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Let's break down and solve the problem based on the text in the image.

### Problem Details:

We are given the following conditions:

- The first digit is less than 6.
- The second digit is twice the first digit.
- The sum of the two digits is more than 10.
- Let \( x \) represent the first digit.

#### Part (a) - Form inequalities representing the situation.

1. **First Condition**: The first digit is less than 6.
   - Since the first digit must be a positive integer less than 6, this gives the inequality:
     \[
     0 \leq x < 6
     \]
   - Also, since we are dealing with digits, \( x \) must be a non-negative integer. Hence, \( x \) is restricted to:
     \[
     0 \leq x \leq 5
     \]

2. **Second Condition**: The second digit is twice the first digit.
   - Let the second digit be \( 2x \). Since \( 2x \) must be a digit, it must also be less than or equal to 9 (because digits range from 0 to 9). This gives the inequality:
     \[
     2x \leq 9
     \]
   - Solving for \( x \):
     \[
     x \leq \frac{9}{2} \implies x \leq 4.5
     \]
   - Since \( x \) must be an integer, we have:
     \[
     x \leq 4
     \]

3. **Third Condition**: The sum of the two digits is more than 10.
   - The sum of the digits is \( x + 2x = 3x \), and we are told this sum must be greater than 10:
     \[
     3x > 10
     \]
   - Solving for \( x \):
     \[
     x > \frac{10}{3} \implies x > 3.33
     \]
   - Since \( x \) must be an integer, this means \( x \geq 4 \).

#### Part (b) - Find the integral values of \( x \).

Now, we combine the inequalities from all the conditions:

- From the first condition: \( 0 \leq x \leq 5 \)
- From the second condition: \( x \leq 4 \)
- From the third condition: \( x \geq 4 \)

Therefore, the only value of \( x \) that satisfies all these conditions is:
\[
x = 4
\]

### Final Answer:
- The integral value of \( x \) is \( 4 \).
  
Would you like more details on any part of the solution?

Here are some related questions to expand the topic:
1. How do you represent inequalities graphically on a number line?
2. What is the process for solving inequalities with multiple conditions?
3. How can this problem be extended if both digits are allowed to be negative?
4. How would you solve the problem if the second digit was three times the first digit?
5. How would you solve a similar problem if the sum of the digits needed to be less than 10?

**Tip**: Always combine inequalities carefully by considering their ranges to avoid overlooking important restrictions!

Two digits are such that, the first digit is less than 6 and the second digit is twice the first digit. The sum of the two digits is more than 10. Taking x to represent the first digit, form all the inequalities in x representing the given situation and find the integral values of x.

Learn how to form inequalities based on digit conditions where the first digit is less than 6, the second digit is twice the first digit, and the sum of the digits exceeds 10. Solve for the integral values of x with step-by-step solutions suitable for middle school students.

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