \]
  2. \[ x - 2 = -3 \]

Step 3: Solve each equation.

  1. For the first equation:
    \[ x - 2 = 3 \]
    Add 2 to both sides:
    \[ x = 3 + 2 = 5 \]
  1. For the second equation:
    \[ x - 2 = -3 \]
    Add 2 to both sides:
    \[ x = -3 + 2 = -1 \]

Solutions:
\[ x = 5 \text{ or } x = -1 \]

Conclusion

Solving absolute value equations can seem tricky at first, but by breaking them down into two separate cases, you can find the solution step by step. Practice with different examples to become more comfortable with this method!