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Anna is no more than 3 years older than 2 times Jamie's age. Jamie is at least 14 and Anna is at most 35.

Which system of linear inequalities can be used to find the possible ages of Anna, [tex]$a$[/tex], and Jamie, [tex]$j$[/tex]?

A. [tex]a \geq 3 + 2j[/tex]; [tex]j \geq 14[/tex], [tex]a \leq 35[/tex]
B. [tex]a \leq 3 + 2j[/tex]; [tex]j \geq 14[/tex], [tex]a \leq 35[/tex]
C. [tex]a \geq 3 + 2j[/tex]; [tex]j \leq 14[/tex], [tex]a \leq 35[/tex]
D. [tex]a \leq 3 + 2j[/tex]; [tex]j \leq 14[/tex], [tex]a \leq 35[/tex]



Answer :

GinnyAnswer
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.