Problem 11

Explanation

We can solve this problem using a system of linear inequalities.

Defining Variables

Let:

• $F =$ Father's age
• $M =$ Mother's age
• $C =$ Child's age

Translating Information into Mathematical Model

1. "The difference between Mother's and Father's ages is not more than 5 years":

   $$|F - M| \le 5$$

   This can be written as:

   $$-5 \le F - M \le 5$$

2. "The difference between Father's and their child's age is 35 years":

   $$F - C = 35 \Rightarrow F = C + 35$$

3. "The difference between Mother's and the child's age is x":

   $$x = M - C$$

   (We assume $M > C$ since Mother is definitely older than the child).

Solving the Inequality

Substitute $F = C + 35$ into the inequality:

Since $x = M - C$, the equation becomes:

Subtract $35$ from all sides:

Multiply all sides by $-1$ (remember, inequality signs reverse):

Thus, the correct value of $x$ is $30 \le x \le 40$.