Solving the absolute value equation |x-2| + 6 = 10

We are given the equation:

$$
|x - 2| + 6 = 10
$$

Step 1: Isolate the absolute value expression

Subtract 6 from both sides:

$$
|x - 2| = 10 - 6
$$
$$
|x - 2| = 4
$$

Step 2: Solve the absolute value equation

The equation $|x - 2| = 4$ means that the expression inside the absolute value can be either 4 or -4. So we set up two separate equations:

1. $x - 2 = 4$
2. $x - 2 = -4$

Step 3: Solve each equation

Equation 1:
$$
x - 2 = 4 \Rightarrow x = 4 + 2 = 6
$$

Equation 2:
$$
x - 2 = -4 \Rightarrow x = -4 + 2 = -2
$$

Step 4: Write the solution

The solutions are:
$$
x = 6 \quad \text{or} \quad x = -2
$$

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✔ Final Answer:

$$
\boxed{x = 6 \quad \text{or} \quad x = -2}
$$

Let me know if you'd like a graph or explanation of why absolute value equations have two solutions!