To solve the problem, we need to translate the verbal conditions into a system of linear inequalities.
1. Understanding the conditions:
- Anna is no more than 3 years older than 2 times Jamie's age. Mathematically, this condition can be expressed as:
[tex]\[
a \leq 3 + 2j
\][/tex]
This means Anna's age [tex]\(a\)[/tex] is less than or equal to 3 years plus twice Jamie's age [tex]\(j\)[/tex].
- Jamie is at least 14. Mathematically, this can be expressed as:
[tex]\[
j \geq 14
\][/tex]
This means Jamie's age [tex]\(j\)[/tex] is greater than or equal to 14.
- Anna is at most 35. Mathematically, this can be expressed as:
[tex]\[
a \leq 35
\][/tex]
This means Anna's age [tex]\(a\)[/tex] is less than or equal to 35.
2. Combining the inequalities:
Now we will combine these conditions to form a system of inequalities:
[tex]\[
\begin{cases}
a \leq 3 + 2j \\
j \geq 14 \\
a \leq 35
\end{cases}
\][/tex]
3. Identifying the correct choice:
Let's match these inequalities with the given choices:
- [tex]\(a \geq 3 + 2j; j \geq 14; a \leq 35\)[/tex]
- [tex]\(a \leq 3 + 2j; j \geq 14; a \leq 35\)[/tex]
- [tex]\(a \geq 3 + 2j; j \leq 14; a \leq 35\)[/tex]
- [tex]\(a \leq 3 + 2j; j \leq 14; a \leq 35\)[/tex]
The correct choice is the one that matches our system of inequalities exactly:
- [tex]\(a \leq 3 + 2j\)[/tex]
- [tex]\(j \geq 14\)[/tex]
- [tex]\(a \leq 35\)[/tex]
That matches with the second option:
[tex]\[
a \leq 3 + 2j; j \geq 14; a \leq 35
\][/tex]
Thus, the system of linear inequalities that can be used to find the possible ages of Anna and Jamie is:
[tex]\[
a \leq 3 + 2j; j \geq 14; a \leq 35
\][/tex]
So, the correct choice is the second one.
