Sure! Let's work through an absolute value problem step by step.

Here's a sample problem:

Solve:
$|2x - 5| = 7$

Step 1: Set up two equations

The definition of absolute value tells us that:

$|A| = B$

means that $A = B$ or $A = -B$.

For our problem, $|2x - 5| = 7$, we break it into two cases:

1. $2x - 5 = 7$
2. $2x - 5 = -7$

Step 2: Solve each equation

Case 1: $2x - 5 = 7$

1. Add 5 to both sides: $2x = 7 + 5$ $2x = 12$
2. Divide by 2: $x = \frac{12}{2} = 6$

Case 2: $2x - 5 = -7$

1. Add 5 to both sides: $2x = -7 + 5$ $2x = -2$
2. Divide by 2: $x = \frac{-2}{2} = -1$

Step 3: Solution

The solutions to the equation are $x = 6$ and $x = -1$.

Step 4: Check the solutions

We should check both solutions by substituting them back into the original equation:

• For $x = 6$:
  $|2(6) - 5| = |12 - 5| = |7| = 7 \quad (\text{True})$
• For $x = -1$:
  $|2(-1) - 5| = |-2 - 5| = |-7| = 7 \quad (\text{True})$

Both values satisfy the original equation, so the final answer is: $x = 6 \quad \text{or} \quad x = -1$

Do you want more details on any specific step? Here are some related questions to consider:

1. How do you solve absolute value inequalities?
2. What happens when there’s no solution for an absolute value equation?
3. Can you explain the geometric interpretation of absolute value equations?
4. How do you handle absolute value equations with more complex expressions inside the absolute value?
5. What’s the difference between absolute value equations and inequalities?

Tip: Always check your solutions in absolute value equations to ensure they satisfy the original equation.