Solving Absolute Value Equations

Absolute value represents the distance of a number from zero on the number line, regardless of direction. To solve absolute value equations, we consider two cases: one for the positive value and one for the negative value.

Example:

Solve the equation $|x - 3| = 5$.

Step 1: Set up the two cases.

• Case 1: $x - 3 = 5$
• Case 2: $x - 3 = -5$

Step 2: Solve each case separately.

• For Case 1:
  $x - 3 = 5$
  $x = 5 + 3$
  $x = 8$

• For Case 2:
  $x - 3 = -5$
  $x = -5 + 3$
  $x = -2$

Step 3: Write the final solution.

The solutions are $x = 8$ and $x = -2$.

Key Questions:

• What does the absolute value represent in a real-world context?
• How do you determine the number of solutions for an absolute value equation?
• Can an absolute value equation have no solutions? Give an example.
• How would you solve $|2x + 1| = 3$?

Keywords:

• Absolute value
• Equations
• Distance
• Solutions
• Cases