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The problem asks to solve the inequality \( |10 - 3x| < 5 \) and find one possible integer value of \( x \).

### Step 1: Break down the absolute value inequality
We know that if \( |A| < B \), it implies \( -B < A < B \). So, we can rewrite the inequality \( |10 - 3x| < 5 \) as:
\[
-5 < 10 - 3x < 5
\]

### Step 2: Solve the compound inequality

#### Part 1: Solve \( -5 < 10 - 3x \)
Subtract 10 from both sides:
\[
-5 - 10 < -3x \quad \Rightarrow \quad -15 < -3x
\]
Now divide by -3 (remember to flip the inequality when dividing by a negative number):
\[
5 > x \quad \Rightarrow \quad x < 5
\]

#### Part 2: Solve \( 10 - 3x < 5 \)
Subtract 10 from both sides:
\[
-3x < 5 - 10 \quad \Rightarrow \quad -3x < -5
\]
Now divide by -3 (again, flip the inequality):
\[
x > \frac{5}{3}
\]

### Step 3: Combine the inequalities
From the two parts, we have:
\[
\frac{5}{3} < x < 5
\]

### Step 4: Identify integer values
The only integer value that satisfies this inequality is \( x = 2 \).

So, one possible integer value of \( x \) is \( \boxed{2} \).

Would you like more details or have any further questions? Here are five related questions:

1. How do you handle absolute value inequalities that involve greater than signs instead of less than?
2. What if the inequality involved a fraction instead of an integer? How would the solution change?
3. How do you graph the solution to absolute value inequalities on a number line?
4. Can there be multiple integer solutions to an absolute value inequality?
5. What happens when the absolute value inequality has a variable on both sides?

**Tip:** When solving absolute value inequalities, always remember to split the inequality into two parts and consider both the positive and negative scenarios.

If |10 - 3x| < 5, what is one possible integer value of x?

Learn how to solve the absolute value inequality |10 - 3x| < 5 and find one possible integer value of x. This step-by-step guide covers key algebraic concepts, including breaking down absolute value inequalities. Suitable for students in Grades 8-10.

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