We are given that:

• Joey earns $12 an hour and gets a raise of $1.50 every six months.
• Blake earns $16 an hour and gets a raise of $0.50 every six months.

The goal is to determine the equation that represents the number of six-month intervals, $x$, it will take for Joey's hourly wage to equal Blake's.

Joey's hourly wage:

Joey's initial wage is $12 per hour. After each six-month interval, his wage increases by $1.50, so his wage after $x$ six-month intervals is: $12 + 1.5x$

Blake's hourly wage:

Blake's initial wage is $16 per hour. After each six-month interval, his wage increases by $0.50, so his wage after $x$ six-month intervals is: $16 + 0.50x$

Setting Joey's and Blake's wages equal:

We need to find when Joey's wage equals Blake's wage, so we set the two expressions equal to each other: $12 + 1.5x = 16 + 0.50x$

This corresponds to option C in the choices given.

Would you like more details on solving the equation? Here are some related questions:

1. How can we solve the equation $12 + 1.5x = 16 + 0.50x$?
2. What is the interpretation of the variable $x$ in this context?
3. How do the wage increments affect the number of six-month intervals needed?
4. How can this type of equation be applied to other real-life salary growth problems?
5. How would the answer change if Blake’s raise was different?

Tip: To solve equations efficiently, always isolate the variable you're solving for by simplifying both sides step-by-step.