We are tasked with solving the equation:

\[
7 - |10 - x| = 3
\]

### Step-by-Step Solution

1. **Isolate the absolute value term:**

\[
|10 - x| = 7 - 3
\]

\[
|10 - x| = 4
\]

2. **Break into two cases:**

Since we are dealing with an absolute value, we consider two cases:

**Case 1: \( 10 - x = 4 \)**

\[
10 - x = 4
\]

Solving for \(x\):

\[
x = 10 - 4 = 6
\]

**Case 2: \( 10 - x = -4 \)**

\[
10 - x = -4
\]

Solving for \(x\):

\[
x = 10 + 4 = 14
\]

### Final Solutions:

The two possible solutions are \( x = 6 \) and \( x = 14 \).

Would you like more details on any part of the process?

Here are five related questions:

1. How do we solve equations with absolute values in general?
2. What happens when the equation has no solutions?
3. Can you explain how to graph absolute value equations?
4. What if the right-hand side of the equation were negative? Would there be any solutions?
5. How do we handle absolute value inequalities?

**Tip:** Always check if the absolute value term can be isolated first when solving equations.

Learn how to solve the absolute value equation 7 − |10 − x| = 3 step by step. This problem explains how to handle absolute value cases and find the two possible solutions for x. Ideal for students in Grades 6-8.

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